Applications of Trigonometry

Transcription

1 5144_Demana_Ch06pp /11/06 9:31 PM Page 501 CHAPTER 6 Applications of Trigonometr 6.1 Vectors in the Plane 6. Dot Product of Vectors 6.3 Parametric Equations and Motion 6.4 Polar Coordinates 6.5 Graphs of Polar Equations 6.6 De Moivre s Theorem and nth Roots Young salmon migrate from the fresh water the are born in to salt water and live in the ocean for several ears. When it s time to spawn, the salmon return from the ocean to the river s mouth, where the follow the organic odors of their homestream to guide them upstream. Researchers believe the fish use currents, salinit, temperature, and the magnetic field of the Earth to guide them. Some fish swim as far as 3500 miles upstream for spawning. See a related problem on page

2 5144_Demana_Ch06pp /11/06 9:31 PM Page CHAPTER 6 Applications of Trigonometr JAMES BERNOULLI ( ) The first member of the Bernoulli famil (driven out of Holland b the Spanish persecutions and settled in Switzerland) to achieve mathematical fame, James defined the numbers now known as Bernoulli numbers. He determined the form (the elastica) taken b an elastic rod acted on at one end b a given force and fied at the other end. Chapter 6 Overview We introduce vectors in the plane, perform vector operations, and use vectors to represent quantities such as force and velocit. Vector methods are used etensivel in phsics, engineering, and applied mathematics. Vectors are used to plan airplane flight paths. The trigonometric form of a comple number is used to obtain De Moivre s theorem and find the nth roots of a comple number. Parametric equations are studied and used to simulate motion. One of the principal applications of parametric equations is the analsis of motion in space. Polar coordinates another of Newton s inventions, although James Bernoulli usuall gets the credit because he published first are used to represent points in the coordinate plane. Planetar motion is best described with polar coordinates. We convert rectangular coordinates to polar coordinates, polar coordinates to rectangular coordinates, and stud graphs of polar equations. 6.1 Vectors in the Plane What ou ll learn about Two-Dimensional Vectors Vector Operations Unit Vectors Direction Angles Applications of Vectors... and wh These topics are important in man real-world applications, such as calculating the effect of the wind on an airplane s path. OBJECTIVE Students will be able to appl the arithmetic of vectors and use vectors to solve real-world problems. MOTIVATE Discuss the difference between the statements: Jose lives 3 miles awa from Mar and Jose lives 3 miles west of Mar. LESSON GUIDE Da 1: Two-Dimensional Vectors; Vector Operations Da : Unit Vectors, Direction Angles; Applications of Vectors Two-Dimensional Vectors Some quantities, like temperature, distance, height, area, and volume, can be represented b a single real number that indicates magnitude or size. Other quantities, such as force, velocit, and acceleration, have magnitude and direction. Since the number of possible directions for an object moving in a plane is infinite, ou might be surprised to learn that two numbers are all that we need to represent both the magnitude of an object s velocit and its direction of motion. We simpl look at ordered pairs of real numbers in a new wa. While the pair (a, b) determines a point in the plane, it also determines a directed line segment (or arrow ) with its tail at the origin and its head at (a, b) (Figure 6.1). The length of this arrow represents magnitude, while the direction in which it points represents direction. Because in this contet the ordered pair (a, b) represents a mathematical object with both magnitude and direction, we call it the position vector of (a, b), and denote it as a, b to distinguish it from the point (a, b). O (a) (a, b) O FIGURE 6.1 The point represents the ordered pair (a, b). The arrow (directed line segment) represents the vector a, b. a, b (b) (a, b)

3 5144_Demana_Ch06pp /11/06 9:31 PM Page 503 SECTION 6.1 Vectors in the Plane 503 IS AN ARROW A VECTOR? While an arrow represents a vector, it is not a vector itself, since each vector can be represented b an infinite number of equivalent arrows. Still, it is hard to avoid referring to the vector PQ in practice, and we will often do that ourselves. When we sa the vector u PQ, we reall mean the vector u represented b PQ. DEFINITION Two-Dimensional Vector A two-dimensional vector v is an ordered pair of real numbers, denoted in component form as a, b. The numbers a and b are the components of the vector v. The standard representation of the vector a, b is the arrow from the origin to the point (a, b). The magnitude of v is the length of the arrow, and the direction of v is the direction in which the arrow is pointing. The vector 0 0, 0, called the zero vector, has zero length and no direction. S( 1, 6) P(3, 4) It is often convenient in applications to represent vectors with arrows that begin at points other than the origin. The important thing to remember is that an two arrows with the same length and pointing in the same direction represent the same vector. In Figure 6., for eample, the vector 3, 4 is shown represented b RS, an arrow with initial point R and terminal point S, as well as b its standard representation OP. Two arrows that represent the same vector are called equivalent. R( 4, ) O(0, 0) The quick wa to associate arrows with the vectors the represent is to use the following rule. FIGURE 6. The arrows RS and OP both represent the vector 3, 4, as would an arrow with the same length pointing in the same direction. Such arrows are called equivalent. Head Minus Tail (HMT) Rule If an arrow has initial point 1, 1 and terminal point,, it represents the vector 1, 1. EXAMPLE 1 Showing Arrows are Equivalent Show that the arrow from R ( 4, ) to S ( 1, 6) is equivalent to the arrow from P (, 1) to Q (5, 3) (Figure 6.3). S( 1, 6) R( 4, ) Q(5, 3) O P(, 1) FIGURE 6.3 The arrows RS and PQ appear to have the same magnitude and direction. The Head Minus Tail Rule proves that the represent the same vector (Eample 1). SOLUTION Appling the HMT rule, we see that RS represents the vector 1 ( 4), 6 3, 4, while PQ represents the vector 5, 3 ( 1) 3, 4. Although the have different positions in the plane, these arrows represent the same vector and are therefore equivalent. Now tr Eercise 1.

4 5144_Demana_Ch06pp /11/06 9:31 PM Page CHAPTER 6 Applications of Trigonometr P( 1, 1 ) Q(, ) FIGURE 6.4 The magnitude of v is the length of the arrow PQ,which is found using the distance formula: v 1 1. EXPLORATION 1 Vector Archer See how well ou can direct arrows in the plane using vector information and the HMT Rule. 1. An arrow has initial point (, 3) and terminal point (7, 5). What vector does it represent? 5,. An arrow has initial point (3, 5) and represents the vector 3, 6. What is the terminal point? 0, If P is the point (4, 3) and PQ represents, 4, find Q. 6, 7 4. If Q is the point (4, 3) and PQ represents, 4, find P., 1 If ou handled Eploration 1 with relative ease, ou have a good understanding of how vectors are represented geometricall b arrows. This will help ou understand the algebra of vectors, beginning with the concept of magnitude. The magnitude of a vector v is also called the absolute value of v, so it is usuall denoted b v. (You might see v in some tetbooks.) Note that it is a nonnegative real number, not a vector. The following computational rule follows directl from the distance formula in the plane (Figure 6.4). WHAT ABOUT DIRECTION? You might epect a quick computational rule for direction to accompan the rule for magnitude, but direction is less easil quantified. We will deal with vector direction later in the section. Magnitude If v is represented b the arrow from 1, 1 to,, then v 1 1. If v a, b, then v a b. P( 3, 4) Q( 5, ) v O(0, 0) (, ) EXAMPLE Finding Magnitude of a Vector Find the magnitude of the vector v represented b PQ,where P ( 3, 4) and Q ( 5, ). SOLUTION Working directl with the arrow, v ( 5 ( 3)) ( 4). Or, the HMT Rule shows that v,, so v ( ) ) (. (See Figure 6.5.) Now tr Eercise 5. FIGURE 6.5 The vector v of Eample. Vector Operations The algebra of vectors sometimes involves working with vectors and numbers at the same time. In this contet we refer to the numbers as scalars. The two most basic algebraic operations involving vectors are vector addition (adding a vector to a vector) and scalar multiplication (multipling a vector b a number). Both operations are easil represented geometricall, and both have immediate applications to man real-world problems.

5 5144_Demana_Ch06pp /11/06 9:3 PM Page 505 SECTION 6.1 Vectors in the Plane 505 WHAT ABOUT VECTOR MULTIPLICATION? There is a useful wa to define the multiplication of two vectors in fact, there are two useful was, but neither one of them follows the simple pattern of vector addition. (You ma recall that matri multiplication did not follow the simple pattern of matri addition either, and for similar reasons.) We will look at the dot product in Section 6.. The cross product requires a third dimension, so we will not deal with it in this course. DEFINITION Vector Addition and Scalar Multiplication Let u u 1, u and v v 1, v be vectors and let k be a real number (scalar). The sum (or resultant ) of the vectors u and v is the vector u v u 1 v 1, u v. The product of the scalar k and the vector u is ku k u 1, u ku 1, ku. The sum of the vectors u and v can be represented geometricall b arrows in two was. In the tail-to-head representation, the standard representation of u points from the origin to u 1, u. The arrow from u 1, u to u 1 v 1, u v represents v (as ou can verif b the HMT Rule). The arrow from the origin to u 1 v 1, u v then represents u v (Figure 6.6a). In the parallelogram representation, the standard representations of u and v determine a parallelogram, the diagonal of which is the standard representation of u v (Figure 6.6b). v u u + v u v u + v (a) (b) FIGURE 6.6 Two was to represent vector addition geometricall: (a) tail-to-head, and (b) parallelogram. u u (1/)u u FIGURE 6.7 Representations of u and several scalar multiples of u. The product ku of the scalar k and the vector u can be represented b a stretch (or shrink) of u b a factor of k. If k > 0, then ku points in the same direction as u; if k < 0, then ku points in the opposite direction (Figure 6.7). EXAMPLE 3 Performing Vector Operations Let u 1, 3 and v 4, 7. Find the component form of the following vectors: (a) u v (b) 3u (c) u ( 1)v SOLUTION Using the vector operations as defined, we have: (a) u v 1, 3 4, 7 1 4, 3 7 3, 10 (b) 3u 3 1, 3 3, 9 (c) u ( 1)v 1, 3 ( 1) 4, 7, 6 4, 7 6, 1 Geometric representations of u v and 3u are shown in Figure 6.8 on the net page. continued

6 5144_Demana_Ch06pp /11/06 9:3 PM Page CHAPTER 6 Applications of Trigonometr (3, 10) v 3u = 3, 9 ( 1, 3) u + v u u = 1, 3 (a) (b) FIGURE 6.8 Given that u 1, 3 and v 4, 7, we can (a) represent u v b the tail-to-head method, and (b) represent 3u as a stretch of u b a factor of 3. Now tr Eercise 13. A WORD ABOUT VECTOR NOTATION Both notations, a, b and ai bj, are designed to conve the idea that a single vector v has two separate components. This is what makes a twodimensional vector two-dimensional. You will see both a, b, c and ai bj ck used for three-dimensional vectors, but scientists stick to the notation for dimensions higher than three. Unit Vectors A vector u with length u 1 is a unit vector. If v is not the zero vector 0, 0, then the vector v 1 u v v v is a unit vector in the direction of v. Unit vectors provide a wa to represent the direction of an nonzero vector. An vector in the direction of v, or the opposite direction, is a scalar multiple of this unit vector u. EXAMPLE 4 Finding a Unit Vector Find a unit vector in the direction of v 3,, and verif that it has length 1. SOLUTION v 3, 3 1 3, so v 1 3, v 1 3 3, The magnitude of this vector is 3, ( 1 3 ) 3 ( ) Thus, the magnitude of v v is 1. Its direction is the same as v because it is a positive scalar multiple of v. Now tr Eercise 1.

7 5144_Demana_Ch06pp /11/06 9:3 PM Page 507 SECTION 6.1 Vectors in the Plane 507 bj FIGURE 6.9 The vector v is equal to ai bj. v sin θ ai v θ v = a, b v cos θ FIGURE 6.10 The horizontal and vertical components of v. The two unit vectors i 1, 0 and j 0, 1 are the standard unit vectors. An vector v can be written as an epression in terms of the standard unit vectors: v a, b a, 0 0, b a 1, 0 b 0, 1 ai bj Here the vector v a, b is epressed as the linear combination ai bj of the vectors i and j. The scalars a and b are the horizontal and vertical components, respectivel, of the vector v. See Figure 6.9. Direction Angles You ma recall from our applications in Section 4.8 that direction is measured in different was in different contets, especiall in navigation. A simple but precise wa to specif the direction of a vector v is to state its direction angle, the angle that v makes with the positive -ais, just as we did in Section 4.3. Using trigonometr (Figure 6.10), we see that the horizontal component of v is v cos and the vertical component is v sin. Solving for these components is called resolving the vector. Resolving the Vector If v has direction angle, the components of v can be computed using the formula v v cos, v sin. v = a, b 6 O 115 FIGURE 6.11 The direction angle of v is 115. (Eample 5) From the formula above, it follows that the unit vector in the direction of v is v u cos, sin. v EXAMPLE 5 Finding the Components of a Vector Find the components of the vector v with direction angle 115 and magnitude 6 Figure SOLUTION If a and b are the horizontal and vertical components, respectivel, of v, then v a, b 6 cos 115, 6 sin 115. So, a 6 cos and b 6 sin Now tr Eercise 9.

8 5144_Demana_Ch06pp /11/06 9:3 PM Page CHAPTER 6 Applications of Trigonometr EXAMPLE 6 Finding the Direction Angle of a Vector Find the magnitude and direction angle of each vector: (a) u 3, (b) v, 5 β u α u = 3, SOLUTION See Figure 6.1. (a) u If is the direction angle of u, then u 3, u cos, u sin. 3 u cos Horizontal component of u v v =, 5 FIGURE 6.1 The two vectors of Eample cos u cos cos ( 1 3 ) is acute. (b) v 5 9. If is the direction angle of v, then v, 5 v cos, v sin. v cos Horizontal component of v 5 cos v ( ) ) ( 5 9 cos 360 cos ( 1 ) Now tr Eercise 33. Applications of Vectors The velocit of a moving object is a vector because velocit has both magnitude and direction. The magnitude of velocit is speed mph FIGURE 6.13 The airplane s path (bearing) in Eample 7. TEACHING NOTE Encourage students to draw pictures to analze the geometr of various situations. v EXAMPLE 7 Writing Velocit as a Vector A DC-10 jet aircraft is fling on a bearing of 65 at 500 mph. Find the component form of the velocit of the airplane. Recall that the bearing is the angle that the line of travel makes with due north, measured clockwise see Section 4.1, Figure 4.. SOLUTION Let v be the velocit of the airplane. A bearing of 65 is equivalent to a direction angle of 5. The plane s speed, 500 mph, is the magnitude of vector v; that is, v 500. See Figure The horizontal component of v is 500 cos 5 and the vertical component is 500 sin 5, so v 500 cos 5 i 500 sin 5 j 500 cos 5, 500 sin , The components of the velocit give the eastward and northward speeds. That is, the airplane travels about mph eastward and about mph northward as it travels at 500 mph on a bearing of 65. Now tr Eercise 41.

9 5144_Demana_Ch06pp /11/06 9:3 PM Page 509 SECTION 6.1 Vectors in the Plane 509 A tpical problem for a navigator involves calculating the effect of wind on the direction and speed of the airplane, as illustrated in Eample 8. EXAMPLE 8 Calculating the Effect of Wind Velocit Pilot Megan McCart s flight plan has her leaving San Francisco International Airport and fling a Boeing 77 due east. There is a 65-mph wind with the bearing 60. Find the compass heading McCart should follow, and determine what the airplane s ground speed will be assuming that its speed with no wind is 450 mph. A 60 C 65 mph θ 450 mph FOLLOW-UP v B D FIGURE 6.14 The -ais represents the flight path of the plane in Eample 8. Have students discuss wh it does not make sense to add a scalar to a vector. ASSIGNMENT GUIDE Da 1: E. 3 7, multiples of 3, 39, 40 Da : E. 9, 3, 34, 37, 4, 43, 45, 46, 49 COOPERATIVE LEARNING Group Activit: E NOTES ON EXERCISES E are problems that students would tpicall encounter in a phsics course. E provide practice with standardized tests. E. 6 and 64 demonstrate connections between vectors and geometr. ONGOING ASSESSMENT Self-Assessment: E. 1, 5, 13, 1, 9, 33, 41, 43, 47 Embedded Assessment: E. 45, 46, 6 SOLUTION See Figure Vector AB represents the velocit produced b the airplane alone, AC represents the velocit of the wind, and is the angle DAB. Vector v AD represents the resulting velocit, so v AD AC AB. We must find the bearing of AB and v. Resolving the vectors, we obtain AC 65 cos 30, 65 sin 30 AB 450 cos, 450 sin AD AC AB 65 cos cos, 65 sin sin Because the plane is traveling due east, the second component of AD must be zero. 65 sin sin 0 sin ( 1 65 sin30 45 ) Thus, the compass heading McCart should follow is Bearing 90 The ground speed of the airplane is v AD 6 5 c o s c o s 0 65 cos cos Using the unrounded value of. McCart should use a bearing of approimatel The airplane will travel due east at approimatel mph. Now tr Eercise 43.

10 5144_Demana_Ch06pp /11/06 9:3 PM Page CHAPTER 6 Applications of Trigonometr D w A EXAMPLE 9 Finding the Effect of Gravit A force of 30 pounds just keeps the bo in Figure 6.15 from sliding down the ramp inclined at 0. Find the weight of the bo. 0 0 C B FIGURE 6.15 The force of gravit AB has a component AC that holds the bo against the surface of the ramp, and a component AD CB that tends to push the bo down the ramp. (Eample 9) SOLUTION We are given that AD 30. Let AB w; then sin 0 C B 3 0. w w Thus, 30 w sin 0 The weight of the bo is about pounds. Now tr Eercise 47. CHAPTER OPENER PROBLEM (from page 501) PROBLEM: During one part of its migration, a salmon is swimming at 6 mph, and the current is flowing downstream at 3 mph at an angle of 7 degrees. How fast is the salmon moving upstream? SOLUTION: Assume the salmon is swimming in a plane parallel to the surface of the water. A θ current salmon swimming in still water B salmon net velocit C In the figure, vector AB represents the current of 3 mph, is the angle CAB,which is 7 degrees, the vector CA represents the velocit of the salmon of 6 mph, and the vector CB is the net velocit at which the fish is moving upstream. So we have AB 3 cos 83, 3 sin ,.98 CA 0, 6 Thus CB CA AB 3 cos 83, 3 sin , 3.0 The speed of the salmon is then CB mph upstream.

11 5144_Demana_Ch06pp /11/06 9:3 PM Page 511 SECTION 6.1 Vectors in the Plane 511 QUICK REVIEW 6.1 (For help, go to Sections 4.3 and 4.7.) In Eercises 1 4, find the values of and ; ; 4.5 (, ) (, ) ; ; In (, ) (, ) Eercises 5 and 6, solve for in degrees. 5. sin ( 1 3 ) cos ( 1 1 ) In Eercises 7 9, the point P is on the terminal side of the angle. Find the measure of if P 5, P 5, P, tan 1 (5 ) A naval ship leaves Port Norfolk and averages 4 knots nautical mph traveling for 3 hr on a bearing of 40 and then 5 hr on a course of 15. What is the boat s bearing and distance from Port Norfolk after 8 hr? Distance: naut mi.; Bearing: SECTION 6.1 EXERCISES In Eercises 1 4, prove that RS and PQ are equivalent b showing that the represent the same vector. 1. R 4, 7, S 1, 5, O 0, 0, and P 3,. R 7, 3, S 4, 5, O 0, 0, and P 3, 3. R, 1, S 0, 1, O 1, 4, and P 1, 4. R, 1, S, 4, O 3, 1, and P 1, 4 In Eercises 5 1, let P,, Q 3, 4, R, 5, and S, 8. Find the component form and magnitude of the vector. 5. PQ 5, ; 9 7. QR 5, 1 ; 6 9. QS, 4 ; QR PS 11, 7 ; RS 4, 13 ; PS 4, 10 ; PR 0, 3 ; 3 1. PS 3PQ 11, 16 ; 377 In Eercises 13 0, let u 1, 3, v, 4, and w, 5. Find the component form of the vector. 13. u v 1, u 1 v 3, u w 3, v 6, u 3w 4, u 4v 10, u 3v 4, u v 1, 7 In Eercises 1 4, find a unit vector in the direction of the given vector. 1. u, 4. v 1, 1 3. w i j 4. w 5i 5j In Eercises 5 8, find the unit vector in the direction of the given vector. Write our answer in (a) component form and (b) as a linear combination of the standard unit vectors i and j. 5. u, 1 6. u 3, 7. u 4, 5 8. u 3, 4 In Eercises 9 3, find the component form of the vector v , , v i 0.89j. 0.71i 0.71j i 0.89j i 0.71j v

12 5144_Demana_Ch06pp /11/06 9:3 PM Page CHAPTER 6 Applications of Trigonometr , ,.9 v 47 In Eercises 33 38, find the magnitude and direction angle of the vector , 4 5; , 5 ; i 4j 5; i 5j 3 4 ; cos 135 i sin 135 j 7; cos 60 i sin 60 j ; 60 In Eercises 39 and 40, find the vector v with the given magnitude and the same direction as u. 39. v, u 3, v 5, u 5, Navigation An airplane is fling on a bearing of 335 at 530 mph. Find the component form of the velocit of the airplane. 3.99, Navigation An airplane is fling on a bearing of 170 at 460 mph. Find the component form of the velocit of the airplane , Flight Engineering An airplane is fling on a compass heading bearing of 340 at 35 mph. A wind is blowing with the bearing 30 at 40 mph. (a) Find the component form of the velocit of the airplane. (b) Find the actual ground speed and direction of the plane. 44. Flight Engineering An airplane is fling on a compass heading bearing of 170 at 460 mph. A wind is blowing with the bearing 00 at 80 mph. (a) Find the component form of the velocit of the airplane. (b) Find the actual ground speed and direction of the airplane. 45. Shooting a Basketball A basketball is shot at a 70 angle with the horizontal direction with an initial speed of 10 m sec. (a) Find the component form of the initial velocit. (b) Writing to Learn Give an interpretation of the horizontal and vertical components of the velocit. 46. Moving a Heav Object In a warehouse a bo is being pushed up a 15 inclined plane with a force of.5 lb, as shown in the figure..5 lb v (a) Find the component form of the force..41, 0.65 (b) Writing to Learn Give an interpretation of the horizontal and vertical components of the force. v Moving a Heav Object Suppose the bo described in Eercise 46 is being towed up the inclined plane, as shown in the figure below. Find the force w needed in order for the component of the force parallel to the inclined plane to be.5 lb. Give the answer in component form..0, Combining Forces Juana and Diego Gonzales, ages si and four respectivel, own a strong and stubborn pupp named Corporal. It is so hard to take Corporal for a walk that the devise a scheme to use two leashes. If Juana and Diego pull with forces of 3 lb and 7 lb at the angles shown in the figure, how hard is Corporal pulling if the pupp holds the children at a standstill? about lb 3 lb lb w In Eercises 49 and 50, find the direction and magnitude of the resultant force. 49. Combining Forces A force of 50 lb acts on an object at an angle of 45. A second force of 75 lb acts on the object at an angle of 30. F lb and Combining Forces Three forces with magnitudes 100, 50, and 80 lb, act on an object at angles of 50, 160, and 0, respectivel. F lb and Navigation A ship is heading due north at 1 mph. The current is flowing southwest at 4 mph. Find the actual bearing and speed of the ship ; 9.6 mph 5. Navigation A motor boat capable of 0 mph keeps the bow of the boat pointed straight across a mile-wide river. The current is flowing left to right at 8 mph. Find where the boat meets the opposite shore. 0.4 mi downstream 53. Group Activit A ship heads due south with the current flowing northwest. Two hours later the ship is 0 miles in the direction 30 west of south from the original starting point. Find the speed with no current of the ship and the rate of the current mph; 7.07 mph 54. Group Activit Epress each vector in component form and prove the following properties of vectors. (a) u v v u (b) u v w u v w (c) u 0 u, where 0 0, 0

13 5144_Demana_Ch06pp /11/06 9:3 PM Page 513 SECTION 6.1 Vectors in the Plane 513 (d) u u 0, where a, b a, b (e) a u v au av (f) a b u au bu (g) ab u a bu (h) a0 0, 0u 0 (i) 1 u u, 1 u u (j) au a u Standardized Test Questions 55. True or False If u is a unit vector, then u is also a unit vector. Justif our answer. 56. True or False If u is a unit vector, then 1 u is also a unit vector. Justif our answer. False. 1/u is not a vector. In Eercises 57 60, ou ma use a graphing calculator to solve the problem. 57. Multiple Choice Which of the following is the magnitude of the vector, 1? D (A) 1 (B) 3 (C) 5 5 (D) 5 (E) Multiple Choice Let u, 3 and v 4, 1. Which of the following is equal to u v? E (A) 6, 4 (B), (C), (D) 6, (E) 6, Multiple Choice Which of the following represents the vector v shown in the figure below? A O 30 3 (A) 3 cos 30, 3 sin 30 (B) 3 sin 30, 3 cos 30 (C) 3 cos 60, 3 sin 60 (D) 3 cos 30, 3 sin 30 (E) 3 cos 30, 3 sin Multiple Choice Which of the following is a unit vector in the direction of v i 3j? C (A) i j (B) i j (C) i 3 10 j 1 (D) 10 i 3 10 j (E) 1 3 i j 8 8 v Eplorations 61. Dividing a Line Segment in a Given Ratio Let A and B be two points in the plane, as shown in the figure. B (a) Prove that BA OA OB,where O is the C origin. A (b) Let C be a point on the line segment BA which divides the segment in the ratio : where 1. That is, B C C A. O Show that OC OA OB. 6. Medians of a Triangle Perform the following steps to use vectors to prove that the medians of a triangle meet at a point O which divides each median in the ratio 1 :. M 1, M,and M 3 are midpoints of the sides of the triangle shown in the figure. A (a) Use Eercise 61 to prove that OM 1 1 OA 1 OB OM 1 OC 1 OB OM 1 3 OA 1 OC (b) Prove that each of OM 1 OC,OM OA,OM3 OB is equal to OA OB OC. (c) Writing to Learn Eplain wh part b establishes the desired result. Etending the Ideas C M 3 M O M 1 B 63. Vector Equation of a Line Let L be the line through the two points A and B. Prove that C, is on the line L if and onl if OC toa 1 t OB,where t is a real number and O is the origin. 64. Connecting Vectors and Geometr Prove that the lines which join one verte of a parallelogram to the midpoints of the opposite sides trisect the diagonal. 55. True. u and u have the same length but opposite directions. Thus, the length of u is also 1.

14 5144_Demana_Ch06pp /11/06 9:3 PM Page CHAPTER 6 Applications of Trigonometr 6. Dot Product of Vectors What ou ll learn about The Dot Product Angle Between Vectors Projecting One Vector onto Another Work... and wh Vectors are used etensivel in mathematics and science applications such as determining the net effect of several forces acting on an object and computing the work done b a force acting on an object. The Dot Product Vectors can be multiplied in two different was, both of which are derived from their usefulness for solving problems in vector applications. The cross product (or vector product or outer product) results in a vector perpendicular to the plane of the two vectors being multiplied, which takes us into a third dimension and outside the scope of this chapter. The dot product (or scalar product or inner product) results in a scalar. In other words, the dot product of two vectors is not a vector but a real number! It is the important information conveed b that number that makes the dot product so worthwhile, as ou will see. Now that ou have some eperience with vectors and arrows, we hope we won t confuse ou if we occasionall resort to the common convention of using arrows to name the vectors the represent. For eample, we might write u PQ as a shorthand for u is the vector represented b PQ. This greatl simplifies the discussion of concepts like vector projection. Also, we will continue to use both vector notations, a, b and ai bj, so ou will get some practice with each. DEFINITION Dot Product The dot product or inner product of u u 1, u and v v 1, v is u v u 1 v 1 u v. DOT PRODUCT AND STANDARD UNIT VECTORS (u 1 i u j) (v 1 i v j) u 1 v 1 u v OBJECTIVE Students will be able to calculate dot products and projections of vectors. MOTIVATE Ask students to guess the meaning of a projection of one vector onto another. LESSON GUIDE Da 1: The Dot Product; Angle Between Vectors Da : Projecting One Vector Onto Another; Work Dot products have man important properties that we make use of in this section. We prove the first two and leave the rest for the Eercises. Properties of the Dot Product Let u, v, and w be vectors and let c be a scalar. 1. u v v u 4. u v w u v u w. u u u u v w u w v w 3. 0 u 0 5. cu v u cv c u v Proof Let u u 1, u and v v 1, v. Propert 1 u v u 1 v 1 u v Definition of u v v 1 u 1 v u v u Commutative propert of real numbers Definition of u v Propert u u u 1 u u 1 u u Definition of u u Definition of u

15 5144_Demana_Ch06pp /11/06 9:3 PM Page 515 SECTION 6. Dot Product of Vectors 515 TEACHING NOTE If ou do not plan to cover Chapter 8 and ou want to cover vectors in threedimensional space, ou can cover the relevant parts of Section 8.6 after ou finish Section 6.. EXAMPLE 1 Finding Dot Products Find each dot product. (a) 3, 4 5, (b) 1, 4, 3 (c) i j 3i 5j SOLUTION (a) 3, 4 5, (b) 1, 4, (c) i j 3i 5j Now tr Eercise 3. DOT PRODUCTS ON CALCULATORS It is reall a waste of time to compute a simple dot product of two-dimensional vectors using a calculator, but it can be done. Some calculators do vector operations outright, and others can do vector operations via matrices. If ou have learned about matri multiplication alread, ou will know wh the matri v [ 1 ] v product [u 1, u ] ields the dot product u 1, u v 1, v as a 1-b-1 matri. (The same trick works with vectors of higher dimensions.) This book will cover matri multiplication in Chapter 7. Propert of the dot product gives us another wa to find the length of a vector, as illustrated in Eample. EXAMPLE Using Dot Product to Find Length Use the dot product to find the length of the vector u 4, 3. SOLUTION It follows from Propert that u u u. Thus, 4, 3 4, 3 4, Now tr Eercise 9. Angle Between Vectors Let u and v be two nonzero vectors in standard position as shown in Figure The angle between u and v is the angle,0 or The angle between an two nonzero vectors is the corresponding angle between their respective standard position representatives. We can use the dot product to find the angle between nonzero vectors, as we prove in the net theorem. v v u u FIGURE 6.16 The angle between nonzero vectors u and v. THEOREM Angle Between Two Vectors If is the angle between the nonzero vectors u and v, then u v cos u v and cos ( 1 u u ) v v

16 5144_Demana_Ch06pp /11/06 9:3 PM Page CHAPTER 6 Applications of Trigonometr Proof We appl the Law of Cosines to the triangle determined b u, v, and v u in Figure 6.16, and use the properties of the dot product. v u u v u v cos v u v u u v u v cos v v v u u v u u u v u v cos v u v u u v u v cos u v u v cos u v cos u v cos ( 1 u u ) v v v =, 5 u =, 3 θ (a) u =, 1 θ EXAMPLE 3 Finding the Angle Between Vectors Find the angle between the vectors u and v. (a) u, 3, v, 5 (b) u, 1, v 1, 3 SOLUTION (a) See Figure 6.17a. Using the Angle Between Two Vectors Theorem, we have u v, 3, 5 11 cos. u v, 3, So, cos (b) See Figure 6.17b. Again using the Angle Between Two Vectors Theorem, we have u v, 1 1, cos. u v, 1 1, So, cos Now tr Eercise 13. v = 1, 3 (b) FIGURE 6.17 The vectors in (a) Eample 3a and (b) Eample 3b. If vectors u and v are perpendicular, that is, if the angle between them is 90, then u v u v cos 90 0 because cos DEFINITION Orthogonal Vectors The vectors u and v are orthogonal if and onl if u v 0.

17 5144_Demana_Ch06pp /11/06 9:3 PM Page 517 SECTION 6. Dot Product of Vectors 517 EXPLORATION EXTENSIONS Now suppose B(, ) is a point that is not on the given circle. If a,what can ou sa about u v? If a, what can ou sa about u v? The terms perpendicular and orthogonal almost mean the same thing. The zero vector has no direction angle, so technicall speaking, the zero vector is not perpendicular to an vector. However, the zero vector is orthogonal to ever vector. Ecept for this special case, orthogonal and perpendicular are the same. EXAMPLE 4 Proving Vectors are Orthogonal Prove that the vectors u, 3 and v 6, 4 are orthogonal. SOLUTION We must prove that their dot product is zero. u v, 3 6, The two vectors are orthogonal. Now tr Eercise 3. A( a, 0) θ B(, ) C(a, 0) FIGURE 6.18 The angle ABC inscribed in the upper half of the circle a. (Eploration 1) EXPLORATION 1 Angles Inscribed in Semicircles Figure 6.18 shows ABC inscribed in the upper half of the circle a. 1. For a, find the component form of the vectors u BA and v BC.,,,. Find u v. What can ou conclude about the angle between these two vectors? Repeat parts 1 and for arbitrar a. Answers will var P u Q R FIGURE 6.19 The vectors u PQ, v PS, and the vector projection of u onto v, PR proj v u. FOLLOW-UP Ask students to name a pair of vectors that are orthogonal but not perpendicular. ASSIGNMENT GUIDE Da 1: E. 1 1, multiples of 3, 30 4, multiples of 3 Da : E. 7 51, multiples of 3, COOPERATIVE LEARNING Group Activit: E. 58, 59 v S Projecting One Vector onto Another The vector projection of u PQ onto a nonzero vector v PS is the vector PR determined b dropping a perpendicular from Q to the line PS (Figure 6.19). We have resolved u into components PR and RQ u PR RQ with PR and RQ perpendicular. The standard notation for PR,the vector projection of u onto v, is PR proj v u. With this notation, RQ u projv u. We ask ou to establish the following formula in the Eercises (see Eercise 58). Projection of u onto v If u and v are nonzero vectors, the projection of u onto v is proj v u u v v v.

18 5144_Demana_Ch06pp /11/06 9:3 PM Page CHAPTER 6 Applications of Trigonometr u v = 5, 5 FIGURE 6.0 The vectors u 6,, v 5, 5, u 1 proj v u, and u u u 1. (Eample 5) 45 u = 6, u u proj v u FIGURE 6.1 If we pull on a bo with force u, the effective force in the direction of v is proj v u, the vector projection of u onto v. F 1 FIGURE 6. The sled in Eample 6. v F EXAMPLE 5 Decomposing a Vector into Perpendicular Components Find the vector projection of u 6, onto v 5, 5. Then write u as the sum of two orthogonal vectors, one of which is proj v u. SOLUTION We write u u 1 u where u 1 proj v u and u u u 1 (Figure 6.0). v u 1 proj v u u v v 0 5, 5, 50 u u u 1 6,, 4, 4 Thus, u 1 u, 4, 4 6, u. Now tr Eercise 5. If u is a force, then proj v u represents the effective force in the direction of v (Figure 6.1). We can use vector projections to determine the amount of force required in problem situations like Eample 6. EXAMPLE 6 Finding a Force Juan is sitting on a sled on the side of a hill inclined at 45. The combined weight of Juan and the sled is 140 pounds. What force is required for Rafaela to keep the sled from sliding down the hill? (See Figure 6..) SOLUTION We can represent the force due to gravit as F 140j because gravit acts verticall downward. We can represent the side of the hill with the vector v cos 45 i sin 45 j i j. The force required to keep the sled from sliding down the hill is F 1 proj v F ( F v v ) v F v v because v 1. So, F 1 F v v 140 ( ) v 70 i j. The magnitude of the force that Rafaela must eert to keep the sled from sliding down the hill is pounds. Now tr Eercise 45. Work If F is a constant force whose direction is the same as the direction of AB, then the work W done b F in moving an object from A to B is W F AB.

19 5144_Demana_Ch06pp /11/06 9:3 PM Page 519 SECTION 6. Dot Product of Vectors 519 NOTES ON EXERCISES E can be completed b using dot products or b using common sense. Encourage students to tr both methods. E involve work done b a force that is not parallel to the direction of motion. E provide practice with standardized tests. ONGOING ASSESSMENT Self-Assessment: E. 3, 9, 13, 3, 5, 45, 53 Embedded Assessment: E. 67, 68 UNITS FOR WORK Work is usuall measured in footpounds or Newton-meters. One Newton-meter is commonl referred to as one Joule. If F is a constant force in an direction, then the work W done b F in moving an object from A to B is W F AB F AB cos where is the angle between F and AB. Ecept for the sign, the work is the magnitude of the effective force in the direction of AB times AB. EXAMPLE 7 Finding Work Find the work done b a 10 pound force acting in the direction 1, in moving an object 3 feet from 0, 0 to 3, 0. SOLUTION The force F has magnitude 10 and acts in the direction 1,, so 1, 10 F 10 1,. 1, 5 The direction of motion is from A 0, 0 to B 3, 0, so AB 3, 0. Thus, the work done b the force is F AB , 3, foot-pounds. 5 5 Now tr Eercise 53. QUICK REVIEW 6. (For help, go to Section 6.1.) In Eercises 1 4, find u. 1. u, u 3i 4j 5 3. u cos 35 i sin 35 j 1 4. u cos 75 i sin 75 j In Eercises 5 8, the points A and B lie on the circle 4. Find the component form of the vector AB. 5. A, 0, B 1, 3 6. A, 0, B 1, 3 3, 3 1, 3 7. A, 0, B 1, 3 1, 3 8. A, 0, B 1, 3 3, 3 In Eercises 9 and 10, find a vector u with the given magnitude in the direction of v. 9. u, v, u 3, v 4i 3j 4 6, , SECTION 6. EXERCISES In Eercises 1 8, find the dot product of u and v. 1. u 5, 3, v 1, 4 7. u 5,, v 8, u 4, 5, v 3, u, 7, v 5, u 4i 9j, v 3i j u i 4j, v 8i 7j u 7i, v i 5j u 4i 11j, v 3j 33 In Eercises 9 1, use the dot product to find u. 9. u 5, u 8, u 4i 4 1. u 3j 3

20 5144_Demana_Ch06pp /11/06 9:3 PM Page CHAPTER 6 Applications of Trigonometr In Eercises 13, find the angle between the vectors. 13. u 4, 3, v 1, u,, v 3, u, 3, v 3, u 5,, v 6, u 3i 3j, v i 3 j u i, v 5j u ( cos 4 ) ( i sin 4 ) ( j, v cos 3 ) ( i sin 3 ) j u ( cos 3 ) ( i sin 3 ) ( j, v 3 cos 5 6 ) ( i 3 sin 5 6 ) j (8, 5) 5 ( 3, 4) 4 3 v 1 u ( 3, 8) ( 1, 9) 9 10 In Eercises 3 4, prove that the vectors u and v are orthogonal. 3. u, 3, v 3, 1 4. u 4, 1, v 1, 4 In Eercises 5 8, find the vector projection of u onto v. Then write u as a sum of two orthogonal vectors, one of which is proj v u. 5. u 8, 3, v 6, 6. u 3, 7, v, 6 7. u 8, 5, v 9, 8. u, 8, v 9, 3 In Eercises 9 and 30, find the interior angles of the triangle with given vertices. 9. 4, 5, 1, 10, 3, , 1, 1, 6, 5, 1 In Eercises 31 and 3, find u v satisfing the given conditions where is the angle between u and v , u 3, v 8 3., u 1, v 40 3 In Eercises 33 38, determine whether the vectors u and v are parallel, orthogonal, or neither. 33. u 5, 3, v 1 0, 3 4 Parallel 34. u, 5, v 1 0, Neither 35. u 15, 1, v 4, 5 Neither 36. u 5, 6, v 1, 10 Orthogonal 37. u 3, 4, v 0, 15 Orthogonal 38. u, 7, v 4, 14 Parallel In Eercises 39 4, find (a) the -intercept A and -intercept B of the line. (b) the coordinates of the point P so that AP is perpendicular to the line and AP 1. (There are two answers.) In Eercises 43 and 44, find the vector(s) v satisfing the given conditions. 43. u, 3, u v 10, v u, 5, u v 11, v Sliding Down a Hill Ojemba is sitting on a sled on the side of a hill inclined at 60. The combined weight of Ojemba and the sled is 160 pounds. What is the magnitude of the force required for Mandisa to keep the sled from sliding down the hill? 46. Revisiting Eample 6 Suppose Juan and Rafaela switch positions. The combined weight of Rafaela and the sled is 15 pounds. What is the magnitude of the force required for Juan to keep the sled from sliding down the hill? pounds 47. Braking Force A 000 pound car is parked on a street that makes an angle of 1 with the horizontal (see figure). 1 (a) Find the magnitude of the force required to keep the car from rolling down the hill pounds (b) Find the force perpendicular to the street pounds

21 5144_Demana_Ch06pp /11/06 9:3 PM Page 51 SECTION 6. Dot Product of Vectors Effective Force A 60 pound force F that makes an angle of 5 with an inclined plane is pulling a bo up the plane.the inclined plane makes an 18 angle with the horizontal (see 5 figure). What is the magnitude of the effective force pulling the bo up the plane? pounds Work Find the work done lifting a 600 pound car 5.5 feet. 14,300 foot-pounds 50. Work Find the work done lifting a 100 pound bag of potatoes 3 feet. 300 foot-pounds 51. Work Find the work done b a force F of 1 pounds acting in the direction 1, in moving an object 4 feet from 0, 0 to 4, foot-pounds 5. Work Find the work done b a force F of 4 pounds acting in the direction 4, 5 in moving an object 5 feet from 0, 0 to 5, foot-pounds 53. Work Find the work done b a force F of 30 pounds acting in the direction, in moving an object 3 feet from 0, 0 to a point in the first quadrant along the line Work Find the work done b a force F of 50 pounds acting in the direction, 3 in moving an object 5 feet from 0, 0 to a point in the first quadrant along the line. 55. Work The angle between a 00 pound force F and AB i 3j is 30. Find the work done b F in moving an object from A to B foot-pounds 56. Work The angle between a 75 pound force F and AB is 60, where A 1, 1 and B 4, 3. Find the work done b F in moving an object from A to B foot-pounds 57. Properties of the Dot Product Let u, v, and w be vectors and let c be a scalar. Use the component form of vectors to prove the following properties. (a) 0 u 0 (b) u v w u v u w (c) u v w u w v w (d) cu v u cv c u v 58. Group Activit Projection of a Vector Let u and v be nonzero vectors. Prove that (a) proj v u ( u v v ) v (b) u proj v u proj v u Group Activit Connecting Geometr and Vectors Prove that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides. 60. If u is an vector, prove that we can write u as u u i i u j j. Standardized Test Questions 61. True or False If u v 0, then u and v are perpendicular. Justif our answer. 6. True or False If u is a unit vector, then u u 1. Justif our answer. True. u u u (1) 1 In Eercises 63 66, ou ma use a graphing calculator to solve the problem. 63. Multiple Choice Let u 1, 1 and v 1, 1. Which of the following is the angle between u and v? D (A) 0 (B) 45 (C) 60 (D) 90 (E) Multiple Choice Let u 4, 5 and v, 3. Which of the following is equal to u v? C (A) 3 (B) 7 (C) 7 (D) 3 (E) Multiple Choice Let u 3, 3 and v, 0. Which of the following is equal to proj v u? A (A) 3, 0 (B) 3, 0 (C) 3, 0 (D) 3, 3 (E) 3, Multiple Choice Which of the following vectors describes a 5 lb force acting in the direction of u 1, 1? B 5 (A) 5 1, 1 (B) 1, 1 (C) 5 1, 1 5 (D) 1, 1 Eplorations (E) 5 1, Distance from a Point to a Line Consider the line L with equation 5 10 and the point P 3, 7. (a) Verif that A 0, and B 5, 0 are the - and -intercepts of L. (b) Find w 1 proj AB AP and w AP proj AB AP. (c) Writing to Learn Eplain wh w is the distance from P to L. What is this distance? (d) Find a formula for the distance of an point P 0, 0 to L. (e) Find a formula for the distance of an point P 0, 0 to the line a b c. Etending the Ideas 68. Writing to Learn Let w cos t u sin t v where u and v are not parallel. (a) Can the vector w be parallel to the vector u? Eplain. (b) Can the vector w be parallel to the vector v? Eplain. (c) Can the vector w be parallel to the vector u v? Eplain. 69. If the vectors u and v are not parallel, prove that au bv cu dv a c, b d.

22 5144_Demana_Ch06pp /11/06 9:3 PM Page 5 5 CHAPTER 6 Applications of Trigonometr 6.3 Parametric Equations and Motion What ou ll learn about Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher... and wh These topics can be used to model the path of an object such as a baseball or a golf ball. t = 0, = 40 t = 4, = 164 t = 1, = 404 t =, = 356 t = 3, = 76 Parametric Equations Imagine that a rock is dropped from a 40-ft tower. The rock s height in feet above the ground t seconds later (ignoring air resistance) is modeled b 16t 40 as we saw in Section.1. Figure 6.3 shows a coordinate sstem imposed on the scene so that the line of the rock s fall is on the vertical line.5. The rock s original position and its position after each of the first 5 seconds are the points.5, 40,.5, 404,.5, 356,.5, 76,.5, 164,.5, 0, which are described b the pair of equations.5, 16t 40, when t 0, 1,, 3, 4, 5. These two equations are an eample of parametric equations with parameter t. As is often the case, the parameter t represents time. Parametric Curves In this section we stud the graphs of parametric equations and investigate motion of objects that can be modeled with parametric equations. t = 5, = 0 [0, 5] b [ 10, 500] FIGURE 6.3 The position of the rock at 0, 1,, 3, 4, and 5 seconds. OBJECTIVE Students will be able to define parametric equations, graph curves parametricall, and solve application problems using parametric equations. MOTIVATE Have students use a grapher to graph the parametric equations t and t for 5 t 5. Have them write the equation for this graph in the form f(). ( ) LESSON GUIDE Da 1: Parametric Equations; Parametric Curves; Eliminating the Parameter; Lines and Line Segments Da : Simulating Motion with a Grapher DEFINITION Parametric Curve, Parametric Equations The graph of the ordered pairs, where f t, g t are functions defined on an interval I of t-values is a parametric curve. The equations are parametric equations for the curve, the variable t is a parameter, and I is the parameter interval. When we give parametric equations and a parameter interval for a curve, we have parametrized the curve. A parametrization of a curve consists of the parametric equations and the interval of t-values. Sometimes parametric equations are used b companies in their design plans. It is then easier for the compan to make larger and smaller objects efficientl b just changing the parameter t. Graphs of parametric equations can be obtained using parametric mode on a grapher. EXAMPLE 1 Graphing Parametric Equations For the given parameter interval, graph the parametric equations t, 3t. (a) 3 t 1 (b) t 3 (c) 3 t 3 continued

23 5144_Demana_Ch06pp /11/06 9:3 PM Page 53 SECTION 6.3 Parametric Equations and Motion 53 NOTES ON EXAMPLES Eample 1 is important because it shows how a parametric graph is affected b the chosen range of t-values. SOLUTION In each case, set Tmin equal to the left endpoint of the interval and Tma equal to the right endpoint of the interval. Figure 6.4 shows a graph of the parametric equations for each parameter interval. The corresponding relations are different because the parameter intervals are different. Now tr Eercise 7. [ 10, 10] b [ 10, 10] (a) [ 10, 10] b [ 10, 10] (b) [ 10, 10] b [ 10, 10] (c) FIGURE 6.4 Three different relations defined parametricall. (Eample 1) TEACHING NOTE If students are not familiar with parametric graphing, it might be helpful to show them the graph of the linear function f() 3 and compare it to one defined parametricall as t and 3t, using a trace ke to show how t,, and are related. Eliminating the Parameter When a curve is defined parametricall it is sometimes possible to eliminate the parameter and obtain a rectangular equation in and that represents the curve. This often helps us identif the graph of the parametric curve as illustrated in Eample. EXAMPLE Eliminating the Parameter Eliminate the parameter and identif the graph of the parametric curve 1 t, t, t. SOLUTION We solve the first equation for t: 1 t t 1 t 1 1 Then we substitute this epression for t into the second equation: t 1 1 [ 10, 5] b [ 5, 5] FIGURE 6.5 The graph of (Eample ) The graph of the equation is a line with slope 0.5 and -intercept 1.5 Figure 6.5. Now tr Eercise 11.

24 5144_Demana_Ch06pp /11/06 9:33 PM Page CHAPTER 6 Applications of Trigonometr EXPLORATION EXTENSIONS Determine the smallest possible range of t-values that produces the graph shown in Figure 6.5, using the given parametric equations. ALERT Man students will confuse range values of t with range values on the function grapher. Point out that while the scale factor does not affect the wa a graph is drawn, the Tstep does affect the wa the graph is displaed. EXPLORATION 1 Graphing the Curve of Eample Parametricall 1. Use the parametric mode of our grapher to reproduce the graph in Figure 6.5. Use for Tmin and 5.5 for Tma.. Prove that the point 17, 10 is on the graph of Find the corresponding value of t that produces this point. t 8 3. Repeat part for the point 3, 10. t 1 4. Assume that a, b is on the graph of Find the corresponding value of t that produces this point. t 1 a b 5. How do ou have to choose Tmin and Tma so that the graph in Figure 6.5 fills the window? Tmin and Tma 5.5 If we do not specif a parameter interval for the parametric equations f t, g t, it is understood that the parameter t can take on all values which produce real numbers for and. We use this agreement in Eample 3. PARABOLAS The inverse of a parabola that opens up or down is a parabola that opens left or right. We will investigate these curves in more detail in Chapter 8. EXAMPLE 3 Eliminating the Parameter Eliminate the parameter and identif the graph of the parametric curve t, 3t. SOLUTION Here t can be an real number. We solve the second equation for t obtaining t 3 and substitute this value for into the first equation. t ( 3 ) 9 9 Figure 6.4c shows what the graph of these parametric equations looks like. In Chapter 8 we will call this a parabola that opens to the right. Interchanging and we can identif this graph as the inverse of the graph of the parabola 9. Now tr Eercise 15. [ 4.7, 4.7] b [ 3.1, 3.1] FIGURE 6.6 The graph of the circle of Eample 4. EXAMPLE 4 Eliminating the Parameter Eliminate the parameter and identif the graph of the parametric curve cos t, sin t, 0 t. SOLUTION The graph of the parametric equations in the square viewing window of Figure 6.6 suggests that the graph is a circle of radius centered at the origin. We confirm this result algebraicall. continued

25 5144_Demana_Ch06pp /11/06 9:33 PM Page 55 SECTION 6.3 Parametric Equations and Motion 55 4cos t 4sin t 4 cos t sin t 4 1 cos t sin t 1 4 The graph of 4 is a circle of radius centered at the origin. Increasing the length of the interval 0 t will cause the grapher to trace all or part of the circle more than once. Decreasing the length of the interval will cause the grapher to onl draw a portion of the complete circle. Tr it! Now tr Eercise 3. In Eercise 65, ou will find parametric equations for an circle in the plane. Lines and Line Segments We can use vectors to help us find parametric equations for a line as illustrated in Eample 5. A(, 3) FIGURE 6.7 Eample 5 uses vectors to construct a parametrization of the line through A and B. TEACHING NOTE O 1 B(3, 6) P(, ) The parametrization in Eample 5 is not unique. You ma want to have our students find alternate parametrizations. EXAMPLE 5 Finding Parametric Equations for a Line Find a parametrization of the line through the points A, 3 and B 3, 6. SOLUTION Let P, be an arbitrar point on the line through A and B. As ou can see from Figure 6.7, the vector OP is the tail-to-head vector sum of OA and AP. You can also see that AP is a scalar multiple of AB. If we let the scalar be t, we have OP OA AP OP OA t AB,, 3 t 3 ( ), 6 3,, 3 t 5, 3, 5t, 3 3t This vector equation is equivalent to the parametric equations 5t and 3 3t. Together with the parameter interval (, ), these equations define the line. We can confirm our work numericall as follows: If t 0, then and 3, which gives the point A. Similarl, if t 1, then 3 and 6, which gives the point B. Now tr Eercise 7. The fact that t 0 ields point A and t 1 ields point B in Eample 5 is no accident, as a little reflection on Figure 6.7 and the vector equation OP OA t AB should suggest. We use this fact in Eample 6.

26 5144_Demana_Ch06pp /11/06 9:33 PM Page CHAPTER 6 Applications of Trigonometr T=0 X=8.5 Y=5 Start, t = 0 (a) EXAMPLE 6 Finding Parametric Equations for a Line Segment Find a parametrization of the line segment with endpoints A, 3 and B 3, 6. SOLUTION In Eample 5 we found parametric equations for the line through A and B: 5t, 3 3t. We also saw in Eample 5 that t 0 produces the point A and t 1 produces the point B. A parametrization of the line segment is given b 5t, 3 3t, 0 t 1. As t varies between 0 and 1 we pick up ever point on the line segment between A and B. Now tr Eercise 9. T=5 X= 9 Y=5 T=8 X=.7 Y=5 FIGURE 6.8 Three views of the graph C 1 : 1 0.1(t 3 0t 110t 85), 1 5, 0 t 1 in the [ 1, 1] b [ 10, 10] viewing window. (Eample 7) GRAPHER NOTE 5 sec later, t = 5 (b) 3 sec after that, t = 8 (c) The equation t is tpicall used in the parametric equations for the graph C in Figure 6.9. We have chosen t to get two curves in Figure 6.9 that do not overlap. Also notice that the -coordinates of C 1 are constant ( 1 5), and that the -coordinates of C var with time t ( t). Simulating Motion with a Grapher Eample 7 illustrates several was to simulate motion along a horizontal line using parametric equations. We use the variable t for the parameter to represent time. EXAMPLE 7 Simulating Horizontal Motion Gar walks along a horizontal line think of it as a number line with the coordinate of his position in meters given b s 0.1 t 3 0t 110t 85 where 0 t 1. Use parametric equations and a grapher to simulate his motion. Estimate the times when Gar changes direction. SOLUTION We arbitraril choose the horizontal line 5 to displa this motion. The graph C 1 of the parametric equations, C 1 : t 3 0t 110t 85, 1 5, 0 t 1, simulates the motion. His position at an time t is given b the point 1 t, 5. Using trace in Figure 6.8 we see that when t 0, Gar is 8.5 m to the right of the -ais at the point 8.5, 5, and that he initiall moves left. Five seconds later he is 9 m to the left of the -ais at the point 9, 5. And after 8 seconds he is onl.7 m to the left of the -ais. Gar must have changed direction during the walk. The motion of the trace cursor simulates Gar s motion. A variation in t, C : 0.1 t 3 0t 110t 85, t, 0 t 1, can be used to help visualize where Gar changes direction. The graph C shown in Figure 6.9 suggests that Gar reverses his direction at 3.9 seconds and again at 9.5 seconds after beginning his walk. Now tr Eercise 37.

27 5144_Demana_Ch06pp /11/06 9:33 PM Page 57 SECTION 6.3 Parametric Equations and Motion 57 C 1 C 1 T=1 X=5.5 Y=135 [0, 6] b [0, 00] (a) C T=3.9 X= Y= 3.9 [ 1, 1] b [ 15, 15] (a) C T=9.5 X= Y= 9.5 [ 1, 1] b [ 15, 15] (b) T= X=5.5 Y=163 [0, 6] b [0, 00] (b) FIGURE 6.9 Two views of the graph C 1 : 1 0.1(t 3 0t 110t 85), 1 5, 0 t 1 and the graph C : 0.1(t 3 0t 110t 85), t, 0 t 1 in the [ 1, 1] b [ 15, 15] viewing window. (Eample 7) Eample 8 solves a projectile-motion problem. Parametric equations are used in two was: to find a graph of the modeling equation and to simulate the motion of the projectile. T=4 X=5.5 Y=13 [0, 6] b [0, 00] (c) T=5 X=5.5 Y=55 [0, 6] b [0, 00] (d) FIGURE 6.30 Simultaneous graphing of 1 t, 1 16t 76t 75 (height against time) and 5.5, 16t 76t 75 (the actual path of the flare). (Eample 8) EXAMPLE 8 Simulating Projectile Motion A distress flare is shot straight up from a ship s bridge 75 ft above the water with an initial velocit of 76 ft sec. Graph the flare s height against time, give the height of the flare above water at each time, and simulate the flare s motion for each length of time. (a) 1 sec (b) sec (c) 4 sec (d) 5 sec SOLUTION An equation that models the flare s height above the water t seconds after launch is 16t 76t 75. A graph of the flare s height against time can be found using the parametric equations 1 t, 1 16t 76t 75. To simulate the flare s flight straight up and its fall to the water, use the parametric equations 5.5, 16t 76t 75. We chose 5.5 so that the two graphs would not intersect. Figure 6.30 shows the two graphs in simultaneous graphing mode for a 0 t 1, b 0 t, c 0 t 4, and d 0 t 5. We can read that the height of the flare above the water after 1 sec is 135 ft, after sec is 163 ft, after 4 sec is 13 ft, and after 5 sec is 55 ft. Now tr Eercise 39.

28 5144_Demana_Ch06pp /11/06 9:33 PM Page CHAPTER 6 Applications of Trigonometr v 0 0 v 0 cos FIGURE 6.31 Throwing a baseball. [0, 450] b [0, 80] v 0 sin FIGURE 6.3 The fence and path of the baseball in Eample 9. See Eploration for was to draw the wall. EXPLORATION EXTENSIONS Using trial and error, determine the minimum angle, to the nearest 0.05, such that the ball clears the fence. In Eample 8 we modeled the motion of a projectile that was launched straight up. Now we investigate the motion of objects, ignoring air friction, that are launched at angles other than 90 with the horizontal. Suppose that a baseball is thrown from a point 0 feet above ground level with an initial speed of v 0 ft sec at an angle with the horizontal Figure The initial velocit can be represented b the vector v v 0 cos, v 0 sin. The path of the object is modeled b the parametric equations v 0 cos t, 16t v 0 sin t 0. The -component is simpl distance -component of initial velocit time. The -component is the familiar vertical projectile-motion equation using the -component of initial velocit. EXAMPLE 9 Hitting a Baseball Kevin hits a baseball at 3 ft above the ground with an initial speed of 150 ft sec at an angle of 18 with the horizontal. Will the ball clear a 0-ft wall that is 400 ft awa? SOLUTION The path of the ball is modeled b the parametric equations 150 cos 18 t, 16t 150 sin 18 t 3. A little eperimentation will show that the ball will reach the fence in less than 3 sec. Figure 6.3 shows a graph of the path of the ball using the parameter interval 0 t 3 and the 0-ft wall. The ball does not clear the wall. Now tr Eercise 43. EXPLORATION Etending Eample 9 1. If our grapher has a line segment feature, draw the fence in Eample 9.. Describe the graph of the parametric equations 400, 0 t 3, 0 t Repeat Eample 9 for the angles 19, 0, 1, and. 30 ft A In Eample 10 we see how to write parametric equations for position on a moving Ferris wheel using time t as the parameter. 10 ft FIGURE 6.33 The Ferris wheel of Eample 10. EXAMPLE 10 Riding on a Ferris Wheel Jane is riding on a Ferris wheel with a radius of 30 ft. As we view it in Figure 6.33, the wheel is turning counterclockwise at the rate of one revolution ever 10 sec. Assume the lowest point of the Ferris wheel 6 o clock is 10 ft above the ground, and that Jane is at the point marked A 3 o clock at time t 0. Find parametric equations to model Jane s path and use them to find Jane s position sec into the ride. continued

29 5144_Demana_Ch06pp /11/06 9:33 PM Page 59 SECTION 6.3 Parametric Equations and Motion 59 FOLLOW-UP Have students eplain how the parametric equations in Eample 10 were determined. ASSIGNMENT GUIDE Da 1: E. 1 4, 6 30, multiples of 3, Da : E , multiples of 3, COOPERATIVE LEARNING Group Activit: E , 66 NOTES ON EXERCISES E and include a variet of interesting applications. E relate to ccloids and hpoccloids. A Spirograph can be used to help illustrate these curves. E provide practice with standardized tests. ONGOING ASSESSMENT Self-Assessment: E. 7, 11, 15, 3, 7, 9, 37, 39, 43, 51 Embedded Assessment: E. 57, 58, 65, P 30 θ A FIGURE 6.34 A model for the Ferris wheel of Eample 10. SOLUTION Figure 6.34 shows a circle with center 0, 40 and radius 30 that models the Ferris wheel. The parametric equations for this circle in terms of the parameter, the central angle of the circle determined b the arc AP, are 30 cos, sin, 0. To take into account the rate at which the wheel is turning we must describe as a function of time t in seconds. The wheel is turning at the rate of radians ever 10 sec, or 10 5 rad sec. So, 5 t. Thus, parametric equations that model Jane s path are given b 30 cos ( 5 t ), sin ( 5 t ), t 0. We substitute t into the parametric equations to find Jane s position at that time: 30 cos ( 5 ) sin ( 5 ) After riding for sec, Jane is approimatel 68.5 ft above the ground and approimatel 9.3 ft to the right of the -ais using the coordinate sstem of Figure Now tr Eercise 51. Quick Review 6.3 (For help, go to Sections P., P.4, 1.3, 4.1, and 6.1.) In Eercises 1 and, find the component form of the vectors (a) OA, (b) OB, and (c) AB where O is the origin. 1. A 3,, B 4, 6. A 1, 3, B 4, 3 In Eercises 3 and 4, write an equation in point-slope form for the line through the two points. 3. 3,, 4, , 3, 4, 3 In Eercises 5 and 6, find and graph the two functions defined implicitl b each given relation In Eercises 7 and 8, write an equation for the circle with given center and radius. 7. 0, 0, 4 8., 5, 3 In Eercises 9 and 10, a wheel with radius r spins at the given rate. Find the angular velocit in radians per second. 9. r 13 in., 600 rpm 10. r 1 in., 700 rpm ( 3) or 6 8 ( 4) ( 1) or ( 4) 8. ( ) ( 5) rad/sec rad/sec 3

30 5144_Demana_Ch06pp /11/06 9:33 PM Page CHAPTER 6 Applications of Trigonometr SECTION 6.3 EXERCISES In Eercises 1 4, match the parametric equations with their graph. Identif the viewing window that seems to have been used. (a) (b) t, t [Hint: Eliminate t and solve for in terms of.] t, t 3 3, t 1. t 3, t, 5 t 5. t, 4 t, t 3. 5 cos t, 5 sin t 4. 4 cos t, 4 sin t 5. sin t, cos t, 0 t cos t, 3 sin t, 0 t In Eercises 7 3 find a parametrization for the curve. 7. The line through the points, 5 and 4,. 8. The line through the points 3, 3 and 5, The line segment with endpoints 3, 4 and 6, The line segment with endpoints 5, and, 4. (c) 1. 4 cos 3 t, sin 3 t. 3 cos t, sin t 3. cos t cos t, sin t sin t 4. sin t t cos t, cos t t sin t In Eercises 5 and 6, (a) complete the table for the parametric equations and (b) plot the corresponding points. 5. t, 1 3 t t cos t, sin t In Eercises 7 10, graph the parametric equations 3 t, t, in the specified parameter interval. Use the standard viewing window t t t t 4 In Eercises 11 6, eliminate the parameter and identif the graph of the parametric curve t, t 1. 3t, 5 t 13. t 3, 9 4t, 3 t t, t, 1 t t, t 1 [Hint: Eliminate t and solve for in terms of.] 16. t, t t, t 3 t t 1, t [Hint: Eliminate t and solve for in terms of.] (d) / und. 4 5/ t 0 / 3 / The circle with center 5, and radius The circle with center, 4 and radius. Eercises refer to the graph of the parametric equations t, t 0.5, 3 t 3 given below. Find the values of the parameter t that produces the graph in the indicated quadrant. [ 5, 5] b [ 5, 5] 33. Quadrant I 0.5 t 34. Quadrant II t Quadrant III 3 t 36. Quadrant IV t Simulating a Foot Race Ben can sprint at the rate of 4 ft sec. Jerr sprints at 0 ft sec. Ben gives Jerr a 10-ft head start. The parametric equations can be used to model a race. 1 0t, 1 3 4t 10, 5 (a) Find a viewing window to simulate a 100-d dash. Graph simulaneousl with t starting at t 0 and Tstep (b) Who is ahead after 3 sec and b how much? Ben is ahead b ft. 38. Capture the Flag Two opposing plaers in Capture the Flag are 100 ft apart. On a signal, the run to capture a flag that is on the ground midwa between them. The faster runner, however, hesitates for 0.1 sec. The following parametric equations model the race to the flag: 1 10 t 0.1, t, 3

31 5144_Demana_Ch06pp /11/06 9:33 PM Page 531 SECTION 6.3 Parametric Equations and Motion 531 (a) Simulate the game in a 0, 100 b 1, 10 viewing window with t starting at 0. Graph simultaneousl. (b) Who captures the flag and b how man feet? 50 ft 50 ft 39. Famine Relief Air Drop A relief agenc drops food containers from an airplane on a war-torn famine area. The drop was made from an altitude of 1000 ft above ground level. (a) Use an equation to model the height of the containers (during free fall) as a function of time t. 16t 1000 (b) Use parametric mode to simulate the drop during the first 6 sec. (c) After 4 sec of free fall, parachutes open. How man feet above the ground are the food containers when the parachutes open? 744 ft 40. Height of a Pop-up A baseball is hit straight up from a height of 5 ft with an initial velocit of 80 ft sec. (a) Write an equation that models the height of the ball as a function of time t. 16t 80t 5 (b) Use parametric mode to simulate the pop-up. (c) Use parametric mode to graph height against time. [Hint: Let t t.] (d) How high is the ball after 4 sec? 69 ft (e) What is the maimum height of the ball? How man seconds does it take to reach its maimum height? 41. The complete graph of the parametric equations cos t, sin t is the circle of radius centered at the origin. Find an interval of values for t so that the graph is the given portion of the circle. (a) The portion in the first quadrant 0 t (b) The portion above the -ais 0 t (c) The portion to the left of the -ais t 3 4. Writing to Learn Consider the two pairs of parametric equations 3 cos t, 3 sin t and 3 sin t, 3 cos t for 0 t. (a) Give a convincing argument that the graphs of the pairs of parametric equations are the same. (b) Eplain how the parametrizations are different. 43. Hitting a Baseball Consider Kevin s hit discussed in Eample 9. (a) Approimatel how man seconds after the ball is hit does it hit the wall? about.80 sec (b) How high up the wall does the ball hit? 7.18 ft (c) Writing to Learn Eplain wh Kevin s hit might be caught b an outfielder. Then eplain wh his hit would likel not be caught b an outfielder if the ball had been hit at a 0 angle with the horizontal. 44. Hitting a Baseball Kirb hits a ball when it is 4 ft above the ground with an initial velocit of 10 ft sec. The ball leaves the bat at a 30 angle with the horizontal and heads toward a 30-ft fence 350 ft from home plate. (a) Does the ball clear the fence? no (b) If so, b how much does it clear the fence? If not, could the ball be caught? not catchable 45. Hitting a Baseball Suppose that the moment Kirb hits the ball in Eercise 44 there is a 5-ft sec split-second wind gust. Assume the wind acts in the horizontal direction out with the ball. (a) Does the ball clear the fence? es (b) If so, b how much does it clear the fence? If not, could the ball be caught? 1.59 ft 46. Two-Softball Toss Chris and Linda warm up in the outfield b tossing softballs to each other. Suppose both tossed a ball at the same time from the same height, as illustrated in the figure. Find the minimum distance between the two balls and when this minimum distance occurs ft; 1.1 sec 45 ft/sec Linda ft 78 ft 41 ft/sec Chris 47. Yard Darts Ton and Sue are launching ard darts 0 ft from the front edge of a circular target of radius 18 in. on the ground. If Ton throws the dart directl at the target, and releases it 3 ft above the ground with an initial velocit of 30 ft sec at a 70 angle, will the dart hit the target? no 48. Yard Darts In the game of darts described in Eercise 47, Sue releases the dart 4 ft above the ground with an initial velocit of 5 ft sec at a 55 angle. Will the dart hit the target? es 49. Hitting a Baseball Orlando hits a ball when it is 4 ft above ground level with an initial velocit of 160 ft sec. The ball leaves the bat at a 0 angle with the horizontal and heads toward a 30-ft fence 400 ft from home plate. How strong must a split-second wind gust be (in feet per second) that acts directl with or against the ball in order for the ball to hit within a few inches of the top of the wall? Estimate the answer graphicall and solve algebraicall. 50. Hitting Golf Balls Nanc hits golf balls off the practice tee with an initial velocit of 180 ft sec with four different clubs. How far down the fairwa does the ball hit the ground if it comes off the club making the specified angle with the horizontal? (a) 15 (b) 0 (c) 5 (d) ft ft ft ft

32 5144_Demana_Ch06pp /11/06 9:33 PM Page CHAPTER 6 Applications of Trigonometr 51. Analsis of a Ferris Wheel Ron is on a Ferris wheel of radius 35 ft that turns counterclockwise at the rate of one revolution ever 1 sec. The lowest point of the Ferris wheel (6 o clock) is 15 ft above ground level at the point 0, 15 on a rectangular coordinate sstem. Find parametric equations for the position of Ron as a function of time t (in seconds) if the Ferris wheel starts t 0 with Ron at the point 35, Revisiting Eample 5 Eliminate the parameter t from the parametric equations of Eample 5 to find an equation in and for the line. Verif that the line passes through the points A and B of the eample. (3 5) Ccloid The graph of the parametric equations t sin t, 1 cos t is a ccloid. Group Activit In Eercises 55 58, a particle moves along a horizontal line so that its position at an time t is given b s t. Write a description of the motion. [Hint: See Eample 7.] 55. s t t 3t, t s t t 4t, 1 t s t 0.5 t 3 7t t, 1 t s t t 3 5t 4t, 1 t 5 Standardized Test Questions 59. True or False The two sets of parametric equations 1 t 1, 1 3t 1 and 3 t 4 3, t correspond to the same rectangular equation. Justif our answer. 60. True or False The graph of the parametric equations t 1, t 1, 1 t 3 is a line segment with endpoints 0, 1 and, 5. Justif our answer. In Eercises 61 64, solve the problem without using a calculator. 61. Multiple Choice Which of the following points corresponds to t 1 in the parametrization t 4, t 1 t? A [, 16] b [ 1, 10] (a) What is the maimum value of 1 cos t? How is that value related to the graph? (b) What is the distance between neighboring -intercepts? 54. Hpoccloid The graph of the parametric equations cos t cos t, sin t sin t is a hpoccloid. The graph is the path of a point P on a circle of radius 1 rolling along the inside of a circle of radius 3, as illustrated in the figure. 3 C 1 P t (a) Graph simultaneousl this hpoccloid and the circle of radius 3. (b) Suppose the large circle had a radius of 4. Eperiment! How do ou think the equations in part (a) should be changed to obtain defining equations? What do ou think the hpoccloid would look like in this case? Check our guesses. All s should be changed to 3 s. (A) 3, (B) 3, 0 (C) 5, (D) 5, 0 (E) 3, 6. Multiple Choice Which of the following values of t produces the same point as t 3 in the parametrization cos t, sin t? A (A) t 4 (B) t (C) t (D) t 4 7 (E) t Multiple Choice A rock is thrown straight up from level ground with its position above ground at an time t 0 given b 5, 16t 80t 7. At what time will the rock be 91 ft above ground? D (A) 1.5 sec (B).5 sec (C) 3.5 sec (D) 1.5 sec and 3.5 sec (E) The rock never goes that high. 64. Multiple Choice Which of the following describes the graph of the parametric equations 1 t, 3t, t 0? C (A) a straight line (B) a line segment (C) a ra (D) a parabola (E) a circle

33 5144_Demana_Ch06pp /11/06 9:33 PM Page 533 SECTION 6.3 Parametric Equations and Motion 533 Eplorations 65. Parametrizing Circles Consider the parametric equations a cos t, a sin t, 0 t. (a) Graph the parametric equations for a 1,, 3, 4 in the same square viewing window. (b) Eliminate the parameter t in the parametric equations to verif that the are all circles. What is the radius? Now consider the parametric equations h a cos t, k a sin t, 0 t. (c) Graph the equations for a 1 using the following pairs of values for h and k: h 4 3 k (d) Eliminate the parameter t in the parametric equations and identif the graph. (e) Write a parametrization for the circle with center 1, 4 and radius 3. 3 cos t 1; 3 sin t Group Activit Parametrization of Lines Consider the parametrization at b, ct d, where a and c are not both zero. (a) Graph the curve for a, b 3, c 1, and d. (b) Graph the curve for a 3, b 4, c 1, and d 3. (c) Writing to Learn Eliminate the parameter t and write an equation in and for the curve. Eplain wh its graph is a line. (d) Writing to Learn Find the slope, -intercept, and -intercept of the line if the eist? If not, eplain wh not. (e) Under what conditions will the line be horizontal? Vertical? c 0; a Throwing a Ball at a Ferris Wheel A 0-ft Ferris wheel turns counterclockwise one revolution ever 1 sec (see figure). Eric stands at point D, 75 ft from the base of the wheel. At the instant Jane is at point A, Eric throws a ball at the Ferris wheel, releasing it from the same height as the bottom of the wheel. If the ball s initial speed is 60 ft sec and it is released at an angle of 10 with the horizontal, does Jane have a chance to catch the ball? Follow the steps below to obtain the answer. (a) Assign a coordinate sstem so that the bottom car of the Ferris wheel is at 0, 0 and the center of the wheel is at 0, 0. Then Eric releases the ball at the point 75, 0. Eplain wh parametric equations for Jane s path are: 1 0 cos 6 t, sin 6 t, t 0. (b) Eplain wh parametric equations for the path of the ball are: 30t 75, 16t 30 3 t, t 0. (c) Graph the two paths simultaneousl and determine if Jane and the ball arrive at the point of intersection of the two paths at the same time. (d) Find a formula for the distance d t between Jane and the ball at an time t. (e) Writing to Learn Use the graph of the parametric equations 3 t, 3 d t, to estimate the minimum distance between Jane and the ball and when it occurs. Do ou think Jane has a chance to catch the ball? 0 ft 75 ft 68. Throwing a Ball at a Ferris Wheel A 71-ft-radius Ferris wheel turns counterclockwise one revolution ever 0 sec. Ton stands at a point 90 ft to the right of the base of the wheel. At the instant Matthew is at point A (3 o clock), Ton throws a ball toward the Ferris wheel with an initial velocit of 88 ft sec at an angle with the horizontal of 100. Find the minimum distance between the ball and Matthew. about 3.47 ft Etending the Ideas 69. Two Ferris Wheels Problem Chang is on a Ferris wheel of center 0, 0 and radius 0 ft turning counterclockwise at the rate of one revolution ever 1 sec. Kuan is on a Ferris wheel of center 15, 15 and radius 15 turning counterclockwise at the rate of one revolution ever 8 sec. Find the minimum distance between Chang and Kuan if both start out t 0 at 3 o clock. about 4.11 ft 70. Two Ferris Wheels Problem Chang and Kuan are riding the Ferris wheels described in Eercise 69. Find the minimum distance between Chang and Kuan if Chang starts out t 0 at 3 o clock and Kuan at 6 o clock. about ft Eercises refer to the graph C of the parametric equations tc 1 t a, td 1 t b where P 1 a, b and P c, d are two fied points. 71. Using Parametric Equations in Geometr Show that the point P, on C is equal to (a) P 1 a, b when t 0. (b) P c, d when t Using Parametric Equations in Geometr Show that if t 0.5, the corresponding point, on C is the midpoint of the line segment with endpoints a, b and c, d. 73. What values of t will find two points that divide the line segment P 1 P into three equal pieces? Four equal pieces? t 1 3, 3 ; t 1 1, 4, 3 4 A D

34 5144_Demana_Ch06pp /11/06 9:33 PM Page CHAPTER 6 Applications of Trigonometr Pole O 6.4 Polar Coordinates What ou ll learn about Polar Coordinate Sstem Coordinate Conversion Equation Conversion Finding Distance Using Polar Coordinates... and wh Use of polar coordinates sometimes simplifies complicated rectangular equations and the are useful in calculus. θ Polar ais P(r, θ ) FIGURE 6.35 The polar coordinate sstem. Polar Coordinate Sstem A polar coordinate sstem is a plane with a point O, the pole, and a ra from O, the polar ais, as shown in Figure Each point P in the plane is assigned as polar coordinates follows: r is the directed distance from O to P, and is the directed angle whose initial side is on the polar ais and whose terminal side is on the line OP. As in trigonometr, we measure as positive when moving counterclockwise and negative when moving clockwise. If r 0, then P is on the terminal side of. If r 0, then P is on the terminal side of. We can use radian or degree measure for the angle as illustrated in Eample 1. EXAMPLE 1 Plotting Points in the Polar Coordinate Sstem Plot the points with the given polar coordinates. (a) P, 3 (b) Q 1, 3 4 (c) R 3, 45 SOLUTION Figure 6.36 shows the three points. Now tr Eercise 7. OBJECTIVE Students will be able to convert points and equations from polar to rectangular coordinates and vice versa. MOTIVATE Ask students to suggest other methods (besides Cartesian coordinates) of describing the location of a point on a plane. LESSON GUIDE Da 1: Polar Coordinate Sstem; Coordinate Conversion; Equation Conversion (Polar to Rectangular) Da : Equation Conversion (Rectangular to Polar); Finding Distance Using Polar Coordinates ALERT Because of their etensive use of the Cartesian coordinate sstem, man students will be surprised that the polar coordinates of a point are not unique. Emphasize the fact that neither r nor is uniquel defined. O π b Pa, 3 π 3 (a) O 1 3π 4 3π Qa 1, b 4 (b) FIGURE 6.36 The three points in Eample 1. O 45 3 (c) R(3, 45 ) Each polar coordinate pair determines a unique point. However, the polar coordinates of a point P in the plane are not unique. EXAMPLE Finding all Polar Coordinates for a Point If the point P has polar coordinates 3, 3, find all polar coordinates for P. SOLUTION Point P is shown in Figure Two additional pairs of polar coordinates for P are ( 3, 3 ) ( 3, 7 3 ) and ( 3, 3 ) ( 3, 4 3 ). continued

35 5144_Demana_Ch06pp /11/06 9:33 PM Page 535 SECTION 6.4 Polar Coordinates 535 4π 3 O 3 π 3 π b Pa3, 3 FIGURE 6.37 The point P in Eample. We can use these two pairs of polar coordinates for P to write the rest of the possibilities: ( 3, 3 n ) ( 3, 6n 1 3 ) or ( 3, 3 n 1 ) ( 3, 6n 4 3 ) Where n is an integer. Now tr Eercise 3. The coordinates r,, r,, and r, all name the same point. In general, the point with polar coordinates r, also has the following polar coordinates: Finding all Polar Coordinates of a Point Let P have polar coordinates r,. An other polar coordinate of P must be of the form r, n or r, n 1 where n is an integer. In particular, the pole has polar coordinates 0,, where is an angle. Pole O(0, 0) r θ Polar ais P(r, θ ) P(, ) FIGURE 6.38 Polar and rectangular coordinates for P. Coordinate Conversion When we use both polar coordinates and Cartesian coordinates, the pole is the origin and the polar ais is the positive -ais as shown in Figure B appling trigonometr we can find equations that relate the polar coordinates r, and the rectangular coordinates, of a point P. Coordinate Conversion Equations Let the point P have polar coordinates r, and rectangular coordinates,. Then r cos, r, r sin, tan. These relationships allow us to convert from one coordinate sstem to the other. EXAMPLE 3 Converting from Polar to Rectangular Coordinates Find the rectangular coordinates of the points with the given polar coordinates. (a) P 3, 5 6 (b) Q, 00 continued

36 5144_Demana_Ch06pp /11/06 9:33 PM Page CHAPTER 6 Applications of Trigonometr SOLUTION (a) For P 3, 5 6, r 3 and 5 6: Pa3, 5π b 6 3 5π 6 r cos r sin 3 cos 5 5 and 3 sin 6 6 3( 3 ).60 3 ( 1 ) 1.5 (a) The rectangular coordinates for P are 3 3, , 1.5 Figure 6.39a. (b) For Q, 00, r and 00 : Q(, 00 ) 00 r cos r sin and cos sin The rectangular coordinates for Q are approimatel 1.88, 0.68 Figure 6.39b. Now tr Eercise 15. (b) FIGURE 6.39 The points P and Q in Eample 3. NOTES ON EXAMPLES Eample 4 provides an opportunit to monitor students use of the inverse kes on their graphers. You ma need to assist some students in the correct choice of the quadrant for this eample. P( 1, 1) π + tan 1 ( 1) = 3π 4 tan 1 π ( 1) = 4 FIGURE 6.40 The point P in Eample 4a. When converting rectangular coordinates to polar coordinates, we must remember that there are infinitel man possible polar coordinate pairs. In Eample 4 we report two of the possibilities. EXAMPLE 4 Converting from Rectangular to Polar Coordinates Find two polar coordinate pairs for the points with given rectangular coordinates. (a) P 1, 1 (b) Q 3, 0 SOLUTION (a) For P 1, 1, 1 and 1: r tan r and tan 1 1 r tan 1 1 n 4 n We use the angles 4 and Because P is on the ra opposite the terminal side of 4, the value of r corresponding to this angle is negative Figure Because P is on the terminal side of 3 4, the value of r corresponding to this angle is positive. So two polar coordinate pairs of point P are (, 4 ) and (, 3 4 ). (b) For Q 3, 0, 3 and 0. Thus, r 3 and n. We use the angles 0 and. So two polar coordinates pairs for point Q are 3, 0 and 3,. Now tr Eercise 7.

37 5144_Demana_Ch06pp /11/06 9:33 PM Page 537 SECTION 6.4 Polar Coordinates 537 EXPLORATION EXTENSIONS Describe how our grapher chooses what values to give when converting rectangular coordinates to polar coordinates. For eample, according to our grapher, what are the possible values for r and for? FOLLOW-UP Ask whether it is possible for two polar equations that are not algebraicall equivalent to have identical graphs. (Yes) EXPLORATION 1 Using a Grapher to Convert Coordinates Most graphers have the capabilit to convert polar coordinates to rectangular coordinates and vice versa. Usuall the give just one possible polar coordinate pair for a given rectangular coordinate pair. 1. Use our grapher to check the conversions in Eamples 3 and 4.. Use our grapher to convert the polar coordinate pairs, 3, 1,,,, 5, 3, 3,, to rectangular coordinate pairs. (1, 3 ), (0, 1), (, 0), (0, 5), (3, 0) 3. Use our grapher to convert the rectangular coordinate pairs 1, 3, 0,, 3, 0, 1, 0, 0, 4 to polar coordinate pairs. (, 3), (, ), (3, 0), (1, ), (4, 3 ) Equation Conversion We can use the Coordinate Conversion Equations to convert polar form to rectangular form and vice versa. For eample, the polar equation r 4 cos can be converted to rectangular form as follows: r 4 cos r 4r cos 4 r, r cos [ 4.7, 4.7] b [ 3.1, 3.1] FIGURE 6.41 The graph of the polar equation r 4 cos in 0. [, 8] b [ 10, 10] FIGURE 6.4 The graph of the vertical line r 4 sec ( 4). (Eample 5) Subtract 4 and add 4. 4 Factor. Thus the graph of r 4 cos is all or part of the circle with center, 0 and radius. Figure 6.41 shows the graph of r 4 cos for 0 obtained using the polar graphing mode of our grapher. So, the graph of r 4 cos is the entire circle. Just as with parametric equations, the domain of a polar equation in r and is understood to be all values of for which the corresponding values of r are real numbers. You must also select a value for min and ma to graph in polar mode. You ma be surprised b the polar form for a vertical line in Eample 5. EXAMPLE 5 Converting from Polar Form to Rectangular Form Convert r 4 sec to rectangular form and identif the graph. Support our answer with a polar graphing utilit. SOLUTION r 4 sec r 4 Divide b sec. se c 1 r cos 4 cos se c. 4 r cos The graph is the vertical line 4 Figure 6.4. Now tr Eercise 35.

38 5144_Demana_Ch06pp /11/06 9:33 PM Page CHAPTER 6 Applications of Trigonometr EXAMPLE 6 Converting from Rectangular Form to Polar Form Convert 3 13 to polar form. SOLUTION [ 5, 10] b [, 8] FIGURE 6.43 The graph of the circle r 6 cos 4 sin. (Eample 6) (8, 110 ) Substituting r for, r cos for, and r sin for gives the following: r 6r cos 4r sin 0 r r 6 cos 4 sin 0 r 0 or r 6 cos 4 sin 0 (5, 15 ) The graph of r 0 consists of a single point, the origin, which is also on the graph of r 6 cos 4 sin 0. Thus, the polar form is r 6 cos 4 sin. The graph of r 6 cos 4 sin for 0 is shown in Figure 6.43 and appears to be a circle with center 3, and radius 1 3, as epected. Now tr Eercise 43. FIGURE 6.44 The distance and direction of two airplanes from a radar source. (Eample 7) Finding Distance Using Polar Coordinates A radar tracking sstem sends out high-frequenc radio waves and receives their reflection from an object. The distance and direction of the object from the radar is often given in polar coordinates. EXAMPLE 7 Using a Radar Tracking Sstem Radar detects two airplanes at the same altitude. Their polar coordinates are 8 mi, 110 and 5 mi, 15. See Figure How far apart are the airplanes? SOLUTION B the Law of Cosines Section 5.6, d cos d c o s 9 5 d 9.80 ASSIGNMENT GUIDE Da 1: E. 1 30, multiples of 3 Da : E , multiples of 3, COOPERATIVE LEARNING Group Activit: E NOTES ON EXERCISES E emphasize the fact that the polar coordinates of a point are not unique. The airplanes are about 9.80 mi apart. Now tr Eercise 51. We can also use the Law of Cosines to derive a formula for the distance between points in the polar coordinate sstem. See Eercise 61. E provide practice with standardized tests. E show the connection between polar equations and parametric equations. ONGOING ASSESSMENT Self-Assessment: E. 7, 15, 3, 7, 35, 43, 51 Embedded Assessment: E. 61

39 5144_Demana_Ch06pp /11/06 9:33 PM Page 539 SECTION 6.4 Polar Coordinates 539 QUICK REVIEW 6.4 (For help, go to Sections P., 4.3, and 5.6.) In Eercises 1 and, determine the quadrants containing the terminal side of the angles. 1. (a) 5 6 II (b) 3 4 III. (a) 300 I (b) 10 III In Eercises 3 6, find a positive and a negative angle coterminal with the given angle , , , , 480 In Eercises 7 and 8, write a standard form equation for the circle. 7. Center 3, 0 and radius 8. Center 0, 4 and radius 3 In Eercises 9 and 10, use The Law of Cosines to find the measure of the third side of the given triangle ( 3) 4 8. ( 4) 9 SECTION 6.4 EXERCISES In Eercises 1 4, the polar coordinates of a point are given. Find its rectangular coordinates , 3 3 (, ) 5π a 4, b 4 a3, 3 π b 3. ( 1, 3 ) 4. (, 60 ) In Eercises 5 and 6, (a) complete the table for the polar equation and (b) plot the corresponding points. 5. r 3 sin r r csc ( 1, 315 ) r 4 und und., In Eercises 7 14, plot the point with the given polar coordinates. 7. 3, , , , , , , , 135 In Eercises 15, find the rectangular coordinates of the point with given polar coordinates , , , , 14 5 (1.6, 1.18) 19., (, 0) 0. 1, (0, 1) 1., 70 (0, ). 3, 360 ( 3, 0) In Eercises 3 6, polar coordinates of point P are given. Find all of its polar coordinates. 3. P, 6 4. P 1, 4 5. P 1.5, 0 6. P.5, 50 In Eercises 7 30, rectangular coordinates of point P are given. Find all polar coordinates of P that satisf (a) 0 (b) (c) P 1, 1 8. P 1, 3 9. P, P 1,

40 5144_Demana_Ch06pp /11/06 9:33 PM Page CHAPTER 6 Applications of Trigonometr In Eercises 31 34, use our grapher to match the polar equation with its graph. (a) (c) 31. r 5 csc (b) 3. r 4 sin (d) 33. r 4 cos 3 (c) 34. r 4 sin 3 (a) In Eercises 35 4, convert the polar equation to rectangular form and identif the graph. Support our answer b graphing the polar equation. 35. r 3 sec 36. r csc 37. r 3 sin 38. r 4 cos 39. r csc r sec r sin 4 cos 4. r 4 cos 4 sin In Eercises 43 50, convert the rectangular equation to polar form. Graph the polar equation Tracking Airplanes The location, given in polar coordinates, of two planes approaching the Vicksburg airport are 4 mi, 1 and mi, 7. Find the distance between the airplanes mi 5. Tracking Ships The location of two ships from Mas Landing Lighthouse, given in polar coordinates, are 3 mi, 170 and 5 mi, 150. Find the distance between the ships..41 mi 53. Using Polar Coordinates in Geometr A square with sides of length a and center at the origin has two sides parallel to the - ais. Find polar coordinates of the vertices. 54. Using Polar Coordinates in Geometr A regular pentagon whose center is at the origin has one verte on the positive -ais at a distance a from the center. Find polar coordinates of the vertices. Standardized Test Questions 55. True or False Ever point in the plane has eactl two polar coordinates. Justif our answer. 56. True or False If r 1 and r are not 0, and if r 1, and r, represent the same point in the plane, then r 1 r. Justif our answer. (b) (d) In Eercises 57 60, solve the problem without using a calculator. 57. Multiple Choice If r 0, which of the following polar coordinate pairs represents the same point as the point with polar coordinates r,? C (A) r, (B) r, (C) r, 3 (D) r, (E) r, Multiple Choice Which of the following are the rectangular coordinates of the point with polar coordinate, 3? C (A) 3, 1 (B) 1, 3 (C) 1, 3 (D) 1, 3 (E) 1, Multiple Choice Which of the following polar coordinate pairs represent the same point as the point with polar coordinates, 110? A (A), 70 (B), 110 (C), 50 (D), 70 (E), Multiple Choice Which of the following polar coordinate pairs does not represent the point with rectangular coordinates,? E (A), 135 (B), 5 (C), 315 (D), 45 (E), 135 Eplorations 61. Polar Distance Formula Let P 1 and P have polar coordinates r 1, 1 and r,,respectivel. (a) If 1 is a multiple of, write a formula for the distance between P 1 and P. (b) Use the Law of Cosines to prove that the distance between P 1 and P is given b d r 1 r r r 1 ( cos 1 ) (c) Writing to Learn Does the formula in part b agree with the formula s ou found in part a? Eplain. 6. Watching Your -Step Consider the polar curve r 4 sin. Describe the graph for each of the following. (a) 0 (b) (c) 0 3 (d) 0 4 In Eercises 63 66, use the results of Eercise 61 to find the distance between the points with given polar coordinates. 63., 10, 5, , 0, 6, , 5, 5, , 35, 8, Etending the Ideas 67. Graphing Polar Equations Parametricall Find parametric equations for the polar curve r f. Group Activit In Eercises 68 71, use what ou learned in Eercise 67 to write parametric equations for the given polar equation. Support our answers graphicall. 68. r cos 69. r 5 sin 70. r sec 71. r 4 csc

41 5144_Demana_Ch06pp /11/06 9:33 PM Page 541 SECTION 6.5 Graphs of Polar Equations Graphs of Polar Equations What ou ll learn about Polar Curves and Parametric Curves Smmetr Analzing Polar Curves Rose Curves Limaçon Curves Other Polar Curves... and wh Graphs that have circular or clindrical smmetr often have simple polar equations, which is ver useful in calculus. OBJECTIVE Students will be able to graph polar equations and determine the maimum r-value and the smmetr of a graph. MOTIVATE Ask students to use a grapher to compare the graphs of the polar equations r tan and r tan for 0. (The graphs are identical.) LESSON GUIDE Da 1: Polar Curves and Parametric Curves; Smmetr; Analzing Polar Graphs; Rose Curves Da : Limaçon Curves; Other Polar Curves TEACHING NOTE When determining smmetr, sometimes onl one of the replacements will appear to produce an equivalent polar equation. For eample, the graph of r is smmetric about the polar ais, but at first glance this equation does not appear to be equivalent to r. Although the pairs (r, ) that solve each equation are different, the graphs of these two equations are the same. Polar Curves and Parametric Curves Polar curves are actuall just special cases of parametric curves. Keep in mind that polar curves are graphed in the (, ) plane, despite the fact that the are given in terms of r and. That is wh the polar graph of r 4 cos is a circle (see Figure 6.41 in Section 6.4) rather than a cosine curve. In function mode, points are determined b a vertical coordinate that changes as the horizontal coordinate moves left to right. In polar mode, points are determined b a directed distance from the pole that changes as the angle sweeps around the pole. The connection is provided b the Coordinate Conversion Equations from Section 6.4, which show that the graph of r f( ) is reall just the graph of the parametric equations f( ) cos f( ) sin for all values of in some parameter interval that suffices to produce a complete graph. (In man of our eamples, 0 will do.) Since modern graphing calculators produce these graphs so easil in polar mode, we are frankl going to assume that ou do not have to sketch them b hand. Instead we will concentrate on analzing the properties of the curves. In later courses ou can discover further properties of the curves using the tools of calculus. Smmetr You learned algebraic tests for smmetr for equations in rectangular form in Section 1.. Algebraic tests also eist for polar form. Figure 6.45 on the net page shows a rectangular coordinate sstem superimposed on a polar coordinate sstem, with the origin and the pole coinciding and the positive -ais and the polar ais coinciding. The three tpes of smmetr figures to be considered will have are: 1. The -ais (polar ais) as a line of smmetr (Figure 6.45a).. The -ais (the line ) as a line of smmetr (Figure 6.45b). 3. The origin (the pole) as a point of smmetr (Figure 6.45c). All three algebraic tests for smmetr in polar forms require replacing the pair r,, which satisfies the polar equation, with another coordinate pair and determining whether it also satisfies the polar equation. Smmetr Tests for Polar Graphs The graph of a polar equation has the indicated smmetr if either replacement produces an equivalent polar equation. To Test for Smmetr Replace B 1. about the -ais, r, r, or r,.. about the -ais, r, r, or r,. 3. about the origin, r, r, or r,.

42 5144_Demana_Ch06pp /11/06 9:33 PM Page CHAPTER 6 Applications of Trigonometr (r, θ ) (r, π θ ) = ( r, θ ) (r, θ ) (r, θ ) θ θ π θ θ θ + π θ (r, θ) = ( r, π θ ) ( r, θ) = (r, θ + π ) (a) (b) (c) FIGURE 6.45 Smmetr with respect to (a) the -ais (polar ais), (b) the -ais (the line ), and (c) the origin (the pole). [ 6,6] b [ 4, 4] FIGURE 6.46 The graph of r 4 sin 3 is smmetric about the -ais. (Eample 1) TEACHING NOTE It is customar to draw graphs of polar curves in radian mode. ALERT Students are used to specifing different viewing windows to control what the see on the graphing screen. On polar graphs, students should be careful to note their selection of the range values for,, and. Since the environment into which the polar coordinates are patched is the rectangular sstem, students need to be concerned about controlling both the viewing window and the polar environment. Encourage students to eperiment with different input values for both the polar function r and the input variable and to observe the effects on the graphs. EXAMPLE 1 Testing for Smmetr Use the smmetr tests to prove that the graph of r 4 sin 3 is smmetric about the -ais. SOLUTION Figure 6.46 suggests that the graph of r 4 sin 3 is smmetric about the -ais and not smmetric about the -ais or origin. r 4 sin 3 r 4 sin 3 Replace r, b r,. r 4 sin 3 r 4 sin 3 sin is an odd function of. r 4 sin 3 (Same as original.) Because the equations r 4 sin 3 and r 4 sin 3 are equivalent, there is smmetr about the -ais. Now tr Eercise 13. Analzing Polar Graphs We analze graphs of polar equations in much the same wa that we analze the graphs of rectangular equations. For eample, the function r of Eample 1 is a continuous function of. Also r 0 when 0 and when is an integer multiple of 3. The domain of this function is the set of all real numbers. Trace can be used to help determine the range of this polar function (Figure 6.47). It can be shown that 4 r 4. Usuall, we are more interested in the maimum value of r rather than the range of r in polar equations. In this case, r 4 so we can conclude that the graph is bounded. A maimum value for r is a maimum r-value for a polar equation. A maimum r-value occurs at a point on the curve that is the maimum distance from the pole. In Figure 6.47, a maimum r-value occurs at 4, 6 and 4,. In fact, we get a maimum r-value at ever r, which represents the tip of one of the three petals.

43 5144_Demana_Ch06pp /11/06 9:33 PM Page 543 SECTION 6.5 Graphs of Polar Equations R=4 θ = [ 6,6] b [ 5, 3] (a) R= 4 θ = [ 6,6] b [ 5, 3] (b) FIGURE 6.47 The values of r in r 4 sin 3 var from (a) 4 to (b) 4. r = + cos θ [ 4.7, 4.7] b [ 3.1, 3.1] Polar coordinates (a) = + cos [0, π ] b [ 4, 4] Rectangular coordinates (b) FIGURE 6.48 With, the -values in (b) are the same as the directed distance from the pole to (r, ) in (a). To find maimum r-values we must find maimum values of r as opposed to the directed distance r. Eample shows one wa to find maimum r-values graphicall. EXAMPLE Finding Maimum r-values Find the maimum r-value of r cos. SOLUTION Figure 6.48a shows the graph of r cos for 0. Because we are onl interested in the values of r, we use the graph of the rectangular equation cos in function graphing mode (Figure 6.48b). From this graph we can see that the maimum value of r, or, is 4. It occurs when is an multiple of. Now tr Eercise 1. EXAMPLE 3 Finding Maimum r-values Identif the points on the graph of r 3 cos for 0 that give maimum r-values. SOLUTION Using trace in Figure 6.49 we can show that there are four points on the graph of r 3 cos in 0 at maimum distance of 3 from the pole: 3, 0, 3,, 3,, and 3, 3. Figure 6.50a shows the directed distances r as the -values of 1 3 cos, and Figure 6.50b shows the distances r as the -values of 3 cos. There are four maimum values of (i.e., r ) in part (b) corresponding to the four etreme values of 1 (i.e., r) in part (a). Now tr Eercise 3. (3, π ) a 3, 3π b Maimum r-values (3, 0) Maimum r-values π b a 3, [0, π ] b [ 5, 5] [0, π ] b [ 5, 5] (a) (b) FIGURE 6.49 The graph of r 3 cos. (Eample 3) FIGURE 6.50 The graph of (a) 1 3 cos and (b) 3 cos in function graphing mode. (Eample 3)

44 5144_Demana_Ch06pp /11/06 9:33 PM Page CHAPTER 6 Applications of Trigonometr ALERT Some students will find the maimum value of r instead of the maimum value of r. Emphasize that we are looking for the maimum distance from the pole, and r ma be either positive or negative at this point. [ 4.7, 4.7] b [ 3.1, 3.1] FIGURE 6.51 The graph of 8-petal rose curve r 3 sin 4. (Eample 4) TEACHING NOTE Polar graphs are an invitation for students to eplore mathematics. Students will be able to produce elaborate graphs using polar graphing techniques. Encourage students to graph different modifications of r a b cos to see the effects of their choices. Rose Curves The curve in Eample 1 is a 3-petal rose curve and the curve in Eample 3 is a 4-petal rose curve. The graphs of the polar equations r a cos n and r a sin n, where n is an integer greater than 1, are rose curves. If n is odd there are n petals, and if n is even there are n petals. EXAMPLE 4 Analzing a Rose Curve Analze the graph of the rose curve r 3 sin 4. SOLUTION Figure 6.51 shows the graph of the 8-petal rose curve r 3 sin 4. The maimum r-value is 3. The graph appears to be smmetric about the -ais, -ais, and the origin. For eample, to prove that the graph is smmetric about the -ais we replace r, b r, : r 3 sin 4 r 3 sin 4 r 3 sin 4 4 r 3 sin 4 cos 4 cos 4 sin 4 Sine difference identit r 3 0 cos 4 1 sin 4 sin 4 0, cos 4 1 r 3 sin 4 r 3 sin 4 Because the new polar equation is the same as the original equation, the graph is smmetric about the -ais. In a similar wa, ou can prove that the graph is smmetric about the -ais and the origin. (See Eercise 58.) Domain: All reals. Range: 3, 3 Continuous Smmetric about the -ais, the -ais, and the origin. Bounded Maimum r-value: 3 No asmptotes. Now tr Eercise 9. Here are the general characteristics of rose curves. You will investigate these curves in more detail in Eercises 67 and 68.

45 5144_Demana_Ch06pp /11/06 9:33 PM Page 545 SECTION 6.5 Graphs of Polar Equations 545 A ROSE IS A ROSE Budding botanists like to point out that the rose curve doesn t look much like a rose. However, consider the beautiful stainedglass window shown here, which is a feature of man great cathedrals and is called a rose window. Graphs of Rose Curves The graphs of r a cos n and r a sin n, where n 1 is an integer, have the following characteristics: Domain: All reals Range: a, a Continuous Smmetr: Bounded Maimum r-value: a No asmptotes Number of petals: Limaçon Curves The limaçon curves n even, smmetric about -, -ais, origin n odd, r a cos n smmetric about -ais n odd, r a sin n smmetric about -ais n, if n is odd n, if n is even are graphs of polar equations of the form r a b sin and r a b cos, where a 0 and b 0. Limaçon, pronounced LEE-ma-sohn, is Old French for snail. There are four different shapes of limaçons, as illustrated in Figure 6.5. a Limaçon with an inner loop: < 1 b (a) a Cardioid: = 1 b (b) Dimpled limaçon: 1 < a < b (c) a Conve limaçon: b (d) FIGURE 6.5 The four tpes of limaçons. R=6 [ 7, 7] b [ 8, ] θ = FIGURE 6.53 The graph of the cardioid of Eample 5. 1 EXAMPLE 5 Analze the graph of r 3 3 sin. Analzing a Limaçon Curve SOLUTION We can see from Figure 6.53 that the curve is a cardioid with maimum r-value 6. The graph is smmetric onl about the -ais. Domain: All reals. Range: 0, 6 Continuous Smmetric about the -ais. Bounded Maimum r-value: 6 No asmptotes. Now tr Eercise 33.

46 5144_Demana_Ch06pp /11/06 9:33 PM Page CHAPTER 6 Applications of Trigonometr R=5 θ =0 [ 3, 8] b [ 4, 4] FIGURE 6.54 The graph of a limaçon with an inner loop. (Eample 6) 1 EXAMPLE 6 Analzing a Limaçon Curve Analze the graph of r 3 cos. SOLUTION We can see from Figure 6.54 that the curve is a limaçon with an inner loop and maimum r-value 5. The graph is smmetric onl about the -ais. Domain: All reals. Range: 1, 5 Continuous Smmetric about the -ais. Bounded Maimum r-value: 5 No asmptotes. Now tr Eercise 39. Graphs of Limaçon Curves The graphs of r a b sin and r a b cos, where a 0 and b 0, have the following characteristics: Domain: All reals Range: a b, a b Continuous Smmetr: Bounded Maimum r-value: a b No asmptotes r a b sin, smmetric about -ais r a b cos, smmetric about -ais EXPLORATION 1 Limaçon Curves Tr several values for a and b to convince ourself of the characteristics of limaçon curves listed above. [ 30, 30] b [ 0, 0] (a) [ 30, 30] b [ 0, 0] (b) FIGURE 6.55 The graph of r for (a) 0 (set min 0, ma 45, step 0.1) and (b) 0 (set min 45, ma 0, step 0.1). (Eample 7) Other Polar Curves All the polar curves we have graphed so far have been bounded. The spiral in Eample 7 is unbounded. EXAMPLE 7 Analzing the Spiral of Archimedes Analze the graph of r. SOLUTION We can see from Figure 6.55 that the curve has no maimum r-value and is smmetric about the -ais. Domain: All reals. Range: All reals. Continuous Smmetric about the -ais. Unbounded No maimum r-value. No asmptotes. Now tr Eercise 41.

47 5144_Demana_Ch06pp /11/06 9:33 PM Page 547 SECTION 6.5 Graphs of Polar Equations 547 The lemniscate curves are graphs of polar equations of the form r a sin and r a cos. [ 4.7, 4.7] b [ 3.1, 3.1] FIGURE 6.56 The graph of the lemniscate r 4 cos. (Eample 8) FOLLOW-UP Have students tr to confirm the -ais smmetr of Eample 4 b using the replacement (r, ) instead of using ( r, ). Discuss the results. ASSIGNMENT GUIDE Da 1: E. 1 8 all, 15 30, multiples of 3 Da : E. 9 1, 33 48, multiples of 3, 58, COOPERATIVE LEARNING Group Activit: E. 57 NOTES ON EXERCISES E require students to find the distance from the pole to the furthest point on the petal, not the arc length. E provide additional discussion of eamples in the tet. E provide practice with standardized tests. ONGOING ASSESSMENT Self-Assessment: E. 13, 1, 3, 9, 33, 39, 41, 43 Embedded Assessment: E. 59, 60, 73 EXAMPLE 8 Analzing a Lemniscate Curve Analze the graph of r 4 cos for 0,. SOLUTION It turns out that ou can get the complete graph using r cos. You also need to choose a ver small step to produce the graph in Figure Domain: 0, 4 3 4, , Range:, Smmetric about the -ais, the -ais, and the origin. Continuous (on its domain) Bounded Maimum r-value: No asmptotes. Now tr Eercise 43. EXPLORATION Revisiting Eample 8 1. Prove that -values in the intervals 4, 3 4 and 5 4, 7 4 are not in the domain of the polar equation r 4 cos.. Eplain wh r co s produces the same graph as r co s in the interval 0,. 3. Use the smmetr tests to show that the graph of r 4 cos is smmetric about the -ais. 4. Use the smmetr tests to show that the graph of r 4 cos is smmetric about the -ais. 5. Use the smmetr tests to show that the graph of r 4 cos is smmetric about the origin. EXPLORATION EXTENSIONS Graph r 4 sin. How is this graph related to the graph of r 4 cos? QUICK REVIEW 6.5 (For help, go to Sections 1. and 5.3.) In Eercises 1 4, find the absolute maimum value and absolute minimum value in 0, and where the occur cos. 3 cos 3. co s sin In Eercises 7 10, use trig identities to simplif the epression. 7. sin sin 8. cos cos 9. cos cos sin 10. sin sin cos In Eercises 5 and 6, determine if the graph of the function is smmetric about the (a) -ais, (b) -ais, and (c) origin. 5. sin no; no; es 6. cos 4 no; es; no

48 5144_Demana_Ch06pp /11/06 9:33 PM Page CHAPTER 6 Applications of Trigonometr SECTION 6.5 EXERCISES In Eercises 1 and, (a) complete the table for the polar equation, and (b) plot the corresponding points. 1. r 3 cos r r sin r In Eercises 3 6, draw a graph of the rose curve. State the smallest -interval 0 k that will produce a complete graph. 3. r 3 sin 3 4. r 3 cos 5. r 3 cos 6. r 3 sin 5 Eercises 7 and 8 refer to the curves in the given figure. [ 4.7, 4.7] b [ 3.1, 3.1] (a) 7. The graphs of which equations are shown? r 3 is graph (b). r 1 3 cos 6 r 3 sin 8 r 3 3 cos 3 8. Use trigonometric identities to eplain which of these curves is the graph of r 6 cos sin. (a) In Eercises 9 1, match the equation with its graph without using our graphing calculator. [ 4.7, 4.7] b [ 4.1,.1] (a) [ 4.7, 4.7] b [ 3.1, 3.1] (b) [ 4.7, 4.7] b [ 3.1, 3.1] (b) 9. Does the graph of r sin or r cos appear in the figure? Eplain. Graph (b) is r cos. 10. Does the graph of r 3 cos or r 3 cos appear in the figure? Eplain. Graph (c) is r 3 cos. 11. Is the graph in (a) the graph of r sin or r cos? Eplain. Graph (a) is r sin. 1. Is the graph in (d) the graph of r 1.5 cos or r 1.5 sin? Eplain. Graph (d) is r 1.5 sin. In Eercises 13 0, use the polar smmetr tests to determine if the graph is smmetric about the -ais, the -ais, or the origin. 13. r 3 3 sin 14. r 1 cos 15. r 4 3 cos 16. r 1 3 sin 17. r 5 cos 18. r 7 sin r 0. r 1 sin 1 cos In Eercises 1 4, identif the points for 0 where maimum r-values occur on the graph of the polar equation. 1. r 3 cos. r 3 sin 3. r 3 cos 3 4. r 4 sin In Eercises 5 44, analze the graph of the polar curve. 5. r 3 6. r r sin r 3 cos r 5 4 sin 3. r 6 5 cos 33. r 4 4 cos 34. r 5 5 sin 35. r 5 cos 36. r 3 sin 37. r 5 cos 38. r 3 4 sin 39. r 1 cos 40. r sin 41. r 4. r r sin, r 9 cos, 0 In Eercises 45 48, find the length of each petal of the polar curve. 45. r 4 sin 46. r 3 5 cos 47. r 1 4 cos r 3 4 sin 5 In Eercises 49 5, select the two equations whose graphs are the same curve. Then, even though the graphs of the equations are identical, describe how the two paths are different as increases from 0 to. 49. r sin, r 1 3 sin, r sin 50. r 1 1 cos, r 1 cos, r 3 1 cos 51. r 1 1 cos, r 1 cos, r 3 1 cos 5. r 1 sin, r sin, r 3 sin [ 3.7, 5.7] b [ 3.1, 3.1] (c) [ 4.7, 4.7] b [ 4.1,.1] (d)

49 5144_Demana_Ch06pp /11/06 9:33 PM Page 549 SECTION 6.5 Graphs of Polar Equations 549 In Eercises 53 56, (a) describe the graph of the polar equation, (b) state an smmetr that the graph possesses, and (c) state its maimum r-value if it eists. 53. r sin sin 54. r 3 cos sin r 1 3 cos r 1 3 sin Group Activit Analze the graphs of the polar equations r a cos n and r a sin n when n is an even integer. 58. Revisiting Eample 4 Use the polar smmetr tests to prove that the graph of the curve r 3 sin 4 is smmetric about the -ais and the origin. 59. Writing to Learn Revisiting Eample 5 Confirm the range stated for the polar function r 3 3 sin of Eample 5 b graphing 3 3 sin for 0. Eplain wh this works. 60. Writing to Learn Revisiting Eample 6 Confirm the range stated for the polar function r 3 cos of Eample 6 b graphing 3 cos for 0. Eplain wh this works. Standardized Test Questions 61. True or False A polar curve is alwas bounded. Justif our answer. False. The spiral r is unbounded. 6. True or False The graph of r cos is smmetric about the -ais. Justif our answer. In Eercises 63 66, solve the problem without using a calculator. 63. Multiple Choice Which of the following gives the number of petals of the rose curve r 3 cos? D (A) 1 (B) (C) 3 (D) 4 (E) Multiple Choice Which of the following describes the smmetr of the rose graph of r 3 cos? D (A) onl the -ais (B) onl the -ais (C) onl the origin (D) the -ais, the -ais, the origin (E) Not smmetric about the -ais, the -ais, or the origin 65. Multiple Choice Which of the following is a maimum r-value for r 3 cos? B (A) 6 (B) 5 (C) 3 (D) (E) Multiple Choice Which of the following is the number of petals of the rose curve r 5 sin 3? B (A) 1 (B) 3 (C) 6 (D) 10 (E) 15 Eplorations 67. Analzing Rose Curves Consider the polar equation r a cos n for n, an odd integer. (a) Prove that the graph is smmetric about the -ais. (b) Prove that the graph is not smmetric about the -ais. (c) Prove that the graph is not smmetric about the origin. (d) Prove that the maimum r-value is a. (e) Analze the graph of this curve. 68. Analzing Rose Curves Consider the polar equation r a sin n for n an odd integer. (a) Prove that the graph is smmetric about the -ais. (b) Prove that the graph is not smmetric about the -ais. (c) Prove that the graph is not smmetric about the origin. (d) Prove that the maimum r-value is a. (e) Analze the graph of this curve. 69. Etended Rose Curves The graphs of r 1 3 sin 5 and r 3 sin 7 ma be called rose curves. (a) Determine the smallest -interval that will produce a complete graph of r 1 ; of r. (b) How man petals does each graph have? Etending the Ideas In Eercises 70 7, graph each polar equation. Describe how the are related to each other. 70. (a) r 1 3 sin 3 (b) r 3 sin 3( 1 ) (c) r 3 3 sin 3( 4 ) 71. (a) r 1 sec (b) r sec ( 4 ) (c) r 3 sec ( 3 ) 7. (a) r 1 cos (b) r r 1 ( 4 ) (c) r 3 r 1 ( 3 ) 73. Writing to Learn Describe how the graphs of r f, r f, and r f are related. Eplain wh ou think this generalization is true.

50 5144_Demana_Ch06pp /11/06 9:34 PM Page CHAPTER 6 Applications of Trigonometr 6.6 De Moivre s Theorem and nth Roots What ou ll learn about The Comple Plane Trigonometric Form of Comple Numbers Multiplication and Division of Comple Numbers Powers of Comple Numbers Roots of Comple Numbers... and wh This material etends our equation-solving technique to include equations of the form z n c, n an integer and c a comple number. The Comple Plane You might be curious as to wh we reviewed comple numbers in Section P.6, then proceeded to ignore them for the net si chapters. (Indeed, after this section we will prett much ignore them again.) The reason is simpl because the ke to understanding calculus is the graphing of functions in the Cartesian plane, which consists of two perpendicular real (not comple) lines. We are not saing that comple numbers are impossible to graph. Just as ever real number is associated with a point of the real number line, ever comple number can be associated with a point of the comple plane. This idea evolved through the work of Caspar Wessel ( ), Jean-Robert Argand ( ) and Carl Friedrich Gauss ( ). Real numbers are placed along the horizontal ais (the real ais ) and imaginar numbers along the vertical ais (the imaginar ais ), thus associating the comple number a bi with the point (a, b). In Figure 6.57 we show the graph of 3i as an eample. Imaginar ais bi a + bi EXAMPLE 1 Plotting Comple Numbers Plot u 1 3i, v i, and u v in the comple plane. These three points and the origin determine a quadrilateral. Is it a parallelogram? Imaginar ais 3i a (a) + 3i Real ais SOLUTION First notice that u v (1 3i) ( i) 3 i. The numbers u, v, and u v are plotted in Figure 6.58a. The quadrilateral is a parallelogram because the arithmetic is eactl the same as in vector addition (Figure 6.58b). Now tr Eercise 1. Imaginar ais u = 1 + 3i u + v = 3 + i u = 1, 3 u + v = 3, Real ais O Real ais O (b) v = i v =, 1 FIGURE 6.57 Plotting points in the comple plane. IS THERE A CALCULUS OF COMPLEX FUNCTIONS? There is a calculus of comple functions. If ou stud it someda, it should onl be after acquiring a prett firm algebraic and geometric understanding of the calculus of real functions. (a) FIGURE 6.58 (a) Two numbers and their sum are plotted in the comple plane. (b) The arithmetic is the same as in vector addition. (Eample 1) Eample 1 shows how the comple plane representation of comple number addition is virtuall the same as the Cartesian plane representation of vector addition. Another similarit between comple numbers and two-dimensional vectors is the definition of absolute value. (b)

51 5144_Demana_Ch06pp /11/06 9:34 PM Page 551 SECTION 6.6 De Moivre s Theorem and nth Roots 551 OBJECTIVE Students will be able to represent comple numbers in the comple plane and write them in trigonometric form. The will be able to use trigonometric form to simplif some algebraic operations with comple numbers. MOTIVATE Have students find all solutions of the equation z 4 1, where z is a comple number. (z 1, z i) POLAR FORM What s in a cis? Trigonometric (or polar) form appears frequentl enough in scientific tets to have an abbreviated form. The epression cos i sin is often shortened to cis (pronounced kiss ). Thus z r cis. DEFINITION Absolute Value (Modulus) of a Comple Number The absolute value or modulus of a comple number z a bi is z a bi a b. In the comple plane, a bi is the distance of a bi from the origin. Trigonometric Form of Comple Numbers Figure 6.59 shows the graph of z a bi in the comple plane. The distance r from the origin is the modulus of z. If we define a direction angle for z just as we did with vectors, we see that a r cos and b r sin. Substituting these epressions for a and b gives us the trigonometric form (or polar form ) of the comple number z. Imaginar ais z = a + bi LESSON GUIDE Da 1: The Comple Plane; Trigonometric Form of Comple Numbers; Multiplication and Division of Comple Numbers Da : Powers of Comple Numbers; Roots of Comple Numbers r θ a = r cos u b = r sin u Real ais FIGURE 6.59 If r is the distance of z a bi from the origin and is the directional angle shown, then z r (cos i sin ), which is the trigonometric form of z. DEFINITION Trigonometric Form of a Comple Number The trigonometric form of the comple number z a bi is z r cos i sin where a r cos, b r sin, r a b, and tan b a. The number r is the absolute value or modulus of z,and is an argument of z. An angle for the trigonometric form of z can alwas be chosen so that 0, although an angle coterminal with could be used. Consequentl, the angle and argument of a comple number z are not unique. It follows that the trigonometric form of a comple number z is not unique.

52 5144_Demana_Ch06pp /11/06 9:34 PM Page CHAPTER 6 Applications of Trigonometr TEACHING NOTE It ma be useful to review comple numbers as introduced in Section P.6. Imaginar ais θ FIGURE 6.60 The comple number for Eample a. 3 4i θ 1 3i Imaginar ais θ θ Real ais Real ais FIGURE 6.61 The comple number for Eample b. EXAMPLE Finding Trigonometric Forms Find the trigonometric form with 0 for the comple number. (a) 1 3 i SOLUTION (b) 3 4i (a) For 1 3 i, r 1 3 i 1 3. Because the reference angle for is 3 (Figure 6.60), Thus, (b) For 3 4i, ( 3 ) i cos 5 5 i sin i The reference angle for (Figure 6.61) satisfies the equation tan 4 3, so tan Because the terminal side of is in the third quadrant, we conclude that Therefore, i 5 cos 4.07 i sin Now tr Eercise 5. Multiplication and Division of Comple Numbers The trigonometric form for comple numbers is particularl convenient for multipling and dividing comple numbers. The product involves the product of the moduli and the sum of the arguments. (Moduli is the plural of modulus.) The quotient involves the quotient of the moduli and the difference of the arguments. Product and Quotient of Comple Numbers Let z 1 r 1 cos 1 i sin 1 and z r cos i sin. Then 1. z 1 z r 1 r cos 1 i sin z r cos z r 1 i sin 1, r 0. 1

53 5144_Demana_Ch06pp /11/06 9:34 PM Page 553 SECTION 6.6 De Moivre s Theorem and nth Roots 553 TEACHING NOTE The proofs of the product and quotient formulas are good applications of the sum and difference identities studied in Section 5.3 Proof of the Product Formula z 1 z r 1 cos 1 i sin 1 r cos i sin r 1 r cos 1 cos sin 1 sin i sin 1 cos cos 1 sin r 1 r cos 1 i sin 1 You will be asked to prove the quotient formula in Eercise 63. TEACHING NOTE Man of the calculations discussed in this section can be performed using a grapher s built-in functions for converting between rectangular and polar coordinates. EXAMPLE 3 Multipling Comple Numbers Epress the product of z 1 and z in standard form: z 1 5 ( cos i sin 4 4 SOLUTION z 1 z 5 ( cos i sin cos ( ( cos i sin 1 1 ), z 14 ( cos 3 i sin 3 ). ) ( 14 cos 3 i sin 3 ) 3 ) ( i sin 4 ) 3 ) i Now tr Eercise 19. EXAMPLE 4 Dividing Comple Numbers Epress the quotient z 1 z in standard form: z 1 cos 135 i sin 135, SOLUTION 1 z z cos 135 i sin cos 300 i sin 300 z 6 cos 300 i sin 300. cos i sin cos 165 i sin i Now tr Eercise 3.

54 5144_Demana_Ch06pp /11/06 9:34 PM Page CHAPTER 6 Applications of Trigonometr z Imaginar ais r z θ r θ Real ais FIGURE 6.6 A geometric interpretation of z. TEACHING NOTE This section provides a nice opportunit to bring geometr and algebra together. Providing geometric motivations to numerical work helps students connect different mathematical ideas. Imaginar ais i 3 FIGURE 6.63 The comple number in Eample 5. 3 Real ais Powers of Comple Numbers We can use the product formula to raise a comple number to a power. For eample, let z r cos i sin. Then z z z r cos i sin r cos i sin r cos i sin r cos i sin Figure 6.6 gives a geometric interpretation of squaring a comple number: its argument is doubled and its distance from the origin is multiplied b a factor of r, increased if r 1 or decreased if r 1. We can find z 3 b multipling z b z : z 3 z z r cos i sin r cos i sin r 3 cos i sin r 3 cos 3 i sin 3 Similarl, z 4 r 4 cos 4 i sin 4 z 5 r 5 cos 5 i sin 5. This pattern can be generalized to the following theorem, named after the mathematician Abraham De Moivre ( ), who also made major contributions to the field of probabilit. De Moivre s Theorem Let z r cos i sin and let n be a positive integer. Then z n r cos i sin n r n cos n i sin n. EXAMPLE 5 Using De Moivre s theorem Find 1 i 3 3 using De Moivre s theorem. SOLUTION Solve Algebraicall See Figure The argument of z 1 i 3 is 3, and its modulus is 1 i Therefore, z ( cos 3 i sin 3 ) z 3 ( 3 cos 3 3 ) ( i sin 3 3 ) 8 cos i sin 8 1 0i 8 continued

55 5144_Demana_Ch06pp /11/06 9:34 PM Page 555 SECTION 6.6 De Moivre s Theorem and nth Roots 555 Support Numericall Figure 6.64a sets the graphing calculator we use in comple number mode. Figure 6.64b supports the result obtained algebraicall. Now tr Eercise 31. Normal Sci Eng Float Radian Degree Func Par Pol Seq Connected Dot Sequential Simul Real a+bi re^θ i Full Horiz G T (a) (1+i (3))3 (b) 8 FIGURE 6.64 (a) Setting a graphing calculator in comple number mode. (b) Computing (1 i 3 ) 3 with a graphing calculator. NOTES ON EXAMPLES Problems like Eample 6 are frequentl found on tests in math contests. The are eas if a student knows De Moivre s theorem. EXAMPLE 6 Using De Moivre s Theorem Find i 8 using De Moivre s theorem. SOLUTION The argument of z i is 3 4, and its modulus is i Therefore, z cos 3 3 i sin 4 4 z 8 cos ( ) ( i sin ) cos 6 i sin 6 1 i 0 1 Now tr Eercise 35. Roots of Comple Numbers The comple number 1 i 3 in Eample 5 is a solution of z 3 8, and the comple number i in Eample 6 is a solution of z 8 1. The comple number 1 i 3 is a third root of 8 and i is an eighth root of 1. TEACHING NOTE It is worth pointing out that unit simpl means one. nth Root of a Comple Number A comple number v a bi is an nth root of z v n z. If z 1, then v is an nth root of unit. if

56 5144_Demana_Ch06pp /11/06 9:34 PM Page CHAPTER 6 Applications of Trigonometr FOLLOW-UP Ask... The comple number cos 3 i sin 3 is an 8th root of unit. Wh is this number not listed in the solution to Eample 9 (or is it)? (It is the same as cos i sin 1 0i.) ASSIGNMENT GUIDE Da 1: E. 3 30, multiples of 3 Da : E , multiples of 3, 59, COOPERATIVE LEARNING Group Activit: E. 64 NOTES ON EXERCISES E provide an opportunit to show how the product and quotient formulas can simplif calculations when applicable. E involve nth roots of comple numbers. Encourage students to think about the radius of the circle in which the roots fall and the angular spacing between the roots. E provide practice with standardized tests. E give a graphical interpretation of the product of two comple numbers. ONGOING ASSESSMENT Self-Assessment: E. 1, 7, 19, 3, 31, 35, 45, 57, 59 Embedded Assessment: E. 71, 7, 78 We use De Moivre s theorem to develop a general formula for finding the nth roots of a nonzero comple number. Suppose that v s cos i sin is an nth root of z r cos i sin.then v n z [s cos i sin ] n r cos i sin s n cos n i sin n r cos i sin (1) Net, we take the absolute value of both sides: s n cos n i sin na r cos i sin s co s n n s in n r c o s s in s n r s n r s 0, r 0 s n r Substituting s n r into Equation (1), we obtain cos n i sin n cos i sin. Therefore, n can be an angle coterminal with. Consequentl, for an integer k, v is an nth root of z if s n r and n k k. n The epression for v takes on n different values for k 0, 1,, n 1, and the values start to repeat for k n, n 1,. We summarize this result. Finding nth Roots of a Comple Number If z r cos i sin, then the n distinct comple numbers r ( n cos k i sin k n n ), where k 0, 1,,...,n 1, are the nth roots of the comple number z. EXAMPLE 7 Finding Fourth Roots Find the fourth roots of z 5 cos 3 i sin 3. SOLUTION The fourth roots of z are the comple numbers 5 ( 4 cos 3 k i sin 3 k 4 4 for k 0, 1,, 3. ) continued

57 5144_Demana_Ch06pp /11/06 9:34 PM Page 557 SECTION 6.6 De Moivre s Theorem and nth Roots 557 Taking into account that 3 k 4 1 k, the list becomes z cos ( 0 1 ) ( i sin 0 1 ) 4 5 cos i sin 1 1 z 4 5 cos ( 1 ) ( i sin 1 ) 4 5 cos 7 i sin z cos ( 1 ) ( i sin 1 ) 4 5 cos 1 3 i sin z cos ( 1 3 ) i sin ( 1 3 ) 4 5 cos 1 9 i sin Now tr Eercise 45. TEACHING NOTE Eample 8 can also be solved b writing the equation z 3 1 0, factoring, and using the quadratic formula. It is useful for students to see that this method gives the same answer. z FIGURE 6.65 The three cube roots z 1, z, and z 3 of 1 displaed on the unit circle (dashed). (Eample 8) z 1 z 3 [.4,.4] b [ 1.6, 1.6] EXAMPLE 8 Finding Cube Roots Find the cube roots of 1 and plot them. SOLUTION First we write the comple number z 1 in trigonometric form z 1 0i cos i sin. The third roots of z 1 cos i sin are the comple numbers cos k i sin k, 3 3 for k 0, 1,. The three comple numbers are z 1 cos 3 i sin i, z cos i sin 1 0i, 3 3 z 3 cos 4 i sin i. Figure 6.65 shows the graph of the three cube roots z 1, z, and z 3. The are evenl spaced (with distance of 3 radians) around the unit circle. Now tr Eercise 57.

58 5144_Demana_Ch06pp /11/06 9:34 PM Page CHAPTER 6 Applications of Trigonometr EXAMPLE 9 Finding Roots of Unit Find the eight eighth roots of unit. SOLUTION First we write the comple number z 1 in trigonometric form z 1 0i cos 0 i sin 0. The eighth roots of z 1 0i cos 0 i sin 0 are the comple numbers cos 0 k i sin 0 k, 8 8 for k 0, 1,,...,7. z 1 cos 0 i sin 0 1 0i z cos 4 i sin 4 i z 3 cos i sin 0 i Imaginar ais z 4 z 5 z 6 z 3 z 7 z z 8 z 1 Real ais FIGURE 6.66 The eight eighth roots of unit are evenl spaced on a unit circle. (Eample 9) z 4 cos 3 3 i sin i 4 4 z 5 cos i sin 1 0i z 6 cos 5 5 i sin i 4 4 z 7 cos 3 3 i sin 0 i z 8 cos 7 7 i sin i 4 4 Figure 6.66 shows the eight points. The are spaced 8 4 radians apart. Now tr Eercise 59. QUICK REVIEW 6.6 (For help, go to Sections P.5, P.6, and 4.3.) In Eercises 1 and, write the roots of the equation in a bi form i, 3i. 5( 1) i, i In Eercises 3 and 4, write the comple number in standard form a bi i 5 4 4i 4. 1 i 4 4 0i In Eercises 5 8, find an angle in 0 which satisfies both equations. 5. sin 1 and cos sin and cos sin 3 and cos sin and cos 5 4 In Eercises 9 and 10, find all real solutions

59 5144_Demana_Ch06pp /11/06 9:34 PM Page 559 SECTION 6.6 De Moivre s Theorem and nth Roots 559 SECTION 6.6 EXERCISES In Eercises 1 and, plot all four points in the same comple plan i, 3 i, i, i. 3i, 1 i, 3, i, In Eercises 3 1, find the trigonometric form of the comple number where the argument satisfies i 4. i 5. i 6. 3 i 7. i i 9. 3 i i In Eercises 13 18, write the comple number in standard form a bi cos 30 i sin cos 10 i sin cos 60 i sin 60 5 (5 ) 3 i 16. 5( cos 4 i sin 4 ) 5 5 i 17. ( cos 7 7 i sin 6 6 ) 6 i ( cos i sin i 1 1 ) In Eercises 19, find the product of z 1 and z. Leave the answer in trigonometric form. 19. z 1 7 cos 5 i sin 5 14 (cos 155 i sin 155 ) z cos 130 i sin z 1 cos 118 i sin 118 (cos 99 i sin 99 ) z 0.5 cos 19 i sin z 1 5( cos 4 i sin 4 ) z 3( cos 5 5 i sin 3 3 ). z 1 3 ( cos i sin 4 ) z 1 3 ( cos 6 i sin 6 ) In Eercises 3 6, find the trigonometric form of the quotient. 3. cos 30 i sin cos 60 i sin z cos 5 i sin 5 3 cos i sin 45 z 4 5 cos 0 i sin 0 cos 115 i sin cos i sin cos 4 i sin 4 In Eercises 7 30, find the product z 1 z and quotient z 1 z in two was, (a) using the trigonometric form for z 1 and z.and (b) using the standard form for z 1 and z. 7. z 1 3 i and z 1 i 8. z 1 1 i and z 3 i 9. z 1 3 i and z 5 3i 30. z 1 3i and z 1 3 i In Eercises 31 38, use De Moivre s theorem to find the indicated power of the comple number. Write our answer in standard form a bi. 31. ( cos 4 i sin 4 )3 3. 3( cos 3 3 i sin 33. ( cos i sin 4 ) 5 43i ) ( cos 5 5 i sin 6 ) i 5 4 4i i ( i) i ( 1 i 3 )3 1 In Eercises 39 44, find the cube roots of the comple number. 39. cos i sin 40. ( cos 4 i sin 4 ) 41. 3( cos 4 4 i sin 3 3 ) 4. 7( cos 11 i sin i 44. i In Eercises 45 50, find the fifth roots of the comple number. 45. cos i sin 46. 3( cos i sin ) 47. ( cos 6 i sin 6 ) 48. ( cos 4 i sin 4 ) 49. i i In Eercises 51 56, find the nth roots of the comple number for the specified value of n i, n i, n i, n i, n i, n , n 5 In Eercises 57 60, epress the roots of unit in standard form a bi. Graph each root in the comple plane. 57. Cube roots of unit 58. Fourth roots of unit 59. Sith roots of unit 60. Square roots of unit 61. Determine z and the three cube roots of z if one cube root of z is 1 3 i. 8; and 1 3 i 6. Determine z and the four fourth roots of z if one fourth root of z is i. 64; i and i 6 )

60 5144_Demana_Ch06pp /11/06 9:34 PM Page CHAPTER 6 Applications of Trigonometr 63. Quotient Formula Let z 1 r 1 cos 1 i sin 1 and z r cos i sin, r 0. Verif that z 1 z r 1 r cos 1 i sin Group Activit nth Roots Show that the nth roots of the comple number r cos i sin are spaced n radians apart on a circle with radius n r. Standardized Test Questions 65. True or False The trigonometric form of a comple number is unique. Justif our answer. 66. True or False The comple number i is a cube root of i. Justif our answer. True. i 3 i, so i is a cube root of i. In Eercises 67 70, ou ma use a graphing calculator to solve the problem. 67. Multiple Choice Which of the following is a trigonometric form of the comple number 1 3 i? B (A) cos 3 i sin 3 (B) cos i sin 3 3 (C) cos 4 4 i sin 3 3 (D) cos 5 5 i sin 3 3 (E) cos 7 7 i sin Multiple Choice Which of the following is the number of distinct comple number solutions of z 5 1 i? E (A) 0 (B) 1 (C) 3 (D) 4 (E) Multiple Choice Which of the following is the standard form for the product A of cos 4 i sin 4 and cos 7 7 i sin 4 4? (A) (B) (C) i (D) 1 i (E) 1 i 70. Multiple Choice Which of the following is not a fourth root of 1? E (A) i (B) i (C) 1 Eplorations (D) 1 (E) i 71. Comple Conjugates The comple conjugate of z a bi is z a bi. Let z r cos i sin. (a) Prove that z r cos i sin. (b) Use the trigonometric form to find z z. r (c) Use the trigonometric form to find z z, if z 0. (d) Prove that z r cos i sin. 7. Modulus of Comple Numbers Let z r cos i sin. (a) Prove that z r. (b) Use the trigonometric form for the comple numbers z 1 and z to prove that z 1 z z 1 z. Etending the Ideas 73. Using Polar Form on a Graphing Calculator The comple number r cos i sin can be entered in polar form on some graphing calculators as re i. (a) Support the result of Eample 3 b entering the comple numbers z 1 and z in polar form on our graphing calculator and computing the product with our graphing calculator. (b) Support the result of Eample 4 b entering the comple numbers z 1 and z in polar form on our graphing calculator and computing the quotient with our graphing calculator. (c) Support the result of Eample 5 b entering the comple number in polar form on our graphing calculator and computing the power with our graphing calculator. 74. Visualizing Roots of Unit Set our graphing calculator in parametric mode with 0 T 8, Tstep 1, Xmin.4, Xma.4, Ymin 1.6, and Yma 1.6. (a) Let cos 8 t and sin 8 t. Use trace to visualize the eight eighth roots of unit. We sa that 8 generates the eighth roots of unit. (Tr both dot mode and connected mode.) (b) Replace 8 in part (a) b the arguments of other eighth roots of unit. Do an others generate the eighth roots of unit? Yes. 6 8, 10 8, 14 8 (c) Repeat parts (a) and (b) for the fifth, sith, and seventh roots of unit, using appropriate functions for and. (d) What would ou conjecture about an nth root of unit that generates all the nth roots of unit in the sense of part (a)? 75. Parametric Graphing Write parametric equations that represent i n for n t. Draw and label an accurate spiral representing i n for n 0, 1,, 3, Parametric Graphing Write parametric equations that represent 1 i n for n t. Draw and label an accurate spiral representing 1 i n for n 0, 1,, z 1 z 3, Eplain wh the triangles formed b 0, 1, and z 1 and b 0, z and z 1 z shown in the figure are similar triangles. z 78. Compass and Straightedge z Construction Using onl a 1 compass and straightedge, construct the location of z 1 z given the location 0 1 of 0, 1, z 1, and z. In Eercises 79 84, find all solutions of the equation (real and comple)

61 5144_Demana_Ch06pp /11/06 9:34 PM Page 561 CHAPTER 6 Ke Ideas 561 CHAPTER 6 Ke Ideas PROPERTIES, THEOREMS, AND FORMULAS Component Form of a Vector 503 The Magnitude or Length of a Vector 504 Vector Addition and Scalar Multiplication 505 Unit Vector in the Direction of the Vector v 506 Dot Product of Two Vectors 514 Properties of the Dot Product 514 Theorem Angle Between Two Vectors 515 Projection of the Vector u onto the Vector v 517 PROCEDURES Head Minus Tail Rule for Vectors 503 Resolving a Vector 507 Work 518 Coordinate Conversion Equations 535 Smmetr Tests for Polar Graphs 541 The Comple Plane 551 Modulus or Absolute Value of a Comple Number 551 Trigonometric Form of a Comple Number 551 De Moivre s Theorem 554 Product and Quotient of Comple Numbers 551 nth Root of a Comple Number 556 GALLERY OF FUNCTIONS Rose Curves: r a cos n and r a sin n [ 6,6] b [ 4, 4] r 4 sin 3 [ 4.7, 4.7] b [ 3.1, 3.1] r 3 sin 4 Limaçon Curves: r a b sin and r a b cos with a 0 and b 0 Limaçon with an inner loop: a b 1 Cardioid: a b 1 Dimpled limaçon: 1 a b Conve limaçon: a b Spiral of Archimedes: Lemniscate Curves: r a sin and r a cos [ 30, 30] b [ 0, 0] r, 0 45 [ 4.7, 4.7] b [ 3.1, 3.1] r 4 cos

62 5144_Demana_Ch06pp /11/06 9:34 PM Page CHAPTER 6 Applications of Trigonometr CHAPTER 6 Review Eercises The collection of eercises marked in red could be used as a chapter test. In Eercises 1 6, let u, 1, v 4,, and w 1, 3 be vectors. Find the indicated epression. 1. u v, 3. u 3w 1, 7 3. u v w u u v 6 6. u w 5 In Eercises 7 10, let A, 1, B 3, 1, C 4,, and D 1, 5. Find the component form and magnitude of the vector. 7. 3AB 3, 6 ; AC BD 8, 3 ; AB CD 6, 5 ; CD AB 4, 9 ; 9 7 In Eercises 11 and 1, find (a) a unit vector in the direction of AB and (b) a vector of magnitude 3 in the opposite direction. 11. A 4, 0, B, 1 1. A 3, 1, B 5, 1 1. (a) 1, 0 (b) 3, 0 In Eercises 13 and 14, find (a) the direction angles of u and v and (b) the angle between u and v. 13. u 4, 3, v, u, 4, v 6, 4 In Eercises 15 18, convert the polar coordinates to rectangular coordinates , 5 (.7, 1.06) , 135 (1.55, 1.55 ) 17., 4 (, ) , 3 4 ( 1.8, 1.8 ) In Eercises 19 and 0, polar coordinates of point P are given. Find all of its polar coordinates. 19. P 1, 3 0. P, 5 6 In Eercises 1 4, rectangular coordinates of point P are given. Find polar coordinates of P that satisf these conditions: (a) 0 (b) (c) P, 3. P 10, 0 3. P 5, 0 4. P 0, In Eercises 5 30, eliminate the parameter t and identif the graph t, 4 3t 6. 4 t, 8 5t, 3 t 5 7. t 3, t cos t, 3 sin t 9. e t 1, e t 30. t 3, ln t, t 0 In Eercises 31 and 3, find a parametrization for the curve. 31. The line through the points 1, and 3, The line segment with endpoints, 3 and 5, 1. Eercises 33 and 34 refer to the comple number z 1 shown in the figure. z If z 1 a bi, find a, b, and z 1. a 3, b 4, z Find the trigonometric form of z 1. In Eercises 35 38, write the comple number in standard form cos 30 i sin cos 150 i sin ( cos 4 4 i sin 3 3 ) cos.5 i sin.5 In Eercises 39 4, write the comple number in trigonometric form where 0. Then write three other possible trigonometric forms for the number i i i 4. i In Eercises 43 and 44, write the comple numbers z 1 z and z 1 z in trigonometric form. 43. z 1 3 cos 30 i sin 30 and z 4 cos 60 i sin z 1 5 cos 0 i sin 0 and z cos 45 i sin 45 In Eercises 45 48, use De Moivre s theorem to find the indicated power of the comple number. Write our answer in (a) trigonometric form and (b) standard form ( cos 4 i sin 4 ) ( cos ( cos 5 3 Imaginar ais 4 5 i sin 3 Real ais ) ( cos i sin 1 i sin 4 4 ) 8 ) 6 In Eercises 49 5, find and graph the nth roots of the comple number for the specified value of n i, n , n , n , n 6

63 5144_Demana_Ch06pp /11/06 9:34 PM Page 563 CHAPTER 6 Review Eercises 563 In Eercises 53 60, decide whether the graph of the given polar equation appears among the four graphs shown. (c) (a) 53. r 3 sin 4 (b) 54. r sin not shown 55. r sin (a) 56. r 3 sin 3 not shown 57. r sin not shown 58. r 1 cos (d) 59. r 3 cos 5 (c) 60. r 3 tan not shown In Eercises 61 64, convert the polar equation to rectangular form and identif the graph. 61. r 6. r sin 63. r 3 cos sin 64. r 3 sec In Eercises 65 68, convert the rectangular equation to polar form. Graph the polar equation In Eercises 69 7, analze the graph of the polar curve. 69. r 5 sin 70. r 4 4 cos 71. r sin 3 7. r sin, Graphing Lines Using Polar Equations (a) Eplain wh r a sec is a polar form for the line a. (b) Eplain wh r b csc is a polar form for the line b. (c) Let m b. Prove that b r sin m cos is a polar form for the line. What is the domain of r? (d) Illustrate the result in part (c) b graphing the line 3 using the polar form from part (c). 74. Flight Engineering An airplane is fling on a bearing of 80 at 540 mph. A wind is blowing with the bearing 100 at 55 mph. (b) (d) (a) Find the component form of the velocit of the airplane. (b) Find the actual speed and direction of the airplane. 75. Flight Engineering An airplane is fling on a bearing of 85 at 480 mph. A wind is blowing with the bearing 65 at 30 mph. (a) Find the component form of the velocit of the airplane. (b) Find the actual speed and direction of the airplane. 76. Combining Forces A force of 10 lb acts on an object at an angle of 0. A second force of 300 lb acts on the object at an angle of 5. Find the direction and magnitude of the resultant force lb; Braking Force A 3000 pound car is parked on a street that makes an angle of 16 with the horizontal (see figure). (a) Find the force required to keep the car from rolling down the hill pounds (b) Find the component of the force perpendicular to the street pounds Work Find the work done b a force F of 36 pounds acting in the direction given b the vector 3, 5 in moving an object 10 feet from 0, 0 to 10, foot-pounds 79. Height of an Arrow Stewart shoots an arrow straight up from the top of a building with initial velocit of 45 ft sec. The arrow leaves from a point 00 ft above level ground. (a) Write an equation that models the height of the arrow as a function of time t. h 16t 45t 00 (b) Use parametric equations to simulate the height of the arrow. (c) Use parametric equations to graph height against time. (d) How high is the arrow after 4 sec? 94 ft (e) What is the maimum height of the arrow? When does it reach its maimum height? 1138 ft; t 7.66 (f) How long will it be before the arrow hits the ground? 80. Ferris Wheel Problem Lucinda is on a Ferris wheel of radius 35 ft that turns at the rate of one revolution ever 0 sec. The lowest point of the Ferris wheel (6 o clock) is 15 ft above ground level at the point 0, 15 of a rectangular coordinate sstem. Find parametric equations for the position of Lucinda as a function of time t in seconds if Lucinda starts t 0 at the point 35, Ferris Wheel Problem The lowest point of a Ferris wheel (6 o clock) of radius 40 ft is 10 ft above the ground, and the center is on the -ais. Find parametric equations for Henr s position as a function of time t in seconds if his starting position t 0 is the point 0, 10 and the wheel turns at the rate of one revolution ever 15 sec. 40 sin t 15, cos t 15

64 5144_Demana_Ch06pp /11/06 9:34 PM Page CHAPTER 6 Applications of Trigonometr 8. Ferris Wheel Problem Sarah rides the Ferris wheel described in Eercise 81. Find parametric equations for Sarah s position as a function of time t in seconds if her starting position t 0 is the point 0, 90 and the wheel turns at the rate of one revolution ever 18 sec. 83. Epiccloid The graph of the parametric equations 4 cos t cos 4t, 4 sin t sin 4t is an epiccloid. The graph is the path of a point P on a circle of radius 1 rolling along the outside of a circle of radius 3, as suggested in the figure. (a) Graph simultaneousl this epiccloid and the circle of radius 3. (b) Suppose the large circle has a radius of 4. Eperiment! How do ou think the equations in part (a) should be changed to obtain defining equations? What do ou think the epiccloid would look like in this case? Check our guesses t C 1 P All 4 s should be changed to 5 s. 84. Throwing a Baseball Sharon releases a baseball 4 ft above the ground with an initial velocit of 66 ft sec at an angle of 5 with the horizontal. How man seconds after the ball is thrown will it hit the ground? How far from Sharon will the ball be when it hits the ground? t 0.71 sec, ft 85. Throwing a Baseball Diego releases a baseball 3.5 ft above the ground with an initial velocit of 66 ft sec at an angle of 1 with the horizontal. How man seconds after the ball is thrown will it hit the ground? How far from Diego will the ball be when it hits the ground? t 1.06 sec, ft 86. Field Goal Kicking Spencer practices kicking field goals 40 d from a goal post with a crossbar 10 ft high. If he kicks the ball with an initial velocit of 70 ft sec at a 45 angle with the horizontal (see figure), will Spencer make the field goal if the kick sails true? It clears the crossbar ft/sec 40 d 87. Hang Time An NFL place-kicker kicks a football downfield with an initial velocit of 85 ft sec. The ball leaves his foot at the 15 ard line at an angle of 56 with the horizontal. Determine the following: (a) The ball s maimum height above the field ft (b) The hang time (the total time the football is in the air). 88. Baseball Hitting Brian hits a baseball straight toward a 15-ft-high fence that is 400 ft from home plate. The ball is hit when it is.5 ft above the ground and leaves the bat at an angle of 30 with the horizontal. Find the initial velocit needed for the ball to clear the fence. just over 15 ft/sec 89. Throwing a Ball at a Ferris Wheel A 60-ft-radius Ferris wheel turns counterclockwise one revolution ever 1 sec. Sam stands at a point 80 ft to the left of the bottom (6 o clock) of the wheel. At the instant Kath is at 3 o clock, Sam throws a ball with an initial velocit of 100 ft sec and an angle with the horizontal of 70. He releases the ball from the same height as the bottom of the Ferris wheel. Find the minimum distance between the ball and Kath ft 90. Yard Darts Gretta and Lois are launching ard darts 0 ft from the front edge of a circular target of radius 18 in. If Gretta releases the dart 5 ft above the ground with an initial velocit of 0 ft sec and at a 50 angle with the horizontal, will the dart hit the target? no

65 5144_Demana_Ch06pp /11/06 9:34 PM Page 565 CHAPTER 6 Project 565 CHAPTER 6 Project Parametrizing Ellipses As ou discovered in the Chapter 4 Data Project, it is possible to model the displacement of a swinging pendulum using a sinusoidal equation of the form a sin b t c d where represents the pendulum s distance from a fied point and t represents total elapsed time. In fact, a pendulum s velocit behaves sinusoidall as well: ab cos b t c, where represents the pendulum s velocit and a, b, and c are constants common to both the displacement and velocit equations. Use a motion detection device to collect distance, velocit, and time data for a pendulum, then determine how a resulting plot of velocit versus displacement (called a phase-space plot) can be modeled using parametric equations. COLLECTING THE DATA Construct a simple pendulum b fastening about 1 meter of string to the end of a ball. Collect time, distance, and velocit readings for between and 4 seconds (enough time to capture at least one complete swing of the pendulum). Start the pendulum swinging in front of the detector, then activate the sstem. The data table below shows a sample set of data collected as a pendulum swung back and forth in front of a CBR where t is total elapsed time in seconds, d distance from the CBR in meters, v velocit in meters second. t d v t d v t d v EXPLORATIONS 1. Create a scatter plot for the data ou collected or the data above.. With our calculator computer in function mode, find values for a, b, c, and d so that the equation a sin b c d (where is distance and is time) fits the distance versus time data plot. 0.8 sin(3.46( 1.47)) Make a scatter plot of velocit versus time. Using the same a, b, and c values ou found in,verif that the equation ab cos b c (where is velocit and is time) fits the velocit versus time data plot. 4. What do ou think a plot of velocit versus distance (with velocit on the vertical ais and distance on the horizontal ais) would look like? Make a rough sketch of our prediction, then create a scatter plot of velocit versus distance. How well did our predicted graph match the actual data plot? 5. With our calculator computer in parametric mode, graph the parametric curve a sin b t c d, ab cos b t c, 0 t where represents distance, represents velocit, and t is the time parameter. How well does this curve match the scatter plot of velocit versus time?

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