Trigonometric. Functions CHAPTER OUTLINE

Transcription

1 bar5969_ch05_ qd 0/5/008 06:53 PM Page 463 pinnacle 0:MHIA065:SE:CH 05: CHAPTER 5 Trigonometric Functions C TRIGONOMETRIC functions seem to have had their origins with the Greeks investigation of the indirect measurement of distances and angles in the celestial sphere. (The ancient Egptians had used some elementar geometr to build the pramids and remeasure lands flooded b the Nile, but neither the nor the ancient Bablonians had developed the concept of angle measure.) The word trigonometr, based on the Greek words for triangle measurement, was first used as the title for a tet b the German mathematician Pitiscus in A.D Modern applications of the trigonometric functions range over man tpes of problems that have little or nothing to do with angles or triangles applications involving periodic phenomena such as sound, light, and electrical waves; business ccles; and planetar motion. OUTLINE 5- Angles and Their Measure 5- Trigonometric Functions: A Unit Circle Approach 5-3 Solving Right Triangles 5-4 Properties of Trigonometric Functions 5-5 More General Trigonometric Functions and Models 5-6 Inverse Trigonometric Functions Chapter 5 Review Chapter 5 Group Activit: A Predator Pre Analsis Involving Mountain Lions and Deer

2 464 CHAPTER 5 TRIGONOMETRIC FUNCTIONS 5- Angles and Their Measure Z Angles Z Degree and Radian Measure Z Converting Degrees to Radians and Vice Versa Z Linear and Angular Speed In this section we introduce the concept of angle and two measures of angles, degree and radian. V Terminal side Initial side Z Figure Angle or angle PVQ or V. Q P n m Z Angles The stud of trigonometr depends on the concept of angle. An angle is formed b rotating (in a plane) a ra m, called the initial side of the angle, around its endpoint until it coincides with a ra n, called the terminal side of the angle. The common endpoint V of m and n is called the verte (Fig. ). A counterclockwise rotation produces a positive angle, and a clockwise rotation produces a negative angle, as shown in Figures (a) and (b). The amount of rotation in either direction is not restricted. Two different angles ma have the same initial and terminal sides, as shown in Figure (c). Such angles are said to be coterminal. Terminal side Initial side positive Initial side Terminal side negative Terminal side (a) (b) (c) Initial side and coterminal Z Figure Angles and rotation. An angle in a rectangular coordinate sstem is said to be in standard position if its verte is at the origin and the initial side is along the positive ais. If the terminal side of an angle in standard position lies along a coordinate ais, the angle is said to be a quadrantal angle. If the terminal side does not lie along a coordinate ais, then the angle is often referred to in terms of the quadrant in which the terminal side lies (Fig. 3).

3 SECTION 5 Angles and Their Measure 465 Z Figure 3 positions. Angles in standard II III Terminal side Initial side I IV II III Terminal side Initial side I IV II III Terminal side Initial side I IV is a quadrantal angle is a third-quadrant angle is a second-quadrant angle (a) (b) (c) Z Degree and Radian Measure Just as line segments are measured in centimeters, meters, inches, or miles, angles are measured in different units. The two most commonl used units for angle measure are degree and radian. Z DEFINITION Degree Measure A positive angle formed b one complete rotation is said to have a measure of 360 degrees (360 ). A positive angle formed b 360 of a complete rotation is said to have a measure of degree ( ). The smbol denotes degrees. Definition is etended to all angles, not just the positive (counterclockwise) ones, in the obvious wa. So, for eample, a negative angle formed b 4 of a complete clockwise rotation has a measure of 90, and an angle for which the initial and terminal sides coincide, without rotation, has a measure of 0. Certain angles have special names that indicate their degree measure. Figure 4 shows a straight angle, a right angle, an acute angle, and an obtuse angle. Z Figure 4 Tpes of angles Straight angle ( rotation) Right angle ( rotation) 4 Acute angle (0 90) Obtuse angle (90 80) (a) (b) (c) (d) Two positive angles are complementar if their sum is 90 ; the are supplementar if their sum is 80. A degree can be divided further using decimal notation. For eample, 4.75 represents an angle of degree measure 4 plus three-quarters of degree. A degree can also be divided further using minutes and seconds just as an hour is divided into

4 466 CHAPTER 5 TRIGONOMETRIC FUNCTIONS minutes and seconds. Each degree is divided into 60 equal parts called minutes, and each minute is divided into 60 equal parts called seconds. Smbolicall, minutes are represented b and seconds b. Thus, 3 4 is a concise wa of writing degrees, 3 minutes, and 4 seconds. Decimal degrees (DD) are useful in some instances and degrees minutes seconds (DMS) are useful in others. You should be able to go from one form to the other as demonstrated in Eample. Z CONVERSION ACCURACY If an angle is measured to the nearest second, the converted decimal form should not go beond three decimal places, and vice versa. EXAMPLE From DMS to DD and Back (A) Convert 47 to decimal degrees. (B) Convert to degree minute second form. SOLUTIONS (A) (B) 47 a ,600 b.787 * ( ) ( ) MATCHED PROBLEM (A) Convert to DD form. (B) Convert to DMS form. Some scientific and some graphing calculators can convert the DD and DMS forms automaticall, but the process differs significantl among the various tpes of calculators. Check our owner s manual for our particular calculator. The conversion methods outlined in Eample show ou the reasoning behind the process, and are sometimes easier to use than the automatic methods for some calculators. *The dashed think boes are used to enclose steps that ma be performed mentall.

5 SECTION 5 Angles and Their Measure 467 Degree measure of angles is used etensivel in engineering, surveing, and navigation. Another unit of angle measure, called the radian, is often preferred in mathematics, scientific work, and engineering applications. Z DEFINITION Radian Measure A positive angle formed b a central angle of a circle has measure radian if the length s of the arc opposite is equal to the radius r of the circle. More generall, if is an positive angle formed b the central angle of a circle, then the radian measure of is given b s r radians where s is the length of the arc opposite and r is the radius of the circle. [Note: s and r must be measured in the same units.] s O r r s r r O r radian REMARKS:. Because the circumference of a circle is proportional to its radius, circles of different radii will give the same radian measure for an angle.. If the radius r, then the radian measure of angle is simpl the arc length s. The circumference of a circle of radius r is r, so the radian measure of a positive angle formed b one complete rotation is s r r r 6.83 radians Just as for degree measure, the definition is etended to appl to all angles; if is a negative angle, its radian measure is given b s r. Note that in the preceding sentence, as well as in Definition, the smbol is used in two was: as the name of the angle and as the measure of the angle. The contet indicates the meaning. EXAMPLE Computing Radian Measure What is the radian measure of a central angle opposite an arc of 4 meters in a circle of radius 6 meters?

6 468 CHAPTER 5 TRIGONOMETRIC FUNCTIONS SOLUTION s r 4 meters 6 meters 4 radians MATCHED PROBLEM What is the radian measure of a central angle opposite an arc of 60 feet in a circle of radius feet? REMARK: It is customar to omit the word radians when giving the radian measure of an angle. But if an angle is measured in an other units, the units must be stated eplicitl. For eample, if 7, then 974. ZZZ EXPLORE-DISCUSS Discuss wh the radian measure of an angle is independent of the size of the circle having the angle as a central angle. Z Converting Degrees to Radians and Vice Versa What is the radian measure of an angle of 80? Let be a central angle of 80 in a circle of radius r. Then the length s of the arc opposite is the circumference C of the circle. Therefore, s C r r and s r r r radians In other words, 80 corresponds to * radians. This is important to remember, because the radian measures of man special angles can be obtained from this correspondence. For eample, 90 is 80 /; therefore, 90 corresponds to / radians. *The constant has a long and interesting histor; a few important dates are listed below: 650 B.C. Rhind Paprus B.C. Archimedes ( ) A.D. 64 Liu Hui A.D. 470 Tsu Ch ung-chih A.D. 674 Leibniz 4( ) (This and other series can be used to compute to an decimal accurac desired.) A.D. 76 Johann Lambert Showed to be irrational ( as a decimal is nonrepeating and nonterminating.)

7 SECTION 5 Angles and Their Measure 469 ZZZ EXPLORE-DISCUSS Write the radian measure of each of the following angles in the form b, a where a and b are positive integers and fraction b is reduced to lowest terms: 5, 30, 45, 60, 75, 90, 05, 0, 35, 50, 65, 80. a Some ke results from Eplore-Discuss are summarized in Figure 5 for eas reference. These correspondences and multiples of them will be used etensivel in work that follows / 60 /3 45 /4 30 / / Z Figure 5 Radian degree correspondences. The following proportion can be used to convert degree measure to radian measure and vice versa. Z RADIAN DEGREE CONVERSION FORMULAS deg 80 rad radians deg 80 radians rad rad radians 80 deg Basic proportion Radians to degrees Degrees to radians [Note: We will omit units in calculations until the final answer. If our calculator does not have a ke labeled, use ] Some scientific and graphing calculators can automaticall convert radian measure to degree measure, and vice versa. Check the owner s manual for our particular calculator.

8 470 CHAPTER 5 TRIGONOMETRIC FUNCTIONS EXAMPLE 3 Radian Degree Conversions (A) Find the radian measure, eact and to three significant digits, of an angle of 75. (B) Find the degree measure, eact and to four significant digits, of an angle of 5 radians. (C) Find the radian measure to two decimal places of an angle of 4. SOLUTIONS (A) (B) (C) rad rad Eact radians deg 5 (75) Eact deg 80 radians rad (5) 4 a4 60 b 4. radians deg (4.) Three significant digits Four significant digits 86.5 Change 4 to DD first. To two decimal places Z Figure 6 Automatic conversion. Figure 6 shows the three preceding conversions done automaticall on a graphing calculator b selecting the appropriate angle mode. MATCHED PROBLEM 3 (A) Find the radian measure, eact and to three significant digits, of an angle of 40. (B) Find the degree measure, eact and to three significant digits, of an angle of radian. (C) Find the radian measure to three decimal places of an angle of 5 3. REMARK: We will write in place of deg and rad when it is clear from the contet whether we are dealing with degree or radian measure. EXAMPLE 4 Engineering A belt connects a pulle of -inch radius with a pulle of 5-inch radius. If the larger pulle turns through 0 radians, through how man radians will the smaller pulle turn? SOLUTION First we draw a sketch (Fig. 7).

9 SECTION 5 Angles and Their Measure 47 Z Figure 7 P 5 in. Q in. When the larger pulle turns through 0 radians, the point P on its circumference will travel the same distance (arc length) that point Q on the smaller circle travels. For the larger pulle, For the smaller pulle, s r s r (5)(0) 50 inches s r 50 5 radians MATCHED PROBLEM 4 In Eample 4, through how man radians will the larger pulle turn if the smaller pulle turns through 4 radians? Z Linear and Angular Speed The average speed v of an object that travels a distance d 30 meters in time t 3 seconds is given b v d t 30 meters 0 meters per second 3 seconds P s 30 m Suppose that a point P moves an arc length of s 30 meters in t 3 seconds on the circumference of a circle of radius r 0 meters (Fig. 8). Then, in those 3 seconds, the point P has moved through an angle of r 0 m s r 30.5 radians 0 We call the average speed of point P, given b Z Figure 8 v s t 0 meters per second

10 47 CHAPTER 5 TRIGONOMETRIC FUNCTIONS the (average) linear speed to distinguish it from the (average) angular speed that is given b t radians per second The simple formula, v r, which relates linear and angular speed, is obtained from the definition of radian measure as follows: s r s r s t r t v r Multipl both sides b r. Divide both sides b t. Substitute v s and t t. These concepts are summarized in the bo. Z LINEAR SPEED AND ANGULAR SPEED Suppose a point P moves through an angle and arc length s, in time t, on the circumference of a circle of radius r. The (average) linear speed of P is v s t and the (average) angular speed is t Furthermore, v r. EXAMPLE 5 Wind Power A wind turbine of rotor diameter 5 meters makes 6 revolutions per minute. Find the angular speed (in radians per second) and the linear speed (in meters per second) of the rotor tip. SOLUTION The radius of the rotor is 5/ 7.5 meters. In minute the rotor moves through an angle of 6() 4 radians. Therefore, the angular speed is t 4 radians 60 seconds 6.49 radians per second

11 SECTION 5 Angles and Their Measure 473 and the linear speed of the rotor tip is v r meters per second MATCHED PROBLEM 5 A wind turbine of rotor diameter meters has a rotor tip speed of 34. meters per second. Find the angular speed of the rotor (in radians per second) and the number of revolutions per minute. EXAMPLE 6 Navigation The traditional unit of distance for air and sea travel is the nautical mile. One nautical mile is the length of one minute of arc on the Earth s equator. Because there are ,600 minutes in a complete circle, the Earth s circumference is,600 nautical miles. (One nautical mile is approimatel.5 miles or 6,076 feet; ordinar miles are also called statute miles to distinguish them from nautical miles). The traditional unit of speed for air and sea travel is the knot: a speed of knot is nautical mile per hour. Recall that an point on the surface of the Earth can be specified b giving its latitude, measured in degrees north or south from the equator, and longitude, measured in degrees east or west from the Greenwich meridian (see Fig. 9). Stockholm, Sweden [ 59 3 N/8 00 E], and Cape Town, South Africa [ S/8 7 E], have nearl the same longitude. If a plane flies from Stockholm to Cape Town in hours, find its linear speed (to the nearest knot) and angular speed (to the nearest tenth of a degree per hour). North Pole Stockholm 593N/800E 59 Equator 8 Quito 04S/7830W Cape Town 3355S/87E Z Figure 9 South Pole Greenwich meridian

12 474 CHAPTER 5 TRIGONOMETRIC FUNCTIONS SOLUTION We consider Earth to be a sphere, so an great circle (that is, an circle on the surface of the sphere having the same center as the equator) has circumference,600 nautical miles. In particular, the circumference of the meridian passing through Stockholm and Cape Town is,600 nautical miles (we ignore the small difference in longitude between the two cities). The central angle between Stockholm and Cape Town (see Fig. 9) is or 93.3 Let s denote the arc length between Stockholm and Cape Town. Because arc lengths on a circle are in the same proportion as their central angles (see Problems 0 and 0 in Eercises 5-) s 93.3, s 5,598 nautical miles Therefore, the linear speed of the plane is Multipl both sides b,600. v s t 5, knots and the angular speed is t per hour. MATCHED PROBLEM 6 A plane flies from Quito, Ecuador [0 4 S/78 30 W], to Kampala, Uganda [0 9 N/3 35 E], in 4 hours. Find the plane s linear speed (to the nearest knot) and angular speed (to the nearest tenth of a degree per hour). Ignore the small difference in latitude between the two cities; Quito and Kampala are both close to the equator. ANSWERS TO MATCHED PROBLEMS. (A) (B) radians (A) 4.9 (B) 57.3 (C) radians radians per second; revolutions per minute knots; 7.9 per hour

13 SECTION 5 Angles and Their Measure Eercises In all problems, if angle measure is epressed b a number that is not in degrees, it is assumed to be in radians.. Eplain the difference between a positive angle and a negative angle.. Eplain the difference between complementar angles and supplementar angles. 3. Would it be better to measure angles b dividing the circumference of a circle into 00 equal parts, rather than 360 equal parts as in degree measure? Eplain. 4. Eplain the connection between an angle of radian, and the radius of a circle. 5. You are watching our nieces ride a Ferris wheel. Eplain how ou could do a mental calculation to estimate their angular speed. 6. Refer to Problem 5. Eplain how ou could do a mental calculation to estimate their linear speed. Find the degree measure of each of the angles in Problems 7, keeping in mind that an angle of one complete rotation corresponds to rotation 8. 5 rotation 9. 4 rotation rotation. 8 rotations. 6 rotations Find the radian measure of a central angle opposite an arc s in a circle of radius r, where r and s are as given in Problems r 4 centimeters, s 4 centimeters 4. r 8 inches, s 6 inches 5. r feet, s 30 feet 6. r 8 meters, s 7 meters 7. r m, s 5 cm 8. r ft, s 3 inches Find the radian measure of each angle in Problems 9 4, keeping in mind that an angle of one complete rotation corresponds to radians rotation 0. 6 rotation. 4 rotation rotation rotations 8 rotations 3 3 Find the eact radian measure, in terms of, of each angle in Problems , 60, 90, 0, 50, , 0, 80, 40, 300, , 90, 35, , 80, 70, , 44, 6, 88, , 7, 08, 44, 80 Find the eact degree measure of each angle in Problems , 3,, 3, 5 3, 3,, 4 3, 5 3, 6, ,, 3, 4,, 3 4, , 5, 3 5, 4 5, 5, 4 5, 6 5, 8 5, In Problems 37 4, determine whether the statement is true or false. If true, eplain wh. If false, give a countereample. 37. If two angles in standard position have the same measure, then the are coterminal. 38. If two angles in standard position are coterminal, then the have the same measure. 39. If two positive angles are complementar, then both are acute. 40. If two positive angles are supplementar, then one is obtuse and the other is acute. 4. If the terminal side of an angle in standard position lies in quadrant I, then the angle is positive. 4. If the initial and terminal sides of an angle coincide, then the measure of the angle is zero. Convert each angle in Problems to decimal degrees to three decimal places

14 476 CHAPTER 5 TRIGONOMETRIC FUNCTIONS Convert each angle in Problems to degree minute second form Find the radian measure to three decimal places for each angle in Problems Find the degree measure to two decimal places for each angle in Problems Indicate whether each angle in Problems is a first-, second-, third-, or fourth-quadrant angle or a quadrantal angle. All angles are in standard position in a rectangular coordinate sstem. (A sketch ma be of help in some problems.) Verball describe the meaning of a central angle in a circle with radian measure. 88. Verball describe the meaning of an angle with degree measure. In Problems 89 94, find all angles in degree measure that satisf the given conditions and is coterminal with and is coterminal with and is coterminal with and is coterminal with and is coterminal with and is coterminal with 35 In Problems 95 00, find all angles in radian measure that satisf the given conditions and is coterminal with / and is coterminal with 5/ and is coterminal with 7/ and is coterminal with / and is coterminal with / and is coterminal with 3/ 0. An arc of a circle of radius r has length s and central angle of radian measure. A second arc of the same circle has length s and central angle of radian measure. Show that 0. Refer to Problem 0. If and are measured in degrees, is the equation valid? Eplain. 03. Justif the following rule of thumb: Seven nautical miles equal 8 statute miles. [Hint: See Eample 6.] 04. Justif the following rule of thumb: A speed of 500 ards in 3 minutes equals 5 knots [Hint: See Eample 6.] APPLICATIONS s s s s 05. ANGULAR SPEED A wheel with diameter 6 feet makes 00 revolutions per minute. Find the angular speed (in radians per second) and the linear speed (in feet per second) of a point on the rim. 06. ANGULAR SPEED A point on the rim of a wheel with diameter 6 feet has a linear speed of 00 feet per second. Find the angular speed (in radians per second) and the number of revolutions per minute. 07. RADIAN MEASURE What is the radian measure of the larger angle made b the hands of a clock at 4:30? Epress the answer eactl in terms of. 08. RADIAN MEASURE What is the radian measure of the smaller angle made b the hands of a clock at :30? Epress the answer eactl in terms of. 09. ENGINEERING Through how man radians does a pulle of 0-centimeter diameter turn when 0 meters of rope are pulled through it without slippage?

15 SECTION 5 Angles and Their Measure ENGINEERING Through how man radians does a pulle of 6-inch diameter turn when 4 feet of rope are pulled through it without slippage?. ASTRONOMY A line from the sun to the Earth sweeps out an angle of how man radians in week? Assume the Earth s orbit is circular and there are 5 weeks in a ear. Epress the answer in terms of and as a decimal to two decimal places.. ASTRONOMY A line from the center of the Earth to the equator sweeps out an angle of how man radians in 9 hours? Epress the answer in terms of and as a decimal to two decimal places. 3. ENGINEERING A trail bike has a front wheel with a diameter of 40 centimeters and a back wheel of diameter 60 centimeters. Through what angle in radians does the front wheel turn if the back wheel turns through 8 radians? 4. ENGINEERING In Problem 3, through what angle in radians will the back wheel turn if the front wheel turns through 5 radians? 5. ANGULAR SPEED If the trail bike of Problem 3 travels at a speed of 0 kilometers per hour, find the angular speed (in radians per second) of each wheel. 6. ANGULAR SPEED If a car travels at a speed of 60 miles per hour, find the angular speed (in radians per second) of a tire that has a diameter of feet. In Problems 7 0, each pair of cities lies nearl on the same meridian. Ignore the small difference in longitude. 7. NAVIGATION Find the distance (to the nearest nautical mile) from Havana, Cuba [ 3 08 N/8 3 W], to Cleveland, Ohio [ 4 30 N/8 4 W]. 8. NAVIGATION Find the distance (to the nearest nautical mile) from Indianapolis, Indiana [ N/86 7 W], to Managua, Nicaragua [ 06 N/86 8 W]. 9. NAVIGATION A plane flies from Lima, Peru [ 06 S/76 55 W], to Washington, D.C. [ N/77 0 W], in 6 hours. Find the plane s linear speed (to the nearest knot) and angular speed (to the nearest tenth of a degree per hour). 0. NAVIGATION A plane flies from Jakarta, Indonesia [6 08 S/06 45 E], to Ulaanbaatar, Mongolia [47 55 N/ E], in 7 hours. Find the plane s linear speed (to the nearest knot) and angular speed (to the nearest tenth of a degree per hour).. CIRCUMFERENCE OF THE EARTH The earl Greeks used the proportion s/c /360, where s is an arc length on a circle, is degree measure of the corresponding central angle, and C is the circumference of the circle (C r). Eratosthenes (40 B.C.), in his famous calculation of the circumference of the Earth, reasoned as follows: He knew at Sene (now Aswan) during the summer solstice the noon sun was directl overhead and shined on the water straight down a deep well. On the same da at the same time, 5,000 stadia (appro. 500 miles) due north in Aleandria, sun ras crossed a vertical pole at an angle of 7.5 as indicated in the figure. Carr out Eratosthenes calculation for the circumference of the Earth to the nearest thousand miles. (The current calculation for the equatorial circumference is 4,90 miles.) Earth Aleandria 7.5 Sene Well Sun ras. CIRCUMFERENCE OF THE EARTH Repeat Problem with the sun crossing the vertical pole in Aleandria at CIRCUMFERENCE OF THE EARTH In Problem, verball eplain how in the figure was determined. 4. CIRCUMFERENCE OF THE EARTH Verball eplain how the radius, surface area, and volume of the Earth can be determined from the result of Problem. The arc length on a circle is eas to compute if the corresponding central angle is given in radians and the radius of the circle is known (s r). If the radius of a circle is large and a central angle is small, then an arc length is often used to approimate the length of the corresponding chord as shown in the figure. If an angle is given in degree measure, converting to radian measure first ma be helpful in certain problems. This information will be useful in Problems 5 8. c s r 5. ASTRONOMY The sun is about mi from the Earth. If the angle subtended b the diameter of the sun on the surface of the Earth is radians, approimatel what is the diameter of the sun to the nearest thousand miles in standard decimal notation? r 6. ASTRONOMY The moon is about 38,000 kilometers from the Earth. If the angle subtended b the diameter of the moon on c s

16 478 CHAPTER 5 TRIGONOMETRIC FUNCTIONS the surface of the Earth is radians, approimatel what is the diameter of the moon to the nearest hundred kilometers? 7. PHOTOGRAPHY The angle of view of a,000-millimeter telephoto lens is.5. At 750 feet, what is the width of the field of view to the nearest foot? 8. PHOTOGRAPHY The angle of view of a 300-millimeter lens is 8. At 500 feet, what is the width of the field of view to the nearest foot? 5- Trigonometric Functions: A Unit Circle Approach Z The Wrapping Function Z Defining the Trigonometric Functions Z Graphing the Trigonometric Functions In this section we introduce the si trigonometric functions in terms of the coordinates of points on the unit circle. Z The Wrapping Function P 0 v (, 0) u Consider a positive angle in standard position, and let P denote the point of intersection of the terminal side of with the unit circle u v (Fig. ).* Let denote the length of the arc opposite on the unit circle. Because the unit circle has radius r, the radian measure of is given b r radians Z Figure In other words, on the unit circle, the radian measure of a positive angle is equal to the length of the intercepted arc; similarl, on the unit circle, the radian measure of a negative angle is equal to the negative of the length of the intercepted arc. Because, we ma consider the real number to be the name of the angle, when convenient. The function W that associates with each real number the point W() P is called the wrapping function. The point P is called a circular point. *We use the variables u and v instead of and so that can be used without ambiguit as an independent variable in defining the wrapping function and the trigonometric functions.

17 SECTION 5 Trigonometric Functions: A Unit Circle Approach 479 (, 0) v (0, ) 0, (, 0) 3 (0, ) Z Figure Circular points on the coordinate aes. u Consider, for eample, the angle in standard position that has radian measure /. Its terminal side intersects the unit circle at the point (0, ). Therefore, W(/) (0, ). Similarl, we can find the circular point associated with an angle that is an integer multiple of / (Fig. ). W(0) (, 0) W a b (0, ) W() (, 0) W a 3 b (0, ) W() (, 0) ZZZ EXPLORE-DISCUSS The name wrapping function stems from visualizing the correspondence as a wrapping of the real number line (shown in blue in Fig. 3) around the unit circle the positive real ais is wrapped counterclockwise, and the negative real ais is wrapped clockwise so that each real number is paired with a unique circular point. v v v 0 (, 0) u 3 0 (, 0) u 3 0 (, 0) u Z Figure 3 The wrapping function. (A) Eplain wh the wrapping function is not one-to-one. (B) In which quadrant is the circular point W()? W(0)? W(00)? v P 6 (a, b) 0 (, 0) P Z Figure 4 u Given a real number, it is difficult, in general, to find the coordinates (a, b) of the circular point W() that is associated with. (It is trigonometr that overcomes this difficult.) For certain real numbers, however, we can find the coordinates (a, b) of W() b using simple geometric facts. For eample, consider /6 and let P denote the circular point W() (a, b) that is associated with. Let P be the reflection of P through the u ais (Fig. 4).

18 480 CHAPTER 5 TRIGONOMETRIC FUNCTIONS Then triangle OPP is equiangular (each angle has measure /3 radians or 60 ) and thus equilateral. Therefore pp, so b /. Because (a, b) lies on the unit circle, we solve for a: a b a a b Substitute b. Subtract 4 from both sides. a 3 4 a 3 Take square roots. a 3 must be discarded. (Wh?) Therefore, W a 3 b a 6, b EXAMPLE Finding Coordinates of Circular Points Find the coordinates of the following circular points: (A) W(/) (B) W(5/) (C) W(/3) (D) W(7/6) (E) W(/4) (, 0) Z Figure v v (0, ) 5 (0, ) 3 (, 0) v u 6 u (, 0) u SOLUTIONS (A) Because the circumference of the unit circle is, / is the radian measure of a negative angle that is 4 of a complete clockwise rotation. Therefore, W(/) (0, ) (Fig. 5). (B) Starting at (, 0) and proceeding counterclockwise, we count quarter-circle steps, /, /, 3/, 4/, and end at 5/. Therefore, the circular point is on the positive vertical ais, and W(5/) (0, ) (see Fig. 5). (C) The circular point W(/3) is the reflection of the point W(/6) (3/, /) through the line u v. To reflect a point through the line u v, ou interchange its coordinates (see Problem 45, in Eercises 5-). Therefore, W(/3) (/, 3/) (Fig. 6). (D) The circular point W(7/6) is the reflection of the point W(/6) (3/, /) through the origin. To reflect a point through the origin ou change the sign of each coordinate (see Section -4). Therefore, W(7/6) (3/, /) (see Fig. 6). (E) The circular point W(/4) (a, b) lies on the line u v, so a b (see Fig. 7 on the net page). Z Figure 6

19 SECTION 5 Trigonometric Functions: A Unit Circle Approach 48 v Since (a, b) lies on the unit circle, (0, ) v u (a, b) 4 (, 0) u a b a a Substitute b a. Divide both sides b. Take the square root of both sides. a a is impossible. (Wh?) Z Figure 7 Therefore, W(/4) (/, / ) (Fig. 7). MATCHED PROBLEM Find the coordinates of the following circular points: (A) W(3) (B) W(7/) (C) W(5/6) (D) W(/3) (E) W(5/4) Some ke results from Eample are summarized in Figure 8. If is an integer multiple of /6 or /4, then the coordinates of W() can be determined easil from Figure 8 b using smmetr properties. For eample, change the sign of the first coordinate of the three points in Quadrant I to obtain the coordinates of their reflections through the v ais in Quadrant II. Similarl, change the sign of both coordinates of the three points in Quadrant I to obtain the coordinates of their reflections through the origin in Quadrant III. And change the sign of the second coordinate of the three points in Quadrant I to obtain the coordinates of their reflections through the u ais in Quadrant IV. Z COORDINATES OF KEY CIRCULAR POINTS v (0, ) ( ) ( ) ( ), 3, 3, (, 0) u Z Figure 8

20 48 CHAPTER 5 TRIGONOMETRIC FUNCTIONS ZZZ EXPLORE-DISCUSS An effective memor aid for recalling the coordinates of the ke circular points in Figure 8 can be created b writing the coordinates of the circular points W(0), W(/6), W(/4), W(/3), and W(/), keeping this order, in a form where each numerator is the square root of an appropriate number and each denominator is. For eample, W(0) (, 0) (4/, 0/). Describe the pattern that results. Z Defining the Trigonometric Functions We define the trigonometric functions in terms of the coordinates of points on the unit circle. This suggests that the trigonometric functions are useful in analzing circular motion, for eample, of satellites, DVD plaers, generators, wheels, and propellers. While true, we will also discover that these functions have man applications that are apparentl unrelated to rotar motion. There are si trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. The values of these functions at a real number are denoted b sin, cos, tan, cot, sec, and csc, respectivel. Z DEFINITION Trigonometric Functions Let be a real number and let (a, b) be the coordinates of the circular point W() that lies on the terminal side of the angle with radian measure. Then: sin b csc b b 0 (a, b) cos a tan b a a 0 sec a cot a b a 0 b 0 W() rad units arc length (, 0) REMARKS:. Note that sin and cos are the second and first coordinates, respectivel, of the point (a, b) on the unit circle.. We assume in Definition that (a, b) is the point on the unit circle that lies on the terminal side of the angle with radian measure. More generall, however, if (a, b) is the point on that terminal side that lies on the circle of radius r 7 0, then: a b r aa r b a b r b Divide both sides b r. So a a lies on the unit circle. r, b r b

21 SECTION 5 Trigonometric Functions: A Unit Circle Approach 483 Therefore (a/r, b/r) is the point on the terminal side of the angle with radian measure (see Problems 47 and 48 in Eercises 5-) that lies on the unit circle (Fig. 9). rad (a, b) (, 0) a, b ( r r ) (r, 0) Z Figure 9 B Definition, sin b r csc r b b 0 cos a r sec r a a 0 tan b a a 0 cot a b b 0 Note that these formulas coincide with those of Definition when r. EXAMPLE Evaluating Trigonometric Functions Find the values of all si trigonometric functions of the angle if (A) W() a 3 5, 4 5 b. (B) The terminal side of contains the point (60, ). SOLUTIONS (A) Note that W() is indeed on the unit circle because B a3 5 b a 4 5 b 9 6 B B 5

22 484 CHAPTER 5 TRIGONOMETRIC FUNCTIONS Using Definition, with a 3 5 and b 4 5, sin b 4 5 csc b 5 4 cos a 3 5 tan b a 4/5 3/5 4 3 sec a 5 3 cot a b 3/5 4/5 3 4 (B) The distance r from (60, ) to (0, 0) is (60) () 3,600 3,7 6 Using Remark following Definition, with a 60, b, and r 6: sin b r 6 cos a r 60 6 tan b a 60 csc r b 6 sec r a 6 60 cot a b 60 ZZZ CAUTION ZZZ Alwas check that values of sin and cos are numbers that are between (or equal to) and, as implied b Definition. Note in particular that this is the case in Eample. MATCHED PROBLEM Find the values of all si trigonometric functions of the angle if (A) W() a 3, 5 3 b. (B) The terminal side of contains the point (3, 84). The domain of both the sine and cosine functions is the set of real numbers R. The range of both the sine and cosine functions is [, ]. This is the set of numbers assumed b b, for sine, and a, for cosine, as the circular point (a, b) moves around the unit circle. The domain of cosecant is the set of real numbers such that b in

23 SECTION 5 Trigonometric Functions: A Unit Circle Approach 485 W() (a, b) is not 0. Similar restrictions are made on the domains of the other three trigonometric functions. We will have more to sa about the domains and ranges of all si trigonometric functions in subsequent sections. Note from Definition that csc is the reciprocal of sin, provided that sin 0. Therefore sin is the reciprocal of csc. Similarl, cos and sec are reciprocals of each other, as are tan and cot. We call these useful facts the reciprocal identities. Z RECIPROCAL IDENTITIES For an real number: csc sin sec cos cot tan sin 0 cos 0 tan 0 In Eample we were able to give a simple geometric argument to find, for eample, that the coordinates of W(7/6) are (3/, /). Therefore, sin (7/6) / and cos (7/6) 3/. These eact values correspond to the approimations given b a calculator [Fig. 0(a)]. For most values of, however, simple geometric arguments fail to give the eact coordinates of W(). But a calculator, set in radian mode, can be used to give approimations. For eample, if /7, then W(/7) (0.90, 0.434) [Fig. 0(b)]. Z Figure 0 (a) (b) (c) Most calculators have function kes for the sine, cosine, and tangent functions, but not for the cotangent, secant, and cosecant. Because the cotangent, secant, and cosecant are the reciprocals of the tangent, cosine, and sine, respectivel, the can be evaluated easil. For eample, cot (/7) /tan (/7).077 [Fig. 0(c)]. Do not use the calculator function kes marked sin, cos, or tan for this purpose these kes are used to evaluate the inverse trigonometric functions of Section 5-6, not reciprocals.

24 486 CHAPTER 5 TRIGONOMETRIC FUNCTIONS EXAMPLE 3 Calculator Evaluation Evaluate to four significant digits. (A) tan.5 (C) sec (/) (B) csc (6.7) (D) The coordinates (a, b) of W() SOLUTIONS (A) (B) tan csc (6.7) /sin (6.7) (C) sec (/) /cos (/).035 (D) W() (cos, sin ) (0.5403, 0.845) MATCHED PROBLEM 3 Evaluate to four significant digits. (A) cot (8.5) (C) csc (4.67) (B) sec (7/8) (D) The coordinates (a, b) of W(00) Z Graphing the Trigonometric Functions The graph of sin is the set of all ordered pairs (, ) of real numbers that satisf the equation. Because sin, b Definition, is the second coordinate of the circular point W(), our knowledge of the coordinates of certain circular points (Table ) gives the following solutions to sin : 0, 0; /, ;, 0; and 3/,.

25 SECTION 5 Trigonometric Functions: A Unit Circle Approach 487 Table 0 3 W() (0, 0) (0, ) (, 0) (0, ) sin 0 0 As increases from 0 to /, the circular point W() moves on the circumference of the unit circle from (0, 0) to (0, ), and so sin [the second coordinate of W()] increases from 0 to. [But this increase is not linear: At /4 (the midpoint of 0 and /) the second coordinate of W() is /, not / (the midpoint of 0 and )]. Similarl, as increases from / to, the circular point W() moves on the circumference of the unit circle from (0, ) to (, 0), and so sin decreases from to 0. These observations are in agreement with the graph of sin, obtained from a graphing calculator in radian mode [Fig. (a)] (a) sin 3 (b) cos 3 (c) tan (d) csc (e) sec (f) cot Z Figure Figure shows the graphs of all si trigonometric functions from 0 to. Because the circular point W() coincides with the circular point W(0), the graphs of the si trigonometric functions from to 4 would be identical to the graphs shown in Figure. The functions sin and cos are bounded; their maimum values are and their minimum values are. The functions tan, cot, sec, and csc are unbounded; the have vertical asmptotes at the values of for which the are undefined. It is instructive to stud and compare the graphs of reciprocal pairs, for eample, cos and sec. Note that sec is undefined when cos equals 0, and that because the maimum positive value of cos is, the minimum positive value of sec is. We will stud the properties of trigonometric functions and their graphs in Section 5-4.

26 488 CHAPTER 5 TRIGONOMETRIC FUNCTIONS A graphing calculator can be used to interactivel eplore the relationship between the unit circle definition of the sine function and the graph of the sine function. Eplore-Discuss 3 provides the details. ZZZ EXPLORE-DISCUSS 3 With our graphing calculator, ou can illuminate the connection between the unit circle definition of the sine function and the graph of the sine function. Set our graphing calculator in radian and parametric modes (parametric equations are discussed in Section B-3, Appendi B). Make the entries as indicated in Figure to obtain the indicated graph ( is entered as Tma and Xma, / is entered as Xscl). Z Figure Use TRACE and move back and forth between the unit circle and the graph of the sine function for various values of T as T increases from 0 to. Discuss what happens in each case. Figure 3 illustrates the case for T 0. Z Figure 3 Repeat the eploration with Y T cos (T) EXAMPLE 4 Zeros and Turning Points Find the zeros and turning points of cos on the interval [/, 3/]. SOLUTION Recall that a turning point is a point on a graph that separates an increasing portion from a decreasing portion, or vice versa. A visual inspection of the graph of cos [Fig. 4(a)] suggests that (0, ) and (, ) are turning points, and that /, /,

27 SECTION 5 Trigonometric Functions: A Unit Circle Approach 489 and 3/ are zeros. These observations are confirmed b noting that as increases from / to 3/, the first coordinate of the circular point W() (that is, cos ) has a maimum value of (when 0), a minimum value of (when ), and has the value 0 when /, /, and 3/ [Fig. 4(b)]. 3 v / 3 3/ (, 0) 3 (0, ) 0 (, 0) u (0, ) (a) (b) Z Figure 4 MATCHED PROBLEM 4 Find all zeros and turning points of csc on the interval (0, 4). EXAMPLE 5 Solving a Trigonometric Equation Find all solutions of the equation sin to three decimal places. SOLUTION 3 Graph sin and and use the intersect command (Fig. 5). The solutions are.339, 0.55, and MATCHED PROBLEM 5 Z Figure 5 3 Find all solutions of the equation cot on the interval (0, ) to three decimal places. ZZZ CAUTION ZZZ A common cause of error is to forget to set a calculator in the correct mode, degree or radian, before graphing or evaluating a function. In radian mode, a calculator will give as the value of sin (/); in degree mode, it will give as the value of sin (/) [because (/).5708 ].

28 490 CHAPTER 5 TRIGONOMETRIC FUNCTIONS ANSWERS TO MATCHED PROBLEMS. (A) (, 0) (B) (0, ) (C) (3/, /) (D) (/, 3/) (E) (/, / ). (A) sin 5 csc 3 (B) 3 5 sin csc cos sec 3 cos tan 5 cot tan (A) 0.48 (B).08 (C).00 (D) (0.863, ) 4. Zeros: none; turning points: (/, ), (3/, ), (5/, ), (7/, ) , 3.46 sec 85 3 cot Eercises. What is the unit circle?. Describe the wrapping function, including its domain and range. 3. Eplain the connection between points on the unit circle and the si trigonometric functions. 4. Eplain wh the function sec is undefined for certain values of. 5. Eplain wh the graph of tan has vertical asmptotes at, 3, 5, Eplain wh ever point on the graph of cos lies on or between the lines and. In Problems 7, find the coordinates of each circular point. 7. W(3/) 8. W(5) 9. W(6) 0. W(5/). W(/4). W(/3) 3. W(/6) 4. W(/6) 5. W(/3) 6. W(/4) 7. W(/3) 8. W(/6) 9. W(3/4) 0. W(7/6). W(3/4). W(0/3) In Problems 3 38, use our answers to Problems 7 to give the eact value of the epression (if it eists). 3. sin (3/) 4. tan (5) 5. cos (6) 6. cot (5/) 7. sec (/4) 8. csc (/3) 9. tan (/6) 30. cos (/6) 3. sin (/3) 3. sec (/4) 33. csc (/3) 34. cot (/6) 35. cos (3/4) 36. tan (7/6) 37. cot (3/4) 38. sin (0/3) In Problems 39 4, find the eact value of the epression given that W() ( 3, 5 3 ). 39. sin 40. cot 4. sec 4. csc In Problems 43 46, find the eact value of the epression given that W() ( 5 7, 8 7). 43. cot 44. cos 45. csc 46. tan In Problems 47 50, find the eact value of the epression given that W() (0, ). 47. cos 48. sec 49. tan 50. csc In Problems 5 6, find the eact value of the epression given that is an angle in standard position and the terminal side of contains the indicated point. 5. cos ; (0, 5) 5. sin ; (9, 0) 53. tan ; (4, 3) 54. cot ; (6, 8)

29 55. csc ; (0.4, 0.09) 56. sec ; (0., 0.35) 97. If sec sec, then. 57. cot ; (8, 5) 58. tan ; (3, 4) 59. sec ; (9, 40) 60. csc ; (35, ) 6. sin ; (, ) 6. cos ; (, 3) In Problems 63 68, in which quadrants must W() lie so that: 63. cos tan sin sec cot csc 6 0 Evaluate Problems to four significant digits using a calculator set in radian mode. 69. cos sin tan (4.644) 7. sec (.555) 73. csc cot sin (cos 0.357) 76. cos (tan 5.83) 77. cos [csc (.408)] 78. sec [ cot (3.566)] Evaluate Problems to four significant digits using a calculator. Make sure our calculator is in the correct mode (degree or radian) for each problem. 79. sin tan cot 8. csc sin tan cot (43.4 ) 86. sec (47.39 ) 87. sin cos In Problems 89 94, determine whether the statement about the wrapping function W is true or false. Eplain. 89. The domain of the wrapping function is the set of all points on the unit circle. 90. The domain of the wrapping function is the set of all real numbers. 9. If W() W( ), then. 9. If, then W() W(). 93. If a and b are real numbers and a b, then there eists a real number such that W() (a, b). 94. If a and b are real numbers and a b, then there eists a unique real number such that W() (a, b). In Problems 95 04, determine whether the statement about the trigonometric functions is true or false. Eplain. 95. If is a real number, then cos is the reciprocal of sin. 96. If is a real number, then (cot ) (tan ). SECTION 5 Trigonometric Functions: A Unit Circle Approach If, then cos cos. 99. The functions sin and csc have the same domain. 00. The functions sin and cos have the same domain. 0. The graph of the function cos has infinitel man turning points. 0. The graph of the function tan has infinitel man turning points. 03. The graph of the function cot has infinitel man zeros. 04. The graph of the function csc has infinitel man zeros. In Problems 05 08, find all zeros and turning points of each function on [ 0, 4]. 05. sec 06. sin 07. tan 08. cot Determine the signs of a and b for the coordinates (a, b) of each circular point indicated in Problems First determine the quadrant in which each circular point lies. [Note: /.57, 3.4, 3/ 4.7, and 6.8. ] 09. W() 0. W(). W(3). W(4) 3. W(5) 4. W(7) 5. W(.5) 6. W(4.5) 7. W(6.) 8. W(.8) In Problems 9, for each equation find all solutions for 0, then write an epression that represents all solutions for the equation without an restrictions on. 9. W() (, 0) 0. W() (, 0). W() (/, / ). W() (/, / ) 3. Describe in words wh W() W( 4) for ever real number. 4. Describe in words wh W() W( 6) for ever real number. If W() (a, b), indicate whether the statements in Problems 5 30 are true or false. Sketching figures should help ou decide. 5. W( ) (a, b) 6. W( ) (a, b) 7. W() (a, b) 8. W() (a, b) 9. W( ) (a, b) 30. W( ) (a, b)

30 49 CHAPTER 5 TRIGONOMETRIC FUNCTIONS In Problems 3 34, find the value of each epression to one significant digit. Use onl the accompaning figure below, Definition, and a calculator as necessar for multiplication and division. Check our results b evaluating each directl on a calculator. 3. (A) sin 0.4 (B) cos 0.4 (C) tan (A) sin 0.8 (B) cos 0.8 (C) cot (A) sec. (B) tan 5.9 (C) cot (A) csc.5 (B) cot 5.6 (C) tan b Unit circle 46. Prove that the reflection of the point (a, b) through the line is the point (b, a) b verifing statements (A) and (B): (A) The line through (a, b) and (b, a) is perpendicular to the line. (B) The midpoint of (a, b) and (b, a) lies on the line. In Problems 47 and 48, consider the point P (a, b), where a and b are not both zero, and let O (0, 0). Ra OP S is defined b 47. Show that b a 0 is the equation of the line through O and P. 48. Refer to Problem 47. Show that ever point on OP S satisfies the equation of the line through O and P. APPLICATIONS OP S 5(ka, kb) k 06 If an n-sided regular polgon is inscribed in a circle of radius r, then it can be shown that the area of the polgon is given b a A nr sin n In Problems 35 38, in which quadrants are the statements true and wh? 35. sin 6 0 and cot cos 7 0 and tan cos 6 0 and sec sin 7 0 and csc 6 0 For which values of, 0, is each of Problems not defined? 39. cos 40. sin 4. tan 4. cot 43. sec 44. csc 45. Prove that the reflection of the point (a, b) through the line is the point (b, a) b verifing statements (A) and (B): (A) The line through (a, b) and (b, a) is perpendicular to the line. (B) The midpoint of (a, b) and (b, a) lies on the line. 5 6 In Problems 49 5, compute each area eactl and then to four significant digits. 49. n, r 5 meters 50. n 4, r 3 inches 5. n 3, r 4 inches 5. n 8, r 0 centimeters APPROXIMATING Problems 53 and 54 refer to a sequence of numbers generated as follows: 0 cos a n a n 53. Let a 0.5, and compute the first five terms of the sequence to si decimal places and compare the fifth term with / computed to si decimal places. 54. Repeat Problem 53, starting with a. a a a cos a a 3 a cos a o a n a n cos a n

31 SECTION 5 3 Solving Right Triangles Solving Right Triangles* Z Figure c a b A right triangle is a triangle with one 90 angle (Fig. ). If onl the angles of a right triangle are known, it is impossible to solve for the sides. (Wh?) But if we are given two sides, or one acute angle and a side, then it is possible to solve for the remaining three quantities. This process is called solving the right triangle. We use the trigonometric functions to solve right triangles. If a right triangle is located in the first quadrant as indicated b Figure, then, b similar triangles, the coordinates of the circular point Q are (a/c, b/c). c b Q (a, b) a (, 0) (a, 0) Z Figure Therefore, using the definition of the trigonometric functions, sin b/c and cos a/c. (Calculations using such trigonometric ratios are valid if is measured in either degrees or radians, provided our calculator is set in the correct mode in this section, we use degree measure.) All si trigonometric ratios are displaed in the bo. Z TRIGONOMETRIC RATIOS (a, b) sin b c csc c b c b a 0 90 cos a c tan b a sec c a cot a b *This section provides a significant application of trigonometric functions to real-world problems. However, it ma be postponed or omitted without loss of continuit, if desired. Some ma want to cover the section just before Sections 7- and 7-.

32 494 CHAPTER 5 TRIGONOMETRIC FUNCTIONS Side b is often referred to as the side opposite angle, a as the side adjacent to angle, and c as the hpotenuse. Using these designations for an arbitrar right triangle removed from a coordinate sstem, we have the following: Z RIGHT TRIANGLE RATIOS Hp Opp sin Opp Hp csc Hp Opp Adj 0 90 cos Adj Hp tan Opp Adj sec Hp Adj cot Adj Opp Table Angle to nearest Significant digits for side measure ZZZ EXPLORE-DISCUSS For a given value of, , eplain wh the value of each of the si trigonometric functions is independent of the size of the right triangle that contains. 0 or 0. 3 or or The use of the trigonometric ratios for right triangles is made clear in Eamples through 4. Regarding computational accurac, we use Table as a guide. (The table is also printed inside the cover of this book for eas reference.) We will use rather than in man places, realizing the accurac indicated in Table is all that is assumed. Another word of caution: When using our calculator be sure it is set in degree mode. EXAMPLE Solving a Right Triangle Solve the right triangle with c 6.5 feet and 3.. SOLUTION c 6.5 ft b First draw a figure and label the parts (Fig. 3): 3. a SOLVE FOR Z Figure and are complementar.