Radian Measure and the Unit Circle Aroach How does an odometer or seedometer on an automobile work? The transmission counts how many times the tires rotate (how many full revolutions take lace) er second. A comuter then calculates how far the car has traveled in that second by multilying the number of revolutions by the tire circumference. Distance is given by the odometer, and the seedometer takes the distance er second and converts to miles er hour (or km/h). Realize that the comuter chi is rogrammed to the tire designed for the vehicle. If a erson were to change the tire size (smaller or larger than the original secifications), then the odometer and seedometer would need to be adjusted. Suose you bought a Ford Exedition Eddie Bauer Edition, which comes standard with 7-inch rims (corresonding to a tire with 5.7-inch diameter), and you decide to later ugrade these tires for 9-inch rims (corresonding to a tire with 8.-inch diameter). If the onboard comuter is not adjusted, is the actual seed faster or slower than the seedometer indicator?* In this case, the seedometer would read 9.6% too low. For examle, if your seedometer read 60 mh, your actual seed would be 65.8 mh. In this chater, you will see how the angular seed (rotations of tires er second), radius (of the tires), and linear seed (seed of the automobile) are related. *Section., Examle and Exercises 5 and 5. Courtesy Ford Motor Comany
c0.qxd 8// 7:07 PM Page 9 IN THIS CHAPTER, you will learn a second way to measure angles using radians. You will convert between degrees and radians. You will calculate arc lengths, areas of circular sectors, and angular and linear seeds. Finally, the third definition of trigonometric functions using the unit circle aroach will be given. You will work with the trigonometric functions in the context of a unit circle. RADIAN MEASURE AND THE UNIT CIRCLE APPROACH. Radian Measure. Arc Length and Area of a Circular Sector. Linear and Angular Seeds. Definition of Trigonometric Functions: Unit Circle Aroach The Radian Measure of an Angle Converting Between Degrees and Radians Arc Length Area of a Circular Sector Linear Seed Angular Seed Relationshi Between Linear and Angular Seeds Trigonometric Functions and the Unit Circle (Circular Functions) Proerties of Circular Functions LEARNING OBJECTIVES Convert between degrees and radians. Calculate arc length and the area of a circular sector. Relate angular and linear seeds. Draw the unit circle and label the sine and cosine values for secial angles (in both degrees and radians). 9
c0.qxd 8// 7:07 PM Page 0 SECTION. RADIAN MEASURE SKILLS OBJECTIVES Calculate the radian measure of an angle. Convert between degrees and radians. Calculate trigonometric function values for angles given in radians. CONCEPTUAL OBJECTIVES Understand that degrees and radians are both measures of angles. Realize that radian measure allows us to write trigonometric functions as functions of real numbers. The Radian Measure of an Angle r r = radian r In geometry and most everyday alications, angles are measured in degrees. However, radian measure is another way to measure angles. Using radian measure allows us to write trigonometric functions as functions not only of angles but also of real numbers in general. Recall that in Section. we defined one full rotation as an angle having measure 60. Now we think of the angle in the context of a circle. A central angle is an angle that has its vertex at the center of a circle. When the interceted arc s length is equal to the radius, the measure of the central angle is radian. From geometry, we know that the ratio of the measures of two angles is equal to the ratio of the lengths of the arcs subtended by those angles (along the same circle). r s u u s s r r r s If u radian, then the length of the subtended arc is equal to the radius, s r. This leads to a general definition of radian measure. CAUTION To correctly calculate radians from the formula u s r, the radius and arc length must be exressed in the same units. D EFINITION Radian Measure If a central angle u in a circle with radius r intercets an arc on the circle of length s, then the measure of u, in radians, is given by u (in radians) s r r r s Note: The formula is valid only if s (arc length) and r (radius) are exressed in the same units. 0 Note that both s and r are measured in units of length. When both are given in the same units, the units cancel, giving the number of radians as a dimensionless (unitless) real number.
c0.qxd 8// 7:07 PM Page. Radian Measure One full rotation corresonds to an arc length equal to the circumference r of the circle with radius r. We see then that one full rotation is equal to radians. u full rotation r r EXAMPLE Finding the Radian Measure of an Angle What is the measure (in radians) of a central angle u that intercets an arc of length feet on a circle with radius 0 feet? Write the formula relating radian measure to arc length and radius. Let s feet and r 0 feet. u s r u ft 0. rad 0 ft Study Ti Notice in Examle that the units, feet, cancel, therefore leaving u as a unitless real number, 0.. YOUR TURN EXAMPLE What is the measure (in radians) of a central angle that intercets an arc of length inches on a circle with radius 50 inches? Finding the Radian Measure of an Angle What is the measure (in radians) of a central angle u that intercets an arc of length 6 centimeters on a circle with radius meters? C OMMON M ISTAKE A common mistake is forgetting to first ut the radius and arc length in the same units. CORRECT Write the formula relating radian measure to arc length and radius. INCORRECT Answer: 0.06 rad Classroom Examle.. Find the measure, in radians, of the central angle u that intercets an arc of length yards on a circle of radius 6 yards. Answer: rad CAUTION Units for arc length and radius must be the same in order to use u s r u (in radians) s r Substitute s 6 centimeters and r meters into the radian exression. u 6 cm m Convert the radius () meters to centimeters: meters 00 centimeters u 6 cm 00 cm The units, centimeters, cancel and the result is a unitless real number. u 0.0 rad Substitute s 6 centimeters and r meters into the radian exression. u 6 cm m ERROR (not converting both numerator and denominator to the same units) Classroom Examle.. Find the measure, in radians, of the central angle u that intercets an arc of length yards on a circle of radius 6 feet. Answer: rad YOUR TURN What is the measure (in radians) of a central angle u that intercets an arc of length millimeters on a circle with radius centimeters? Answer: 0. rad
c0.qxd 8// 7:07 PM Page CHAPTER Radian Measure and the Unit Circle Aroach Because radians are unitless, the word radians (or rad) is often omitted. If an angle measure is given simly as a real number, then radians are imlied. WORDS The measure of u is degrees. The measure of u is radians. MATH u u Converting Between Degrees and Radians To convert between degrees and radians, we must first look for a relationshi between them. We start by considering one full rotation around the circle. An angle corresonding to one full rotation is said to have measure 60, and we saw reviously that one full rotation corresonds to u rad. WORDS Write the angle measure (in degrees) that corresonds to one full rotation. Write the angle measure (in radians) that corresonds to one full rotation. Arc length is the circumference of the circle. Substitute s r into u (in radians) s r. Equate the measures corresonding to one full rotation. Divide by. MATH u 60 s r u r r 60 rad 80 rad Divide by 80 or. 80 rad or 80 hr This leads us to formulas aunit conversations, like that convert between degrees 60 min b and radians. Let u d reresent an angle measure given in degrees and u r reresent the corresonding angle measure given in radians. CONVERTING DEGREES TO RADIANS To convert degrees to radians, multily the degree measure by u r u d a 80 b 80. CONVERTING RADIANS TO DEGREES To convert radians to degrees, multily the radian measure by u d u r a 80 b 80.
c0.qxd 8// 7:07 PM Page. Radian Measure Before we begin converting between degrees and radians, let s first get a feel for radians. How many degrees is radian? WORDS Multily radian by 80. Aroximate by.. Use a calculator to evaluate and round to the nearest degree. MATH a 80 b a 80. b 57 rad 57 A radian is much larger than a degree (almost 57 times larger). Let s comare two angles, one measuring 0 radians and the other measuring 0. 0 rad Note that rad/rev.77 revolutions, whereas 0 revolution. y y 0º x x 0 rad EXAMPLE Convert 5 to radians. Multily 5 by 80. Converting Degrees to Radians (5 )a 80 b 5 80 Classroom Examle.. Convert 5 to radians. Answer: Simlify. rad Note: is the exact value. A calculator can be used to aroximate this exression. Scientific and grahing calculators have a button. The decimal aroximation of rounded to three decimal laces is 0.785. Exact Value: Aroximate Value: 0.785 YOUR TURN Convert 60 to radians. Answer: or.07
c0.qxd 8// 7:07 PM Page CHAPTER Radian Measure and the Unit Circle Aroach EXAMPLE Convert 7 to radians. Converting Degrees to Radians Classroom Examle.. a.* Convert 80(n ) to radians, where n is an integer. b. Convert 000 to radians. Answer: a. (n ) b. 00 9 Multily 7 by 80. Simlify (factor out the common ). Use a calculator to aroximate. 7 a 80 b 8 5 8.8 rad Answer: or 8.09 9 YOUR TURN Convert 60 to radians. Classroom Examle..5 Convert to degrees. 6 Answer: 0 Answer: 70 EXAMPLE 5 Converting Radians to Degrees Convert to degrees. Multily Simlify. by 80. YOUR TURN Convert to degrees. 80 0 EXAMPLE 6 Converting Radians to Degrees Convert 0 radians to degrees. 80 Multily 0 radians by. 0 80 Simlify. 800 57 0º = 5º = 5 50º = 6 80º = 7 0º = 6 5 5º = 0º = 90º = 70º = 60º = 5º = 0º = 6 60º = 0º = 6 7 5º = 5 00º = Since 80, we know the following secial angles: 90 60 5 6 0 and we can now draw the unit circle with the secial angles in both degrees and radians.
c0.qxd 8// 7:07 PM Page 5. Radian Measure 5 The following table lists sine and cosine values for secial angles in both degrees and radians. Tangent, secant, cosecant, and cotangent values can all be found from sine and cosine values using quotient and recirocal identities. The table only lists secial angles in quadrant I and quadrantal angles ( 0 u 60 or 0 u ). Values in quadrants II, III, and IV can be found using reference angles and knowledge of the algebraic sign ( or ) of the sine and cosine functions in each quadrant. ANGLE, VALUE OF TRIGONOMETRIC FUNCTION RADIANS DEGREES SIN COS 0 0 0 6 0 5 60 90 0 80 0 70 0 60 0 Technology Ti Set a TI/scientific calculator to radian mode by tying MODE ENTER. (radian) EXAMPLE 7 Evaluate sin a exactly. b Evaluating Trigonometric Functions for Angles in Radian Measure Recognize that or convert to degrees. 60 Find the value of sin 60. Equate sin 60 and sin a b. 80 60 sin 60 sin a b Use a TI/scientific calculator to check the value of sin a and a. b b Press nd ^. YOUR TURN Evaluate cos a exactly. b Answer: If the angle of the trigonometric function to be evaluated has its terminal side in quadrants II, III, or IV, then we use reference angles and knowledge of the algebraic sign ( or ) in that quadrant. We know how to find reference angles in degrees. Now we will find reference angles in radians.
c0.qxd 8// 7:07 PM Page 6 6 CHAPTER Radian Measure and the Unit Circle Aroach TERMINAL SIDE LIES IN... DEGREES RADIANS QI QII QIII QIV a u a 80 u a u 80 a 60 u a u a u a u a u EXAMPLE 8 Finding Reference Angles in Radians Find the reference angle for each angle given. Classroom Examle..8 Find the reference angle for each angle given. 5 a. b. Answer: a. b. a. b. 6 Solution (a): The terminal side of u lies in quadrant II. Recall that radians is of a full revolution, so is of a half of revolution. y x The reference angle is made with the terminal side and the negative x-axis. Solution (b): The terminal side of u lies in quadrant IV. y Recall that is a comlete revolution. Note that is not quite aor. 6 6 b 6 x The reference angle is made with the terminal side and the ositive x-axis. 6 6 6 6 Answer: YOUR TURN Find the reference angle for 5.
c0.qxd 8// 7:07 PM Page 7. Radian Measure 7 EXAMPLE 9 Evaluate cosa 5 exactly. b Evaluating Trigonometric Functions for Angles in Radian Measure Using Reference Angles Technology Ti Use the TI/scientific calculator 5 The terminal side of angle lies in 5 quadrant III since. The reference angle is 5. = 5º 5 y x to check the value for cos a 5 and comare with b. Find the cosine value for the reference angle. Determine the algebraic sign for the cosine function in quadrant III. Combine the algebraic sign of the cosine function in quadrant III with the value of the cosine function of the reference angle. Confirm with a calculator. cosa b cos 5 Negative () cosa 5 b 0.707 0.707 Classroom Examle..9 Evaluate cos a 5 exactly. 6 b Answer: YOUR TURN Evaluate sin a 7 exactly. b Answer: SECTION. SUMMARY In this section, a second measure of angles was introduced, which allows us to write trigonometric functions as functions of real numbers. A central angle of a circle has radian measure equal to the ratio of the arc length interceted by the angle to the radius of the circle, u s. r Radians and degrees are related by the relation that 80. One radian is aroximately equal to 57. Careful attention must be aid to what mode (degrees or radians) calculators are set when evaluating trigonometric functions. To evaluate a trigonometric function for nonacute angles in radians, we use reference angles (in radians) and knowledge of the algebraic sign of the trigonometric function. To convert from radians to degrees, multily the 80 radian measure by. To convert from degrees to radians, multily the degree measure by 80.
c0.qxd 8// 7:07 PM Page 8 8 CHAPTER Radian Measure and the Unit Circle Aroach SECTION. EXERCISES SKILLS In Exercises 0, find the measure (in radians) of a central angle that intercets an arc on a circle of radius r with indicated arc length s.. r 0 cm, s cm. r 0 cm, s cm. r in., s in.. r 6 in., s in. 5. r 00 cm, s 0 mm 6. r m, s cm 7. r s in., in. 8. r s cm, cm 9. r.5 cm, s 5 mm 0. r.6 cm, s 0. mm In Exercises, convert each angle measure from degrees to radians. Leave answers in terms of.. 0. 60. 5. 90 5. 5 6. 70 7. 8. 00 9. 70 0. 0. 780. 50. 0. 75 0 In Exercises 5 8, convert each angle measure from radians to degrees. 7 5. 6. 7. 8. 9. 0.. 6 6 8 9 7 9.. 9. 6 5. 6. 7. 7 8. 8 0 6 5 9 In Exercises 9, convert each angle measure from radians to degrees. Round answers to the nearest hundredth of a degree. 9. 0.. 0.85..7..7989. 5.98 In Exercises 5 50, convert each angle measure from degrees to radians. Round answers to three significant digits. 5. 7 6. 65 7. 8. 7 9. 56.5 50. 98.7 In Exercises 5 58, find the reference angle for each of the following angles in terms of both radians and degrees. 7 5 5 7 9 5. 5. 5. 5. 55. 56. 57. 58. In Exercises 59 8, find the exact value of the following exressions. Do not use a calculator. 5 59. sin a b 60. cos a 6 b 6. sin a7 b 6. 6. sin a b 6. cos a7 6 b 65. sin a b 66. 67. sin a 68. cos a 69. cos a5 70. 6 b b b 7. tan a 7. tan a5 7. tan a 7. 6 b b 6 b 75. tan a5 76. tan a 77. cot a 78. 6 b b b 79. sec(5) 80. cot a 8. sin a 8. b b cos a b cos a 6 b sin a 5 6 b tan a 5 6 b csc a b cos a b 8. cos a7 8. sin a 8 6 b b
c0.qxd 8/6/ 9:57 AM Page 9. Radian Measure 9 A P P L I C AT I O N S 9. Srinkler. A water srinkler can reach an arc of 5 feet, 0 feet from the srinkler as shown. Through how many radians does the srinkler rotate? For Exercises 85 and 86, refer to the following: Two electronic signals that are not co-hased are called out of hase. Two signals that cancel each other out are said to be 80 out of hase, or the difference in their hases is 80. 85. Electronic Signals. How many radians out of hase are two signals whose hase difference is 70? 86. Electronic Signals. How many radians out of hase are two signals whose hase difference is 0? 87. Construction. In China, you find circular clan homes called tulou. Some tulou are three or four stories high and exceed 70 meters in diameter. If a wedge or section on the third floor of such a building has a central angle measuring 6, how many radians is this? 5 ft 0 ft Kin Cheung/Reuters/Landov 9. Srinkler. A srinkler is set to reach an arc of 5 feet, 5 feet from the srinkler. Through how many radians does the srinkler rotate? 95. Engine. If a car engine is said to be running at 500 RPMs (revolutions er minute), through how many radians is the engine turning every second? 96. Engine. If a car engine is said to rotate 5,000 er second, through how many radians does the engine turn each second? For Exercises 97 and 98, refer to the following: A traction slint is commonly used to treat comlete long bone fractures of the leg. The angle between the leg and torso is an oblique angle u. The reference angle a is the acute angle between the leg in traction and the bed. 88. Construction. In China, you find circular clan homes called tulou. Some tulou are three or four stories high and exceed 70 meters in diameter. If a wedge or section on the third floor of such a building has a central angle measuring 7, how many radians is this? 89. Clock. How many radians does the second hand of a clock turn in minutes? 90. Clock. How many radians does the second hand of a clock turn in minutes and 5 seconds? 9. London Eye. The London Eye has casules (each caable of holding 5 assengers with an unobstructed view of London). What is the radian measure of the angle made between the center of the wheel and the sokes aligning with each casule? 9. Sace Needle. The sace needle in Seattle has a restaurant that offers views of Mount Rainier and Puget Sound. The restaurant comletes one full rotation in aroximately 5 minutes. How many radians will the restaurant have rotated in 5 minutes?, find the measure of the reference angle in both radians and degrees. 97. Health/Medicine. If u, find the measure of the reference angle in both radians and degrees. 98. Health/Medicine. If u
c0.qxd 8// 7:08 PM Page 0 0 CHAPTER Radian Measure and the Unit Circle Aroach For Exercises 99 0, refer to the following: A water molecule is comosed of one oxygen atom and two hydrogen atoms and exhibits a bent shae with the oxygen atom at the center. H Net negative charge Attraction of bonding electrons to the oxygen creates local negative O and ositive article charges 05º H + + Net ositive charge 99. Chemistry. The angle between the O-H bonds in a water molecule is aroximately 05. Find the angle between the O-H bonds of a water molecule in radians. 00. Chemistry. The angle between the S-O bonds in sulfur dioxide (SO ) is aroximately 0. Find the angle between the S-O bonds of sulfur dioxide in radians. 0. Chemistry/Environment. Nitrogen dioxide (NO ) is a toxic gas and rominent air ollutant. The angle between the N-O bond is.. Find the angle between the N-O bonds in radians. O N.º 0. Chemistry/Environment. Methane (CH ) is a chemical comound and otent greenhouse gas. The angle between the C-H bonds is 09.5. Find the angle between the C-H bonds in radians. H C H 09.5º H 9.7 m H O 08.70 m CATCH THE MISTAKE In Exercises 0 06, exlain the mistake that is made. 0. What is the measure (in radians) of a central angle u that intercets an arc of length 6 centimeters on a circle with radius meters? Write the formula for radians. Substitute s 6, r. u 6 Write the angle in terms of radians. u rad This is incorrect. What mistake was made? 0. What is the measure (in radians) of a central angle u that intercets an arc of length inches on a circle with radius foot? Write the formula for radians. u s r Substitute s, r. u Write the angle in terms of radians. u rad This is incorrect. What mistake was made? 05. Evaluate 6 tan(5) 5 sec a. b Evaluate tan(5) and sec a b. Substitute the values of the trigonometric functions. Simlify. This is incorrect. What mistake was made? 06. Aroximate with a calculator cos() tan(65) sin(). Round to three decimal laces. Evaluate the trigonometric functions individually. cos() 0.7 Substitute the values into the exression. cos() tan(65) sin() 0.7.5 0.08 Simlify. tan(5) 6 tan(5) 5 sec a b 6() 5() 6 tan(5) 5 sec a b 6 tan(65).5 cos() tan(65) sin().680 sec a b This is incorrect. What mistake was made? sin() 0.08
c0.qxd 8// 7:08 PM Page. Arc Length and Area of a Circular Sector CONCEPTUAL In Exercises 07 0, determine whether each statement is true or false. 07. An angle with measure radians is a quadrant II angle. 08. Angles exressed exactly in radian measure are always given in terms of. 09. For an angle with ositive measure, it is ossible for the numerical values of the degree and radian measures to be equal. CHALLENGE 0. The sum of the angles with radian measure in a triangle is.. Find the sum of comlementary angles in radian measure.. How many comlete revolutions does an angle with measure 9 radians make?. The distance between Atlanta, Georgia, and Boston, Massachusetts, is aroximately 900 miles along the curved surface of the Earth. The radius of the Earth is aroximately 000 miles. What is the central angle with vertex at the center of the Earth and sides of the angles intersecting the surface of the Earth in Atlanta and Boston?. The radius of the Earth is aroximately 600 kilometers. If a central angle, with vertex at the center of the Earth, intersects the surface of the Earth in London (UK) and Rome (Italy) with a central angle of 0. radians, what is the distance along the Earth s surface between London and Rome? Round to the nearest hundred kilometers. 5. At 8:0, what is the radian measure of the smaller angle between the hour hand and minute hand? 6. At 9:05, what is the radian measure of the larger angle between the hour hand and minute hand? 7. Find the exact value for 5 cos ax for x. b sin(x) 5 8. Find the exact value for cos ax for x. b sin a x 6 b 5 TECHNOLOGY 9. With a calculator set in radian mode, find sin. With a 80 calculator set in degree mode, find sin a Why do b. your results make sense? 0. With a calculator set in radian mode, find cos 5. With a 80 calculator set in degree mode, find cos a5 Why do b. your results make sense? SECTION. ARC LENGTH AND AREA OF A CIRCULAR SECTOR SKILLS OBJECTIVES Calculate the length of an arc along a circle. Find the area of a circular sector. Solve alication roblems involving circular arc lengths and sectors. CONCEPTUAL OBJECTIVE Understand that to use the arc length formula, the angle measure must be in radians. In Section., radian measure was defined in terms of the ratio of a circular arc of length s and length of the circle s radius r. u (in radians) s r In this section (.) and the next (.), we look at alications of radian measure that involve calculating arc lengths and areas of circular sectors and calculating angular and linear seeds.
c0.qxd 8// 7:08 PM Page CHAPTER Radian Measure and the Unit Circle Aroach Arc Length From geometry we know the length of an arc of a circle is roortional to its central angle. In Section., we learned that for the secial case when the arc length is equal to the circumference of the circle, the angle measure in radians corresonding to one full rotation is. Let us now assume that we are given the central angle and we want to find the arc length. WORDS Write the definition of radian measure. Multily both sides of the equation by r. Simlify. MATH u s r r u s r r r u s The formula s r u is true only when u is in radians. To develo a formula when u is in degrees, we multily u by to convert the angle measure to radians. 80 Study Ti To use the relationshi s r u the angle u must be in radians. D EFINITION Arc Length If a central angle u in a circle with radius r intercets an arc on the circle of length s, then the arc length s is given by s r u r s ru d a 80 b u r u d is in radians. is in degrees. Classroom Examle.. Find the arc length of a sector determined by central angle on a circle with radius 6 meters. Answer: m EXAMPLE Finding Arc Length When the Angle Has Radian Measure In a circle with radius 0 centimeters, an arc is interceted by a central angle with 7 measure Find the arc length.. Write the formula for arc length when the angle has radian measure. s ru r Substitute r 0 centimeters and u r 7. s (0 cm)a 7 b Simlify. s 5 cm Answer: 5 in. YOUR TURN In a circle with radius 5 inches, an arc is interceted by a central angle with measure. Find the arc length.
c0.qxd 8// 7:08 PM Page. Arc Length and Area of a Circular Sector EXAMPLE Finding Arc Length When the Angle Has Degree Measure In a circle with radius 7.5 centimeters, an arc is interceted by a central angle with measure 76. Find the arc length. Aroximate the arc length to the nearest centimeter. Write the formula for arc length when the angle has degree measure. s r u d a 80 b Substitute r 7.5 centimeters and u d 76. Evaluate the result with a calculator. Round to the nearest centimeter. s (7.5 cm)(76 )a 80 b s 9.98 cm s 0 cm YOUR TURN In a circle with radius 0 meters, an arc is interceted by a central angle with measure. Find the arc length. Aroximate the arc length to the nearest meter. Answer: 9 m EXAMPLE Path of International Sace Station The International Sace Station (ISS) is in an aroximate circular orbit 00 kilometers above the surface of the Earth. If the ground station tracks the sace station when it is within a 5 central angle of this circular orbit about the center of the Earth above the tracking antenna, how many kilometers does the ISS cover while it is being tracked by the ground station? Assume that the radius of the Earth is 600 kilometers. Round to the nearest kilometer. 00 km 5º ISS 600 km Write the formula for arc length when the angle has degree measure. Recognize that the radius of the orbit is r 600 00 6800 kilometers and that u d 5. Evaluate with a calculator. Round to the nearest kilometer. s r u d a 80 b s (6800 km)(5 )a 80 b s 50.708 km s 5 km The ISS travels aroximately 5 kilometers during the ground station tracking. YOUR TURN If the ground station in Examle could track the ISS within a 60 central angle of its circular orbit about the center of the Earth, how far would the ISS travel during the ground station tracking? Answer: 7 km
c0.qxd 8// 7:08 PM Page CHAPTER Radian Measure and the Unit Circle Aroach Classroom Examle.. Consider two gears working together such that the smaller gear has a radius of 0 centimeters, while the larger gear has a radius measuring 5 centimeters. Through how many degrees does the small gear rotate when the large gear makes one comlete rotation? Answer: 900 EXAMPLE Gears Gears are inside many devices like automobiles and ower meters. When the smaller gear drives the larger gear, then tyically the driving gear is rotated faster than a larger gear would be if it were the drive gear. In general, smaller ratios of radius of the driving gear to the driven gear are called for when machines are exected to yield more ower. The smaller gear has a radius of centimeters, and the larger gear has a radius of 6. centimeters. If the smaller gear rotates 70, how many degrees has the larger gear rotated? Round the answer to the nearest degree. 6. cm cm Technology Ti When solving for u d, be sure to use a air of arentheses for the roduct in the denominator. u d 80 7 cm 6(6. cm) 80 7 6(6.) Study Ti u d Notice that when calculating in Examle, the centimeter units cancel but its degree measure remains. Recognize that the small gear arc length the large gear arc length. Smaller Gear Write the formula for arc length when the angle has degree measure. Substitute the values for the smaller gear: r centimeters and u d 70. Simlify. Larger Gear Remember that the larger gear s arc length is equal to the smaller gear s arc length. Write the formula for arc length when the angle has degree measure. Substitute s a7 and 6 b centimeter r 6. centimeters. Solve for u d. Simlify. s r u d a 80 b s smaller ( cm)(70 )a 80 b s smaller a 7 6 b cm s a 7 6 b cm s r u d a 80 b a 7 6 cmb (6. cm)u d a 80 b u d 80 7 cm 6(6. cm) u d 79.6875 Round to the nearest degree. The larger gear rotates aroximately 80. u d 80 Area of a Circular Sector A restaurant lists a iece of French silk ie as having 00 calories. How does the chef arrive at that number? She calculates the calories of all the ingredients that went into making the entire ie and then divides by the number of slices the ie yields. For examle, if an entire ie has 00 calories and it is sliced into 8 equal ieces, then each
c0.qxd 8// 7:08 PM Page 5. Arc Length and Area of a Circular Sector 5 iece has 00 calories. Although that examle involves volume, the idea is the same with areas of sectors of circles. Circular sectors can be thought of as ieces of a ie. Recall that arc lengths of a circle are roortional to the central angle (in radians) and the radius. Similarly, a circular sector is a ortion of the entire circle. Let A reresent the area of the sector of the circle and u r reresent the central angle (in radians) that forms the sector. Then, let us consider the entire circle whose area is r and the angle that reresents one full rotation has measure (radians). r r s WORDS Write the ratio of the area of the sector to the area of the entire circle. Write the ratio of the central angle r to the measure of one full rotation. MATH A r u r The ratios must be equal (roortionality of sector to circle). Multily both sides of the equation by r. Simlify. r A r u r A r u r r A r u r D EFINITION Area of a Circular Sector The area of a sector of a circle with radius r and central angle u is given by A r u r u r is in radians. Study Ti To use the relationshi A r u the angle must be in radians. A r u d a 80 b u d is in degrees. EXAMPLE 5 Finding the Area of a Circular Sector When the Angle Has Radian Measure Find the area of the sector associated with a single slice of izza if the entire izza has a -inch diameter and the izza is cut into 8 equal ieces. The radius is half the diameter. r 7 in. Classroom Examle..5 Find the area of the sector with diameter 6 feet and 7 central angle. 8 Answer: 8 ft Find the angle of each slice if the izza is cut into 8 ieces ( of the comlete revolution). 8 Write the formula for circular sector area in radians. u r 8 A r u r
c0.qxd 8// 7:08 PM Page 6 6 CHAPTER Radian Measure and the Unit Circle Aroach Substitute r 7 inches and u r into the area equation. Simlify. Aroximate the area with a calculator. A (7 in.) a b A 9 8 in. A 9 in. Answer: 8 in. 5 in. YOUR TURN Find the area of a slice of izza (cut into 8 equal ieces) if the entire izza has a 6-inch diameter. Classroom Examle..6 Find the exact area of the sector with diameter. inches and central angle 5. Answer: 9 60 in. EXAMPLE 6 Finding the Area of a Circular Sector When the Angle Has Degree Measure Srinkler heads come in all different sizes deending on the angle of rotation desired. If a srinkler head rotates 90 and has enough ressure to kee a constant 5-foot sray, what is the area of the sector of the lawn that gets watered? Round to the nearest square foot. Write the formula for circular sector area in degrees. Substitute r 5 feet and d 90 into the area equation. A r u d a 80 b A (5 ft) (90 )a 80 b Simlify. Round to the nearest square foot. A a 65 b ft 90.87 ft A 9 ft Answer: 50 ft ft YOUR TURN If a srinkler head rotates 80 and has enough ressure to kee a constant 0-foot sray, what is the area of the sector of the lawn it can water? Round to the nearest square foot. SECTION. SUMMARY SMH In this section, we used the roortionality concet (both the arc length and area of a sector are roortional to the central angle of a circle). The definition of radian measure was used to develo formulas for the arc length of a circle when the central angle is given in either radians or degrees. The formula for the area of a sector of a circle was also develoed for the cases in which the central angle is given in either radians or degrees. s r u r u r is in radians. A r u r u r is in radians. s r u d a 80 b u d is in degrees. A r u d a 80 b u d is in degrees.
c0.qxd 8// 7:08 PM Page 7. Arc Length and Area of a Circular Sector 7 SECTION. EXERCISES SKILLS In Exercises, find the exact length of each arc made by the indicated central angle and radius of each circle.. u, r mm. u, r 5 cm. u r 8 ft., 5. u, 6. u in. 7. u, r 8 m 8. 7 r.5 m, r 0 u r 6 yd 8, u, r 5 m 9. u 8, r 500 km 0. u, r 800 km. u 8, r cm. u 0, r 0 cm In Exercises, find the exact length of each radius given the arc length and central angle of each circle.. s 5 u. s 5 u 5. s u 6. ft, 0 6 m, 5 in., 5 7. s yd, u 8. s in., u 9. s u 0 0. 5 5 9 yd, s 5 u 9 km, s 6 cm,. s 8. s u 0. s km, u 5 o. s ft, u 5 o mi, u 0 m, 6 80 u 5 In Exercises 5 6, use a calculator to aroximate the length of each arc made by the indicated central angle and radius of each circle. Round answers to two significant digits. 5. u., r 0. mm 6. u., r 5.5 cm 7. u r 8 yd 8. 5, u r 6 ft 0, 9. u.95, r 0 mi 0. u 7 mm. u 79.5, r.55 m. 8, r 7 u 9.7, r 0.6 m. u 9, r 500 km. u, r 00 km 5. u 57, r ft 6. u 7, r 58 in. In Exercises 7 8, find the area of the circular sector given the indicated radius and central angle. Round answers to three significant digits. 7. u 6, r 7 ft 8. u 5, r in. 9. u r. km 8, 0. u 5 r mi 6,. u cm, r 0. u m, r. u 56, r. cm. u 7, r.5 mm 5. u., r.5 ft 6. u, r.0 ft 7. u.8 o, r.6 mi 8. u 60, r 5 km
c0.qxd 9/8/ 8 8: AM Page 8 C H A P T E R Radian Measure and the Unit Circle Aroach A P P L I C AT I O N S David Ball/Index Stock/Photolibrary 9. Low Earth Orbit Satellites. A low Earth orbit (LEO) satellite is in an aroximate circular orbit 00 kilometers above the surface of the Earth. If the ground station tracks the satellite when it is within a 5 cone above the tracking antenna (directly overhead), how many kilometers does the satellite cover during the ground station track? Assume the radius of the Earth is 600 kilometers. Round your answer to the nearest kilometer. 50. Low Earth Orbit Satellites. A low Earth orbit (LEO) satellite is in an aroximate circular orbit 50 kilometers above the surface of the Earth. If the ground station tracks the satellite when it is within a 0 cone above the tracking antenna (directly overhead), how many kilometers does the satellite cover during the ground station track? Assume the radius of the Earth is 600 kilometers. Round your answer to the nearest kilometer. 5. Big Ben. The famous clock tower in London has a minute hand that is feet long. How far does the ti of the minute hand of Big Ben travel in 5 minutes? Round your answer to the nearest foot. Getty Images, Inc. 5. Big Ben. The famous clock tower in London has a minute hand that is feet long. How far does the ti of the minute hand of Big Ben travel in 5 minutes? Round your answer to two decimal laces. 5. London Eye. The London Eye is a wheel that has casules and a diameter of 00 feet. What is the distance someone has traveled once they reach the highest oint for the first time? 5. London Eye. Assuming the wheel stos at each casule in Exercise 5, what is the distance someone has traveled from the oint he or she first gets in the casule to the oint at which the Eye stos for the sixth time during the ride? 55. Gears. The smaller gear shown below has a radius of 5 centimeters, and the larger gear has a radius of. centimeters. If the smaller gear rotates 0, how many degrees has the larger gear rotated? Round the answer to the nearest degree. 56. Gears. The smaller gear has a radius of inches, and the larger gear has a radius of 5 inches (see the figure above). If the smaller gear rotates 0, how many degrees has the larger gear rotated? Round the answer to the nearest degree. 57. Bicycle Low Gear. If a bicycle has 6-inch diameter wheels, the front chain drive has a radius of. inches, and the back drive has a radius of inches, how far does the bicycle travel for every one rotation of the cranks (edals)?
c0.qxd 8// 7:08 PM Page 9. Arc Length and Area of a Circular Sector 9 58. Bicycle High Gear. If a bicycle has 6-inch diameter wheels, the front chain drive has a radius of inches, and the back drive has a radius of inch, how far does the bicycle travel for every one rotation of the cranks (edals)? 59. Odometer. A Ford Exedition Eddie Bauer Edition comes standard with 7-inch rims (which corresonds to a tire with 5.7-inch diameter). Suose you decide to later ugrade these tires for 9-inch rims (corresonding to a tire with 8.-inch diameter). If you do not get your onboard comuter reset for the new tires, the odometer will not be accurate. After your new tires have actually driven 000 miles, how many miles will the odometer reort the Exedition has been driven? Round to the nearest mile. 60. Odometer. For the same Ford Exedition Eddie Bauer Edition in Exercise 59, after you have driven 50,000 miles, how many miles will the odometer reort the Exedition has been driven if the comuter is not reset to account for the new oversized tires? Round to the nearest mile. 6. Srinkler Coverage. A srinkler has a 0-foot sray and covers an angle of 5. What is the area that the srinkler waters? 6. Srinkler Coverage. A srinkler has a -foot sray and covers an angle of 60. What is the area that the srinkler waters? 6. Windshield Wier. A windshield wier that is inches long (blade and arm) rotates 70. If the rubber art is 8 inches long, what is the area cleared by the wier? Round to the nearest square inch. 6. Windshield Wier. A windshield wier that is inches long (blade and arm) rotates 65. If the rubber art is 7 inches long, what is the area cleared by the wier? Round to the nearest square inch. 65. Bicycle Wheel. A bicycle wheel 6 inches in diameter travels 5 in 0.05 seconds. Through how many revolutions does the wheel turn in 0 seconds? 66. Bicycle Wheel. A bicycle wheel 6 inches in diameter travels in 0.075 seconds. Through how many revolutions does the wheel turn in 0 seconds? Getty Images, Inc. 67. Bicycle Wheel. A bicycle wheel 6 inches in diameter travels 0 inches in 0.0 seconds. What is the seed of the wheel in revolutions er second? 68. Bicycle Wheel. A bicycle wheel 6 inches in diameter travels at four revolutions er second. Through how many radians does the wheel turn in 0.5 seconds? For Exercises 69 and 70, refer to the following: Sniffers outside a chemical munitions disosal site monitor the atmoshere surrounding the site to detect any toxic gases. In the event that there is an accidental release of toxic fumes, the data rovided by the sniffers make it ossible to determine both the distance d that the fumes reach as well as the angle of sread u that swee out a circular sector. 69. Environment. If the maximum angle of sread is 05 and the maximum distance at which the toxic fumes were detected was 9 miles from the site, find the area of the circular sector affected by the accidental release. 70. Environment. To rotect the ublic from the fumes, officials must secure the erimeter of this area. Find the erimeter of the circular sector in Exercise 69. For Exercises 7 and 7, refer to the following: The structure of human DNA is a linear double helix formed of nucleotide base airs (two nucleotides) that are stacked with sacing of. angstroms (. 0 m), and each base air is rotated 6 with resect to an adjacent air and has 0 base airs er helical turn. The DNA of a virus or a bacterium, however, is a circular double helix (see the figure below) with the structure varying among secies. Twists (Source: htt://www.biohysics.org/portals// PDFs/Education/Vologodskii.df.) 7. Biology. If the circular DNA of a virus has 0 twists (or turns) er circle and an inner diameter of.5 nanometers, find the arc length between consecutive twists of the DNA. 7. Biology. If the circular DNA of a virus has 0 twists (or turns) er circle and an inner diameter of.0 nanometers, find the arc length between consecutive twists of the DNA.
c0.qxd 8// 7:08 PM Page 50 50 CHAPTER Radian Measure and the Unit Circle Aroach CATCH THE MISTAKE In Exercises 7 and 7, exlain the mistake that is made. 7. A circle with radius 5 centimeters has an arc that is made from a central angle with measure 65. Aroximate the arc length to the nearest millimeter. Write the formula for arc length. s r u Substitute r 5 centimeters and u 65 into the formula. s (5 cm)(65) Simlify. s 5 cm This is incorrect. What mistake was made? 7. For a circle with radius r. centimeters, find the area of the circular sector with central angle measuring u 5. Round the answer to three significant digits. Write the formula for area of a circular sector. A r u r Substitute r. centimeters and A u 5 into the formula. (. cm) (5 ) Simlify. A 60.5 cm This is incorrect. What mistake was made? CONCEPTUAL In Exercises 75 78, determine whether each statement is true or false. 75. The length of an arc with central angle 5 in a unit 79. If a smaller gear has radius r and a larger gear has radius circle is 5. r and the smaller gear rotates u what is the degree measure of the angle the larger gear rotates? 76. The length of an arc with central angle in a unit circle 80. If a circle with radius r has an arc length s associated is. with a articular central angle, write the formula for the area of the sector of the circle formed by that central 77. If the radius of a circle doubles, then the arc length angle, in terms of the radius and arc length. (associated with a fixed central angle) doubles. 78. If the radius of a circle doubles, then the area of the sector (associated with a fixed central angle) doubles. CHALLENGE For Exercises 8 8, refer to the following: You may think that a baseball field is a circular sector but it is not. If it were, the distances from home late to left field, center field, and right field would all be the same (the radius). Where the infield dirt meets the outfield grass and along the fence in the outfield are arc lengths associated with a circle of radius 95 feet and with a vertex located at the itcher s mound (not home late). Infield / Outfield Grass Line: 95-ft radius from front of itching rubber Third base -ft radius Infield Foul line Second base Home late -ft radius Pitching mound 9-ft radius 90 ft between bases Foul line First base -ft radius 8. What is the area enclosed in the circular sector with radius 95 feet and central angle 50? Round to the nearest hundred square feet. 8. Aroximate the area of the infield by adding the area in blue to the result in Exercise 8. Neglect the area near first and third bases and the foul line. Round to the nearest hundred square feet. 8. If a batter wants to bunt a ball so that it is fair (in front of home late and between the foul lines) but kee it in the dirt (in the sector in front of home late), within how small of an area is the batter trying to kee his bunt? Round to the nearest square foot. 8. Most bunts would fall within the blue triangle in the diagram on the left. Assume the catcher only fields bunts that fall in the sector described in Exercise 8 and the itcher only fields bunts that fall on the itcher s mound. Aroximately how much area do the first baseman and third baseman each need to cover? Round to the nearest square foot.
c0.qxd 8// 7:08 PM Page 5 SECTION. LINEAR AND ANGULAR SPEEDS SKILLS OBJECTIVES Calculate linear seed. Calculate angular seed. Solve alication roblems involving angular and linear seeds. CONCEPTUAL OBJECTIVE Relate angular seed to linear seed. In the chater oener about a Ford Exedition with standard 7-inch rims, we learned that the onboard comuter that determines distance (odometer reading) and seed (seedometer) combines the number of tire rotations and the size of the tire. Because the onboard comuter is set for 7-inch rims (which corresonds to a tire with 5.7-inch diameter), if the owner decided to ugrade to 9-inch rims (corresonding to a tire with 8.-inch diameter), the comuter would have to be udated with this new information. If the comuter is not udated with the new tire size, both the odometer and seedometer readings will be incorrect. You will see in this section that the angular seed (rotations of tires er second), radius (of the tires), and linear seed (seed of the automobile) are related. In the context of a circle, we will first define linear seed, then angular seed, and then relate them using the radius. Linear Seed It is imortant to note that although velocity and seed are often used as synonyms, seed is how fast you are traveling, whereas velocity is the seed in which you are traveling and the direction you are traveling. In hysics the difference between seed and velocity is that velocity has direction and is written as a vector (Chater 7), and seed is the magnitude of the velocity vector, which results in a real number. In this chater, seed will be used. Recall the relationshi between distance, rate, and time: d rt. Rate is seed, and in words this formula can be rewritten as distance seed time or seed distance time It is imortant to note that we assume seed is constant. If we think of a car driving around a circular track, the distance it travels is the arc length s, and if we let v reresent seed and t reresent time, we have the formula for seed around a circle (linear seed): v s t s D EFINITION Linear Seed If a oint P moves along the circumference of a circle at a constant seed, then the linear seed v is given by v s t where s is the arc length and t is the time. 5
c0.qxd 8// 7:08 PM Page 5 5 CHAPTER Radian Measure and the Unit Circle Aroach EXAMPLE Linear Seed Classroom Examle..* A car travels at a constant seed around a circular track with circumference equal to.5 miles. How many las would the car need to comlete in 0 minutes in order to average a linear seed of 75 miles er hour? Answer: 6 las A car travels at a constant seed around a circular track with circumference equal to miles. If the car records a time of 5 minutes for 9 las, what is the linear seed of the car in miles er hour? Calculate the distance traveled around the circular track. Substitute t 5 minutes and s 8 miles into v s t. Convert the linear seed from miles er minute to miles er hour. s (9 las)a mi b 8 mi la v 8 mi 5 min 8 mi min v a b a60 b 5 min hr Simlify. v 7 mh Answer: 05 mh YOUR TURN A car travels at a constant seed around a circular track with circumference equal to miles. If the car records a time of minutes for 7 las, what is the linear seed of the car in miles er hour? Angular Seed To calculate linear seed, we find how fast a osition along the circumference of a circle is changing. To calculate angular seed, we find how fast the central angle is changing. Study Ti The units of angular seed will be in radians er unit time (e.g., radians er minute). D EFINITION Angular Seed If a oint P moves along the circumference of a circle at a constant seed, then the central angle that is formed with the terminal side assing through oint P also changes over some time t at a constant seed. The angular seed (omega) is given by v u t where is given in radians EXAMPLE Angular Seed Classroom Examle.. A lighthouse in the middle of a channel rotates its light in a circular motion with constant seed. If the beacon of light comletes three rotations every seconds, find its angular seed in radians er minute. Answer: 0 rad/min A lighthouse in the middle of a channel rotates its light in a circular motion with constant seed. If the beacon of light comletes one rotation every 0 seconds, what is the angular seed of the beacon in radians er minute? Calculate the angle measure in radians associated with one rotation. Substitute u and t 0 seconds into v u t. u (rad) v 0 sec s
c0.qxd 8// 7:08 PM Page 5. Linear and Angular Seeds 5 Convert the angular seed from radians er second to radians er minute. Simlify. (rad) v 0 sec v rad /min 60 sec min YOUR TURN If the lighthouse in Examle is adjusted so that the beacon rotates one time every 0 seconds, what is the angular seed of the beacon in radians er minute? Answer: v rad/min Relationshi Between Linear and Angular Seeds In the chater oener, we discussed the Ford Exedition with 7-inch standard rims that would have odometer and seedometer errors if the owner decided to ugrade to 9-inch rims without udating the onboard comuter. That is because angular seed (rotations of tires er second), radius (of the tires), and linear seed (seed of the automobile) are related. To see how, let us start with the definition of arc length (Section.), which comes from the definition of radian measure (Section.). WORDS Write the definition of radian measure. Write the definition of arc length (u in radians). Divide both sides by t. Rewrite the right side of the equation. MATH u s r s ru s t ru t s t r u t Recall the definitions of linear and angular seeds. v s and s Substitute and into v rv t r u v s t t t t t. R ELATING LINEAR AND ANGULAR SPEEDS If a oint P moves at a constant seed along the circumference of a circle with radius r, then the linear seed v and the angular seed v are related by v rv or v v r r y P s x Study Ti This relationshi between linear seed and angular seed assumes the angle is given in radians. Note: This relationshi is true only when u is given in radians.
c0.qxd 8// 7:08 PM Page 5 5 CHAPTER Radian Measure and the Unit Circle Aroach We now will investigate the Ford Exedition scenario with ugraded tires. Notice that tires of two different radii with the same angular seed have different linear seeds since v rv. The larger tire (larger r) has the faster linear seed. Study Ti. in..85 in. We could have solved Examle the following way: 75 mh x 5.7 in. 8. in. 8. in. x 75 mh 5.7 in. 8.96 mh EXAMPLE Relating Linear and Angular Seeds A Ford F-50 comes standard with tires that have a diameter of 5.7 inches. If the owner decided to ugrade to tires with a diameter of 8. inches without having the onboard comuter udated, how fast will the truck actually be traveling when the seedometer reads 75 miles er hour? The comuter in the F-50 thinks the tires are 5.7 inches in diameter and knows the angular seed. Use the rogrammed tire diameter and seedometer reading to calculate the angular seed. Then use that angular seed and the ugraded tire diameter to get the actual seed (linear seed). STEP Calculate the angular seed of the tires. Write the formula for the angular seed. v v r Substitute v 75 miles er hour and r 5.7 75 mi/hr v.85 inches into the formula..85 in. mile 580 feet 6,60 inches. 75(6,60) in./hr v.85 in. Simlify. v 69,805 rad hr STEP Calculate the actual linear seed of the truck. Write the linear seed formula. v rv Substitute r 8.. inches v (. in.)a69,805 rad and v 69,805 radians er hour. hr b Simlify. v 5,,5 in. hr mile 580 feet 6,60 inches. v 5,,5 in. hr mi 6,60 in. v 8.96 mi hr Although the seedometer indicates a seed of 75 miles er hour, the actual seed is aroximately 8 miles er hour. Answer: Aroximately 6 mh YOUR TURN Suose the owner of the F-50 in Examle decides to downsize the tires from their original 5.7-inch diameter to a.-inch diameter. If the seedometer indicates a seed of 65 miles er hour, what is the actual seed of the truck?
c0.qxd 8// 7:08 PM Page 55. Linear and Angular Seeds 55 SECTION. SUMMARY In this section, circular motion was defined in terms of linear seed (seed along the circumference of a circle) v and angular seed (seed of angle rotation) v. Linear seed: v s t Angular seed: v u t, where u is given in radians. Linear and angular seeds associated with circular motion are related through the radius r of the circle. v rv or v v r It is imortant to note that these formulas hold true only when angular seed is given in radians er unit of time. SECTION. EXERCISES SKILLS In Exercises 0, find the linear seed of a oint that moves with constant seed in a circular motion if the oint travels along the circle of arc length s in time t. Label your answer with correct units.. s m, t 5 sec. s ft, t min. s 68,000 km, t 50 hr. s 7,5 mi, t days 5. s.75 nm (nanometers), t 0.5 ms (milliseconds) 6. s.6 m (microns), t 9 ns (nanoseconds) 7. s 6 in., t min 8. s 5 cm, t 8 hr 9. s, t 5. sec 0. s. mm, t. min 0 m In Exercises 0, find the distance traveled (arc length) of a oint that moves with constant seed v along a circle in time t.. v.8 m/sec, t.5 sec. v 6. km/hr, t.5 hr. v.5 mi/hr, t 0 min. v 5.6 ft/sec, t min 5. v 60 mi/hr, t 5 min 6. v 7 km/hr, t 0 min 7. v 750 km/min, t days 8. v 0 ft/sec, t 7 min 9. v ft/s, t min 0. v 6 km/hr, t 0 min In Exercises, find the angular seed associated with rotating a central angle in time t.. u 5, t 0 sec. u t. u 00, t 5 min., 6 sec 5. u 7, t hr 6. u 8., t 0.5 hr 7. u 00, t 5 sec 8. u, t 0 min u 60, t 0. sec 9. u 780, t min 0. u 0, t 6 min. u 900, t.5 sec. u 50, t 5.6 sec
c0.qxd 8// 7:08 PM Page 56 56 CHAPTER Radian Measure and the Unit Circle Aroach In Exercises, find the linear seed of a oint traveling at a constant seed along the circumference of a circle with radius r and angular seed.. rad v sec, r 9 in.. v 5. v rad 0 sec, r 5 mm 6. v 7. rad v r.5 in. 5 sec, 8. v 6 rad 9. v, r 7 0. v rad, r 0. in. sec yd 8 min. v 0 rad, r 0 cm. v 7. rad, r.6 mm sec sec In Exercises 5, find the distance a oint travels along a circle s, over a time t, given the angular seed, and radius of the circle r. Round to three significant digits. rad. r 5 cm, v rad t 0 sec. r mm, v 6 t sec 6 sec, sec, 5. r 5. in., v rad t 0 min 6. 5 sec, rad 7. r m, v t 00 sec 8. r 6.5 cm, v sec, 9. rad r 0 cm, v, t 5 sec 0 sec 50. r 5 cm, v 5. r 5 in., v 5 rotations er second, t 5 min (exress distance in miles*) 5. r 7 in., v 6 rotations er second, t 0 min (exress distance in miles*) * mi 580 ft rad sec, 5 rad 6 sec, 8 rad 5 sec, r. ft, r 8 cm r ft r.5 cm v rad sec, t min rad, t 50.5 min 5 sec 5 rad sec, t 9 min APPLICATIONS 5. Tires. A car owner decides to ugrade from tires with a diameter of. inches to tires with a diameter of 6. inches. If she doesn t udate the onboard comuter, how fast will she actually be traveling when the seedometer reads 65 mh? 5. Tires. A car owner decides to ugrade from tires with a diameter of.8 inches to tires with a diameter of 7.0 inches. If she doesn t udate the onboard comuter, how fast will she actually be traveling when the seedometer reads 70 mh? 55. Planets. The Earth rotates every hours (actually hours, 56 minutes, and seconds) and has a diameter of 796 miles. If you re standing on the equator, how fast are you traveling in miles er hour (how fast is the Earth sinning)? Comute this using hours and then with hours, 56 minutes, seconds as time of rotation. 56. Planets. The lanet Juiter rotates every 9.9 hours and has a diameter of 88,86 miles. If you re standing on its equator, how fast are you traveling in miles er hour? 57. Carousel. A boy wants to jum onto a moving carousel that is sinning at the rate of five revolutions er minute. If the carousel is 60 feet in diameter, how fast must the boy run, in feet er second, to match the seed of the carousel and jum on? 58. Carousel. A boy wants to jum onto a layground carousel that is sinning at the rate of 0 revolutions er minute. If the carousel is 6 feet in diameter, how fast must the boy run, in feet er second, to match the seed of the carousel and jum on?
c0.qxd 8// 7:08 PM Page 57. Linear and Angular Seeds 57 59. Music. Some eole still have their honograh collections and lay the records on turntables. A honograh record is a vinyl disc that rotates on the turntable. If a -inch-diameter record rotates at revolutions er minute, what is the angular seed in radians er minute? 60. Music. Some eole still have their honograh collections and lay the records on turntables. A honograh record is a vinyl disc that rotates on the turntable. If a -inch-diameter record rotates at revolutions er minute, what is the linear seed of a oint on the outer edge in inches er minute? 6. Bicycle. How fast is a bicyclist traveling in miles er hour if his tires are 7 inches in diameter and his angular seed is 5 radians er second? 6. Bicycle. How fast is a bicyclist traveling in miles er hour if his tires are inches in diameter and his angular seed is 5 radians er second? 6. Electric Motor. If a -inch-diameter ulley that s being driven by an electric motor and running at 600 revolutions er minute is connected by a belt to a 5-inch-diameter ulley to drive a saw, what is the seed of the saw in revolutions er minute? 6. Electric Motor. If a.5-inch-diameter ulley that s being driven by an electric motor and running at 800 revolutions er minute is connected by a belt to a -inch-diameter ulley to drive a saw, what is the seed of the saw in revolutions er minute? For Exercises 65 and 66, refer to the following: NASA exlores artificial gravity as a way to counter the hysiologic effects of extended weightlessness for future sace exloration. NASA s centrifuge has a 58-foot-diameter arm. Niall McDiarmid/Alamy 65. NASA. If two humans are on oosite (red and blue) ends of the centrifuge and their linear seed is 00 miles er hour, how fast is the arm rotating? 66. NASA. If two humans are on oosite (red and blue) ends of the centrifuge and they rotate one full rotation every second, what is their linear seed in feet er second? For Exercises 67 and 68, refer to the following: To achieve similar weightlessness as that on NASA s centrifuge, ride the Gravitron at a carnival or fair. The Gravitron has a diameter of meters, and in the first 0 seconds it achieves zero gravity and the floor dros. 67. Gravitron. If the Gravitron rotates times er minute, find the linear seed of the eole riding it in meters er second. 68. Gravitron. If the Gravitron rotates 0 times er minute, find the linear seed of the eole riding it in kilometers er hour. 69. Clock. What is the linear seed of a oint on the end of a 0-centimeter second hand given in meters er second? 70. Clock. What is the angular seed of a oint on the end of a 0-centimeter second hand given in radians er second? Courtesy NASA Patrick Reddy/America -7/Getty Images, Inc.
c0.qxd 8// 7:08 PM Page 58 58 CHAPTER Radian Measure and the Unit Circle Aroach CATCH THE MISTAKE In Exercises 7 and 7, exlain the mistake that is made. 7. If the radius of a set of tires on a car is 5 inches and the tires rotate 80 er second, how fast is the car traveling (linear seed) in miles er hour? 7. If a bicycle has tires with radius 0 inches and the tires rotate 90 er second, how fast is the bicycle traveling (linear seed) in miles er hour? Write the formula for linear seed. Let r 5 inches and v 80 er second. Simlify. Let mile 580 feet 6,60 inches and hour 600 seconds. v rv v (5 in.)(80 /sec) v 700 in./sec v a 700 600 b mh 6,60 Write the formula for linear seed. Let r 0 inches and v 80 er second. Simlify. Let mile 580 feet 6,60 inches and hour 600 seconds. v rv v (0 in.)(80 /sec) v 800 in./sec v a 800 600 b mh 6,60 Simlify. v 5. mh This is incorrect. The correct answer is aroximately.7 miles er hour. What mistake was made? Simlify. v 0. mh This is incorrect. The correct answer is aroximately.8 miles er hour. What mistake was made? CONCEPTUAL In Exercises 7 and 7, determine whether each statement is true or false. 7. Angular and linear seed are inversely roortional. 7. Angular and linear seed are directly roortional. 75. In the chater oener about the Ford Exedition, if the standard tires have radius r and the ugraded tires have radius r, assuming the owner does not get the onboard comuter adjusted, find the actual seed the Ford is traveling, v, in terms of the indicated seed on the seedometer, v. 76. For the Ford in Exercise 75, find the actual mileage the Ford has traveled, s, in terms of the indicated mileage on the odometer, s. In Exercises 77 and 78, use the diagram below: The large gear has a radius of 6 centimeters, the medium gear has a radius of centimeters, and the small gear has a radius of centimeter. cm cm 6 cm 77. If the small gear rotates revolution er second, what is the linear seed of a oint traveling along the circumference of the large gear? 78. If the small gear rotates.5 revolutions er second, what is the linear seed of a oint traveling along the circumference of the large gear? CHALLENGE 79. A boy swings a red ball attached to a 0-foot string around his head as fast as he can. He then icks u a blue ball attached to a 5-foot string and swings it at the same angular seed. How does the linear velocity of the blue ball comare to that of the red ball. 80. One of the cars on a Ferris wheel, 00 feet in diameter, goes all of the way around in 5 seconds. What is the linear seed of a oint halfway between the car and the hub?
c0.qxd 8// 7:08 PM Page 59 SECTION. DEFINITION OF TRIGONOMETRIC FUNCTIONS: UNIT CIRCLE APPROACH SKILLS OBJECTIVES Draw the unit circle illustrating the secial angles and label the sine and cosine values. Determine the domain and range of trigonometric (circular) functions. Classify circular functions as even or odd. CONCEPTUAL OBJECTIVES Understand that trigonometric functions using the unit circle aroach are consistent with both of the revious definitions (right triangle trigonometry and trigonometric functions of nonacute angles in the Cartesian lane). Relate x-coordinates and y-coordinates of oints on the unit circle to the values of the cosine and sine functions. Visualize eriodic roerties of trigonometric (circular) functions. Recall that the first definition of trigonometric functions we develoed was in terms of ratios of sides of right triangles (Section.). Then, in Section., we suerimosed right triangles on the Cartesian lane, which led to a second definition of trigonometric functions (for any angle) in terms of ratios of x- and y-coordinates of a oint and the distance from the origin to that oint. In this section, we inscribe right triangles into the unit circle in the Cartesian lane, which will yield a third definition of trigonometric functions. It is imortant to note that all three definitions are consistent with one another. Trigonometric Functions and the Unit Circle (Circular Functions) (0, ) y Recall that the equation for the unit circle (radius of centered at the origin) is given by x y. We will use the term circular function later in this section, but it is imortant to note that a circle is not a function (it does not ass the vertical line test). If we form a central angle u in the unit circle such that the terminal side lies in quadrant I, we can use the revious two definitions of the sine and cosine functions when r (i.e., on the unit circle) and noting that we can form a right triangle with legs of lengths x and y and hyotenuse r. (, 0) r = x (0, ) (x, y) y x (, 0) TRIGONOMETRIC RIGHT TRIANGLE CARTESIAN FUNCTION TRIGONOMETRY PLANE sin u cos u oosite hyotenuse y y adjacent hyotenuse x x y r y y x r x x 59
c0.qxd 8// 7:08 PM Page 60 60 CHAPTER Radian Measure and the Unit Circle Aroach Notice that any oint (x, y) on the unit circle can be written as (cos u, sin u), where u is the measure of a trigonometric angle defined in Chater. If we recall the unit circle coordinate values for secial angles (Section.), we can now summarize the exact values for the sine and cosine functions in the illustration below. Study Ti (cos u, sin u) reresents a oint (x, y) on the unit circle. (x, y) = (cos, sin ) ( ), y (0, ) (, ) 90º 0º 60º 5 5º 6 0º 6 50º (, 0) (, ) 80º 0 ( 7 0º, 6 5º ) 5 ( 0º, 70º ) (, ) ( ), (, ) 5º ) 0º 0 60º (, (, 0) 0º 5º 6 (, 00º 7 ) 5 (, ) (0, ) ( ), x The following observations are consistent with roerties of trigonometric functions we ve studied already: sin u 0 in quadrant I and quadrant II, where y 0. cos u 0 in quadrant I and quadrant IV, where x 0. The equation of the unit circle x y leads also to the Pythagorean identity cos u sin u that we derived in Section.. Circular Functions Using the unit circle relationshi (x, y) (cos u, sin u), where u is the central angle whose terminal side intersects the unit circle at the oint (x, y), we can now define the remaining trigonometric functions using this unit circle aroach and the quotient and recirocal identities. Because the trigonometric functions are defined in terms of the unit circle, the trigonometric functions are often called circular functions. Recall that u s, and since r r, we know that u s. y (, 0) (0, ) (x, y) s x (, 0) (0, )
c0.qxd 8// :6 PM Page 6. Definition of Trigonometric Functions: Unit Circle Aroach 6 D EFINITION Let (x, y) be any oint on the unit circle (x y ). If u is the real number that reresents the distance from the oint (, 0) along the circumference of the circle to the oint (x, y), then sin u y csc u y y 0 Trigonometric Functions: Unit Circle Aroach cos u x sec u x x 0 tan u y x cot u x y x 0 y 0 The coordinates of the oints on the unit circle can be written as (cos u, sin u), and since is a real number, the trigonometric functions are often called circular functions. Classroom Examle.. Comute: a. cos a b b. cot a b c. sec a 7 6 b Answer: a. b. 0 c. Technology Ti EXAMPLE Finding Exact Trigonometric (Circular) Function Values Find the exact values for each of the following using the unit circle definition. a. sin a7 b. cos a5 c. tan a b 6 b b Use a TI calculator to confirm the values for sin a 7 b, cos a5 and tan a 6 b, b. Solution (a): 7 The angle corresonds to the coordinates a on the unit circle., b The value of the sine function is the y-coordinate. sin a 7 b Solution (b): 5 The angle corresonds to the coordinates a on the unit circle., 6 b The value of the cosine function is the x-coordinate. cos a 5 6 b Solution (c): The angle corresonds to the coordinates (0, ) on the unit circle. The value of the cosine function is the x-coordinate. cos a b 0 The value of the sine function is the y-coordinate. The tangent function is the ratio of the sine to cosine functions. Substitute cos a and sin a b 0 b. sin a b tan a sin (/) b cos (/) tan a b 0 Technology Ti Since tan a is undefined, the TI b calculator will dislay an error message. tan a b is undefined YOUR TURN Find the exact values for each of the following using the unit circle definition. Answer: a. b. c. a. sin a5 b. cos a7 c. tan a 6 b b b
c0.qxd 8// 7:08 PM Page 6 6 CHAPTER Radian Measure and the Unit Circle Aroach Classroom Examle.. Solve cos u on [0, ]. Answer:, 5 EXAMPLE Solving Equations Involving Trigonometric (Circular) Functions Use the unit circle to find all values of u, 0 u, for which sin u. Since the value of the sine function is negative, u must lie in quadrants III or IV. The value of sine is the y-coordinate. The angles corresonding to sin are 7 6 and. 6 There are two values for u that are greater than or equal to zero and less than or equal to that satisfy the equation sin u. ( y (0, ) (, 90º ) ( ), 60º 0º 5º 5 5º 6 0º 6 50º (, 0) 0º 0 80º 0º 0 60º 0º (, ) u 7 6, 6 (, ) (, ) x (, 0) ( 7 6 5º 5º 6, 5 00º 7 ) 0º 5 ( 70º, ) (0, ) (, ) (, ) (, ) (, ) Answer: u, YOUR TURN Find all values of u, 0 u, for which cos u. Proerties of Circular Functions WORDS The coordinates of any oint (x, y) that lies on the unit circle satisfies the equation x y. Since x cos u and y sin u, the following trigonometric inequalities hold. MATH x and y cos u and sin u State the domain and range of the cosine and sine functions. Domain: (, ) Range: [, ] cos u Since cot u and csc u sin u sin u, the values for u that make sin u 0 must be eliminated from the domain of the Domain: u n, where n is cotangent and cosecant functions. an integer sin u Since tan u and sec u cos u cos u, the values for u that make cos u 0 must be eliminated from the domain of the tangent and secant functions. (n ) Domain: u n, where n is an integer
c0.qxd 8// 7:08 PM Page 6. Definition of Trigonometric Functions: Unit Circle Aroach 6 The following box summarizes the domains and ranges of the trigonometric (circular) functions. DOMAINS AND RANGES OF THE TRIGONOMETRIC (CIRCULAR) FUNCTIONS For any real number and integer n, FUNCTION DOMAIN RANGE sin u cos u tan u cot u sec u (, ) (, ) (n ) all real numbers such that u n all real numbers such that u n (n ) all real numbers such that u n [, ] [, ] (, ) (, ) (,, ) csc u all real numbers such that u n (,, ) Recall from algebra that even and odd functions have both an algebraic and a grahical interretation. Even functions are functions for which f(x) f(x) for all x in the domain of f, and the grah of an even function is symmetric about the y-axis. Odd functions are functions for which f(x) f(x) for all x in the domain of f, and the grah of an odd function is symmetric about the origin. y (x, y) = (cos, sin) x (x, y) = (cos( ), sin( )) = (cos, sin) The cosine function is an even function. The sine function is an odd function. cos u cos(u) sin(u) sin u
c0.qxd 8// 7:08 PM Page 6 6 CHAPTER Radian Measure and the Unit Circle Aroach EXAMPLE Using Proerties of Trigonometric (Circular) Functions Technology Ti Use a TI/scientific calculator 5 to confirm the value of cos a 6 b. Evaluate cos a 5 6 b. The cosine function is an even function. cos a 5 6 b cos a5 6 b Use the unit circle to evaluate cosine. cos a 5 6 b cos a 5 6 b Answer: YOUR TURN Evaluate sin a 5 6 b. Study Ti Set the calculator to radian mode before evaluating circular functions in radians. Alternatively, convert the radian measure to degrees before evaluating the trigonometric function value. Classroom Examle.. Evaluate exactly: a. sin a 5 b b. cos a 7 b Answer: a. b. Classroom Examle.. Evaluate cos a 8 using a b calculator. Answer: 0.659 It is imortant to note that although trigonometric (circular) functions can be evaluated exactly for some secial angles, a calculator can be used to aroximate trigonometric (circular) functions for any value. EXAMPLE Evaluating Trigonometric (Circular) Functions with a Calculator Use a calculator to evaluate sin a 7 Round the answer to four decimal laces. b. C OMMON CORRECT Evaluate with a calculator. 0.9659586 Round to four decimal laces. sin a 7 b 0.9659 M ISTAKE INCORRECT Evaluate with a calculator. 0.097976 ERROR (calculator in degree mode) Many calculators automatically reset to degree mode after every calculation, so be sure to always check what mode the calculator indicates. Answer: 0.765 YOUR TURN Use a calculator to evaluate tan a 9 Round the answer to four decimal laces. 5 b.
c0.qxd 8// 7:08 PM Page 65. Definition of Trigonometric Functions: Unit Circle Aroach 65 EXAMPLE 5 Even and Odd Trigonometric (Circular) Functions Show that the secant function is an even function. Show that sec(u) sec u. Secant is the recirocal of cosine. Cosine is an even function, so cos(u) cos u. Secant is the recirocal of cosine, sec u cos u. sec(u) cos(u) sec(u) cos u sec(u) cos u sec u Since sec(u) sec u, the secant function is an even function. Classroom Examle..5 Prove that the cosecant function is an odd function. Answer: csc(u) sin(u) sin u csc u SECTION. SUMMARY In this section, we have defined trigonometric functions in terms of the unit circle. The coordinates of any oint (x, y) that lies on the unit circle satisfy the equation x y. The Pythagorean identity cos u sin u follows immediately from the unit circle equation if (x, y) (cos u, sin u), where u is the central angle whose terminal side intersects the unit circle at the oint (x, y). The cosine function is an even function, cos(u) cos u, and the sine function is an odd function, sin(u) sin u. SECTION. EXERCISES SKILLS In Exercises, find the exact values of the indicated trigonometric functions using the unit circle.. sin a5. cos a 5 b b. cos a7. sin a 7 6 b 6 b 5. sin a 6. cos a b b 7. tan a7 8. cot a 7 b b 9. seca5 0. 6 b csca 6 b. sec 5. csc 00. tan 0. cot 0 (, ) ( y, ) (0, ) 90º (, ( ), ) 60º 0º 5º ( 5 5º (, ), ) 6 0º 6 50º (, 0) 0º 0 x 80º 0 60º (, 0) ( 7 0º 0º 6 5º 5º 6, ) 5 00º 7 0º 5 ( 70º, (, ) ) (0, ) ( ( ),, ) (, )
c0.qxd 8// 7:08 PM Page 66 66 CHAPTER Radian Measure and the Unit Circle Aroach In Exercises 5 0, use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions. 5. sin a 6. sin a5 7. sin a 8. b b b sin a 7 6 b 9. cos a b 0. cos a5 b. cos a5 6 b.. sin(5 ). sin(80 ) 5. sin(70 ) 6. 7. cos(5 ) 8. cos(5 ) 9. cos(90 ) 0. cos a 7 b sin(60 ) cos(0 ) In Exercises 50, use the unit circle to find all of the exact values of that make the equation true in the indicated interval.. cos u 0 u.,. sin u 0 u., cos u, sin u, 0 u 0 u 5. cos u 6., 0 u sin u, 0 u 7. cos u, 0 u 8. sin u, 0 u 9. sin u 0, 0 u 0. sin u, 0 u. cos u, 0 u. cos u 0, 0 u. tan u, 0 u. cot u, 0 u 5. sec u, 0 u 6. csc u, 0 u 7. csc u is undefined, 0 u 8. sec u is undefined, 0 u 9. tan u is undefined, 0 u 50. cot u is undefined, 0 u In Exercises 5 58, aroximate the trigonometric function values. Round answers to four decimal laces. 5. cos a7 5. sin a5 5. cot a 5. b 9 b 5 b tan a 7 b 55. sin 56. cos 7 57. tan(.5) 58. csc APPLICATIONS For Exercises 59 and 60, refer to the following: The average daily temerature in Peoria, Illinois, can be (x ) redicted by the formula T 50 8 cos, where 65 x is the number of the day in a nonlea year (January, February, etc.) and T is in degrees Fahrenheit. 59. Atmosheric Temerature. What is the exected temerature on February 5? 60. Atmosheric Temerature. What is the exected temerature on August 5? For Exercises 6 and 6, refer to the following: The human body temerature normally fluctuates during the day. Assume a erson s body temerature can be redicted by the formula T 99. 0.5 sin ax where x is the number of b, hours since midnight and T is in degrees Fahrenheit. 6. Body Temerature. What is the erson s temerature at 6:00 A.M.? 6. Body Temerature. What is the erson s temerature at 9:00 P.M.?
c0.qxd 8// 7:08 PM Page 67. Definition of Trigonometric Functions: Unit Circle Aroach For Exercises 6 and 6, refer to the following: The height of the water in a harbor changes with the tides. On a articular day, it can be determined by the formula h(x) 5.8 sin c (x )d, where x is the number of hours 6 since midnight and h is the height of the tide in feet. 67 65. Yo-Yo Dieting. A woman has been yo-yo dieting for years. Her weight changes throughout the year as she gains and loses weight. Her weight in a articular month can be determined by the formula w(x) 5 0 cos a xb, 6 where x is the month and w is in ounds. If x corresonds to January, how much does she weigh in June? 66. Yo-Yo Dieting. How much does the woman in Exercise 65 weigh in December? Bill Brooks/Alamy 67. Seasonal Sales. The average number of guests visiting the Magic Kingdom at Walt Disney World er day is given by n(x) 0,000 0,000 sin c (x ) d, where n is the number of guests and x is the month. If January corresonds to x, how many eole, on average, are visiting the Magic Kingdom er day in February? 68. Seasonal Sales. How many guests are visiting the Magic Kingdom in Exercise 67 in December? 69. Temerature. The average high temerature for a certain city is given by the equation T 60 0 cos a tb, where 6 T is degrees Fahrenheit and t is time in months. What is the average temerature in June (t 6)? 70. Temerature. The average high temerature for a certain city is given by the equation T 65 5 cos a tb, where 6 T is degrees Fahrenheit and t is time in months. What is the average temerature in October (t 0)? 7. Gear. The vertical osition in centimeters of a tooth on a gear is given by the function y sin(0t), where t is time in seconds. Find the vertical osition after.5 seconds. 7. Gear. The vertical osition in centimeters of a tooth on a gear is given by the equation y 5 sin(.6t), where t is time in seconds. Find the vertical osition after 0 seconds. 7. Oscillating Sring. A weight is attached to a sring and then ulled down and let go to begin a vertical motion. The osition of the weight in inches from equilibrium is given by 7 7 the equation y 5 sin a t b, where t is time in seconds after the sring is let go. Find the osition of the weight.5 seconds after being let go. 6. Tides. What is the height of the tide at :00 P.M.? 6. Tides. What is the height of the tide at 5:00 A.M.? 7. Oscillating Sring. A weight is attached to a sring and then ulled down and let go to begin a vertical motion. The osition of the weight in inches from equilibrium is given by the equation y 5 sin a.6t b, where t is time in seconds after the sring is let go. Find the osition of the weight 5 seconds after being let go.
c0.qxd 8// 7:08 PM Page 68 68 CHAPTER Radian Measure and the Unit Circle Aroach For Exercises 75 and 76, refer to the following: During the course of treatment of an illness, the concentration of a drug in the bloodstream in micrograms er microliter fluctuates during the dosing eriod of 8 hours according to the model C(t) 5..7 sina t b, 0 t 8 Note: This model does not aly to the first dose of the medication. 75. Health/Medicine. Find the concentration of the drug in the bloodstream at the beginning of a dosing eriod. 76. Health/Medicine. Find the concentration of the drug in the bloodstream 6 hours after taking a dose of the drug. In Exercises 77 and 78, refer to the following: By analyzing available emirical data, it has been determined that the body temerature of a articular secies fluctuates during a -hour day according to the model T(t) 6.. cos c (t )d, 0 t where T reresents temerature in degrees Celsius and t reresents time in hours measured from :00 a.m. (midnight). 77. Biology. Find the aroximate body temerature at midnight. Round your answer to the nearest degree. 78. Biology. Find the aroximate body temerature at :5.m. Round your answer to the nearest degree. CATCH THE MISTAKE In Exercises 79 and 80, exlain the mistake that is made. 79. Use the unit circle to evaluate tan a 5 exactly. 6 b Tangent is the ratio of sine to cosine. Use the unit circle to identify sine and cosine. Substitute values for sine and cosine. Simlify. 5 tan a 5 sin a 6 b 6 b cos a 5 6 b sin a 5 6 b tan a 5 6 b tan a 5 6 b and cos a 5 6 b 80. Use the unit circle to evaluate sec a exactly. 6 b sec a 6 b Secant is the recirocal of cosine. Use the unit circle to evaluate cosine. Substitute the value for cosine. cos a 6 b sec a 6 b Simlify. sec a 6 b This is incorrect. What mistake was made? cos a 6 b This is incorrect. What mistake was made? CONCEPTUAL In Exercises 8 8, determine whether each statement is true or false. 8. sin(n u) sin u, for n an integer. 8. cos(n u) cos u, for n an integer. 8. (n ) sin u when u, for n an integer. 8. cos u when u n, for n an integer. 85. Is y csc x an even or an odd function? Justify your answer. 86. Is y tan x an even or an odd function? Justify your answer. 87. Find all the values of u, 0 u, for which the equation is true: sin u cos u. 88. Find all the values of u (u is any real number) for which the equation is true: sin u cos u.
c0.qxd 8// 7:08 PM Page 69. Definition of Trigonometric Functions: Unit Circle Aroach 69 CHALLENGE 89. How many times is the exression ƒ cos(t) ƒ true for 0 t? 90. How many times is the exression ` sin a true for tb` 0 t 0? 9. For what values of x, such that 0 x, is the exression ƒ cos t ƒ ƒ sin t ƒ true? 9. For what values of x, such that 0 x, is the exression ƒ sec t ƒ ƒ cos t ƒ true? 9. Find values of x such that 0 x and both of the following are true: sin x and cos x. 9. Find values of x such that 0 x and both of the following are true: tan x and sec x 0. TECHNOLOGY 95. Use a calculator to aroximate sin. What do you exect sin( ) to be? Verify your answer with a calculator. 96. Use a calculator to aroximate cos 7. What do you exect cos(7 ) to be? Verify your answer with a calculator. For Exercises 97 and 98, refer to the following: A grahing calculator can be used to grah the unit circle with arametric equations (these will be covered in more detail in Section 8.5). For now, set the calculator in arametric and radian modes and let X cos T Y sin T Set the window so that 0 t, ste, X, 5 and Y. 97. To aroximate cos a use the trace function to move b, 5 stes aof to the right of t 0 and read the 5 eachb x-coordinate. 98. To aroximate sin a use the trace function to move b, 5 stes aof to the right of t 0 and read the 5 eachb y-coordinate.
c0.qxd 8// 7:08 PM Page 70 CHAPTER INQUIRY-BASED LEARNING PROJECT Mr. Wilson is looking to exand his watering trough for his horses. His neighbor, Dr. Parkinson, suggests considering something other than the square trough he currently has. Jokingly, she says, Mr. Wilson, think outside the box. Uon the advice of his mathematics rofessor neighbor, Mr. Wilson decides to itch out the sides of his troughs, forming a traezoidal cross section using his current barn as one of the sides. (Reread the Inquiry-Based Learning Project in Chater.) Your goal is to maximize the cross section of his trough. To do this, you will first use theta ( u) as your variable and look at how the area changes as u changes. Barn Barn Current Future ft ft ft ft trough trough ft. Fill in the chart (use two decimals). To get started, when u is 0, the traezoid is just the original square trough. As u increases, the original square becomes two triangles and one rectangle. Use right triangle trigonometry to calculate the various bases and height. Note: The base for the triangles differs from the base of the rectangles. Also, you do not need to do every single traezoid by hand. Do as many as you think is necessary to understand how to write the area as a function of u. The ability to write out the area function is the rimary goal. Theta ( u) 0 5 5 5 5 5 55 65 75 85 ft. From your chart values, describe what haens to the area of the trough as u increases.. Is the maximum area for the trough necessarily included in this chart? Exlain.. Write the area A( u) as a function of u using sin u and cos u. Again, look to how you calculated areas in the chart for direction. Also be sure to use your calculator s table to check your roblems done by hand and vice versa. 5. Grah this function on a reasonable domain and be sure to indicate what the domain is. 6. Exlain the meaning of the y-intercet in this scenario. 7. Summarize your findings for Mr. Wilson. Remember, you were given a charge to build the biggest trough ossible. How are you going to do it and what is the new and imroved area? 8. After looking at your results from Chater s Inquiry-Based Learning Project, exlain why many eole consider the otimum u to be a counterintuitive result. 70
c0.qxd 8// 7:08 PM Page 7 MODELING OUR WORLD Tire selection affects fuel economy in automobiles. The more miles er gallon consumers can obtain in their automobiles, the less gasoline we consume (money) and hence burn (ollution/greenhouse gases). Tire size (both diameter and tread width) can affect gas mileage, deending on what kind of driving you do (highway vs. city and flat vs. hilly). Go back and reread the Chater oener about the Ford Exedition and the consequences (seedometer and odometer) of altering the tires. Assume the original tires have a diameter of 6 inches and the new tires have a diameter of 8 inches.. If you know you drove 5,000 miles in a year (according to your GPS Navigation System), what would your odometer actually read (assume the onboard comuter was not adjusted when the new tires were ut on the Exedition)?. If your seedometer reads 85 miles er hour, what is your actual seed?. If your onboard comuter is saying you are getting 6 miles er gallon, what is your actual gas mileage?. Assuming gasoline costs $ er gallon, how much money would you be saving by increasing your tires inches in diameter? 5. Find a function that models your gasoline savings er year as a function of increase in diameter of tires. 6. Do the gasoline savings seem worth the investment in larger tires? 7
c0.qxd 8// 7:08 PM Page 7 CHAPTER REVIEW 6. Inverse Trigonometric Functions 7 SECTION CONCEPT KEY IDEAS/FORMULAS. Radian measure The radian measure of an angle u (in radians) s r s (arc length) and r (radius) must have the same units. r r s Converting between degrees and radians Degrees to radians: Radians to degrees: u r u d a 80 b u d u r a 80 b. Arc length and area of a circular sector r s r CHAPTER REVIEW Arc length Area of circular sector is in radians, or is in degrees is in radians, or A, where u d is in degrees r u d a 80 b. Linear and angular seeds Uniform circular motion Linear seed s r u r, where u r s r u d a b, where 80 A r u r, where u r u d Linear seed: seed around the circumference of a circle Angular seed: rotation seed of angle Linear seed v is given by v s t where s is the arc length and t is time. Angular seed Angular seed v is given by v u t where u is given in radians. Relationshi between linear and angular seeds v rv or v v r It is imortant to note that these formulas hold true only when angular seed is given in radians er unit of time. 7
c0.qxd 8// 7:08 PM Page 7 SECTION CONCEPT KEY IDEAS/FORMULAS. Definition of trigonometric functions: Unit circle aroach Trigonometric functions and the unit circle (circular functions) (, 0) y (0, ) (cos, sin) r = x (, 0) (0, ) Proerties of circular functions (, ) ( y, ) (0, ) 90º (, ( ), ) 60º 0º 5º ( 5 5º (, ), ) 6 0º 6 50º (, 0) 0º 0 x 80º 0 60º (, 0) ( 7 0º 0º 6 5º 5º 6, ) 5 00º 7 0º 5 ( 70º, (, ) ) (0, ) ( ( ),, ) Cosine is an even function: Sine is an odd function: (, ) cos(u) cos u sin(u) sin u CHAPTER REVIEW FUNCTION DOMAIN RANGE sin u cos u tan u cot u sec u csc u * n is an ineger. (, ) (, ) (n ) u n u n (n ) u n u n [, ] [, ] (, ) (, ) (,, ) (,, ) 7
c0.qxd 8// 7:08 PM Page 7 CHAPTER REVIEW EXERCISES REVIEW EXERCISES. Radian Measure Convert from degrees to radians. Leave your answers exact in terms of.. 5. 0. 0. 80 5. 6 6. 08 7. 50 8. 600 9. 50 0. 5 Convert from radians to degrees. 5.... 6 5 7 5. 6. 7. 8. 9 0 9. 5 0. 8 Find the reference angle of each angle given (in radians). 7 5 7.... 6 6. Arc Length and Area of a Circular Sector Find the arc length interceted by the indicated central angle of the circle with the given radius. Round to two decimal laces. 5. u 6. u 5, r 0 in., r 5 cm 6 7. u 00, r 5 in. 8. u 6, r ft Find the measure of the angle whose interceted arc and radius of a circle are given. 9. r in., s 6 in. 0. r 0 ft, s 7 in.. r 6 ft, s ft. r 8 m, s m. r 5 ft, s 0 in.. r km, s m Find the measure of each radius given the arc length and central angle of each circle. 5. u 5 8, s in. 6. u, s km 7. u 50, s m 8. u 6, s in. 9. 5 u 0, s 8 yd 0. u 80, s 5 ft 6 Find the area of the circular sector given the indicated radius and central angle.. u, r mi. u 5, r 9 in.. u 60, r 60 m. u 8, r 6 cm. Linear and Angular Seeds Find the linear seed of a oint that moves with constant seed in a circular motion if the oint travels arc length s in time t. 5. s ft, t 9 sec 6. s 580 ft, t min 7. s 5 mi, t min 8. s cm, t 0.5 sec Find the distance traveled by a oint that moves with constant seed v along a circle in time t. 9. v 5 mi/hr, t day 50. v 6 ft/sec, t min 5. v 80 mi/hr, t 5 min 5. v.5 cm/hr, t 6 sec Find the angular seed (radians/second) associated with rotating a central angle in time t. 5. u 6, t 9 sec 5. u, t 0.05 sec 55. u 5, t 0 sec 56. u 0, t sec Find the linear seed of a oint traveling at a constant seed along the circumference of a circle with radius r and angular seed. 57. v 5 rad 58. v rad, r 0 in. 6 sec, r m 0 sec Find the distance s a oint travels along a circle over a time t, given the angular seed and radius of the circle r. 59. 60. 6. 6. r 0 ft, v rad, t 0 sec sec r 6 in., v rad sec, t 6 sec r yd, v rad, t 0 sec sec r 00 in., v rad 8 sec, t min Alications 6. A ladybug is clinging to the outer edge of a child s sinning disk. The disk is inches in diameter and is sinning at 60 revolutions er minute. How fast is the ladybug traveling? 6. How fast is a motorcyclist traveling in miles er hour if his tires are 0 inches in diameter and his angular seed is 0 radians er second? 7
c0.qxd 8// 7:08 PM Page 75 Review Exercises 75. Definition of Trigonometric Functions: Unit Circle Aroach Find the exact values of the indicated trigonometric functions. 65. tan a5 66. cos a 5 6 b 6 b 67. sin a 68. 6 b sec a 6 b 69. cot a5 70. csc a 5 b b 7. sin a 7. cos a b b 7. cos 7. tan 5 75. cos 60 76. sin 0 77. sin a5 78. cos a 5 6 b b 79. cos(0 ) 80. sin(5 ) Find all of the exact values of that make the equation true in the indicated interval. 8. sin u, 0 u 8. cos u, 0 u 8. tan u 0, 0 u 8. sin u, 0 u Technology Exercises Section. Find the measure (in degrees, minutes, and nearest seconds) of a central angle that intercets an arc on a circle with radius r with indicated arc length s. Use the TI calculator commands ANGLE and DMS to change to degrees, minutes, and seconds. 85. r. ft, s 9.7 ft 86. r 56.9 cm, s 9. cm Section. For Exercises 87 and 88, refer to the following: A grahing calculator can be used to grah the unit circle with arametric equations (these will be covered in more detail in Section 8.). For now, set the calculator in arametric and radian modes and let X cos T Y sin T Set the window so that 0 T, ste, X, 5 and Y. To aroximate the sine or cosines of a T value, use the TRACE key, enter the T value, and read the corresonding coordinates from the screen. 87. Use the above stes to aroximate cos a to four b decimal laces. 88. Use the above stes to aroximate sin a 5 to four decimal 6 b laces. REVIEW EXERCISES
c0.qxd 8// 7:08 PM Page 76 CHAPTER PRACTICE TEST PRACTICE TEST. Find the measure (in radians) of a central angle u that intercets an arc on a circle with radius r 0 centimeters and arc length s millimeters.. Convert to degree measure.. Convert 60 to radian measure. Leave the answer exact in terms of.. Convert 7 to radian measure. Round to two decimal laces. 5. What is the reference angle to u 7? 6. Find the radius of the minute hand on a clock if a oint on the end travels 0 centimeters in 0 minutes. 7. Betty is walking around a circular walking ath. If the radius of the ath is 0.50 miles and she has walked through an angle of 0, how far has she walked? 8. Calculate the arc length on a circle with central angle u and radius r 8 yards. 5 9. A srinkler has a 5-foot sray and it covers an angle of 0. What is the area that the srinkler waters? Round to the nearest square foot. 0. A bicycle with tires of radius r 5 inches is being ridden by a boy at a constant seed the tires are making five rotations er second. How many miles will he ride in 5 minutes? ( mi 580 ft). The smaller gear in the diagram below has a radius of centimeters, and the larger gear has a radius of 5. centimeters. If the smaller gear rotates 5, how many degrees has the larger gear rotated? Round answer to the nearest degree.. Samuel rides 55 feet on a merry-go-round that is 0 feet in diameter in a clockwise direction. Through what angle has Samuel rotated?. Layla is building an ornamental wall that is in the shae of a iece of a circle feet in diameter. If the central angle of the circle is 0, how long is the rock wall?. A blueberry ie is made in a 9-inch-diameter ie an. If a -inch-radius circle is cut out of the middle for decoration, what is the area of each iece of ie if the ie is cut into 8 equal ieces? 5. Tom s hands go in a 9-inch-radius circular attern as he rows his boat across a lake. If his hands make a comlete rotation every.5 seconds, what are the angular seed and linear seed of his hands? In Exercises 6 0, if ossible, find the exact value of the indicated trigonometric function using the unit circle. 6. sin a7 7. tan a7 8. csc a 6 b b b 9. cot a 0. sec a 7 b b. What is the measure in radians of the smaller angle between the hour and minute hands at 0:0?. Find all of the exact values of u that make the equation sin u true in the interval 0 u.. Find all of the exact values of u that make the equation tan u true in the interval 0 u.. Sales of a seasonal roduct s vary according to the time of year sold given as t. If the equation that models sales is s 500 5 cos a t, what were the sales in March 6 b (t )? 5. The manager of a -hour lant tracks roductivity throughout the day and finds that the equation 50 cos a accurately models outut t b from his workers at time t, where is the number of units roduced by the workers and t is the time in hours after midnight. What is the lant s outut at 5:00 in the evening? 76
c0.qxd 8// 7:08 PM Page 77 CHAPTERS CUMULATIVE TEST. In a triangle with angles a, b, and g, and 0. Given the angle 99.99 in standard osition, state the a b g 80, if a 5, b 5, find g. quadrant of this angle.. In a 0-60 -90 triangle, if the hyotenuse is meters,. Given the angle 70 in standard osition, find the axis what are the lengths of the two legs? of this angle.. If D (9x 6) and G (7x ) in the diagram. The angle u in standard osition has the terminal side below, find the measures D and G. defined by the line x y 0, x 0. Calculate the values for the six trigonometric functions of u. m n A B. Given the angle u 900 in standard osition, calculate, C D m if ossible, the values for the six trigonometric functions of u.. If cos u E F, 9 and the terminal side of u lies in G H n quadrant III, find csc u. 5. Evaluate the exression sin 50 sec(50 ).. Height of a Woman. If a 6-foot volleyball layer has a 6. Find the ositive measure of u (rounded to the nearest -foot -inch shadow, how long a shadow will her -foot degree) if tan u.85 and the terminal side of u lies in 6-inch daughter cast? quadrant III. 5. Use the triangle below to find cos u. 7. Given cot u 5 use the recirocal identity to find tan u., 6. Write csc 0 in terms of its cofunction. 7. Perform the oeration B A, where A 9 r 5s and B 7 r 9s. 8. Use a calculator to aroximate sec (78 5r). Round the answer to four decimal laces. 9. Given a 7. and a miles, use the right triangle diagram to solve the right triangle. Write the answer for angle measures in decimal degrees. 5 c 9 a b 8. If cos u 6 and the terminal side of u lies in quadrant IV, find sin u. 9. Find sin u and cos u if tan u 6 and the terminal side of u lies in quadrant III. 0. Find the measure (in radians) of a central angle u that intercets an arc on a circle of radius r.6 centimeters with arc length s millimeters.. Clock. How many radians does the second hand of a clock turn in minute, 5 seconds?. Find the exact length of the radius with arc length s 9 meters and central angle u 7. 7. Find the distance traveled (arc length) of a oint that moves with constant seed v.6 meters er second along a circle in. seconds.. Bicycle. How fast is a bicyclist traveling in miles er hour if his tires are inches in diameter and his angular seed is 5 radians er second? 5. Find all of the exact values of u, when tan u and 0 u. CUMULATIVE TEST 77