4.2 TRIGONOMETRIC FUNCTIONS : T UNIT C The Unit Circle

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292Chapter 4Trigonometry4.2 TRIGONOMETRIC FUNCTIONS: THE UNIT CIRCLEWhat you should learn Identify a unit circle and describeits relationship to real numbers. Evaluate trigonometric functionsusing the unit circle. Use the domain and period toevaluate sine and cosine functions. Use a calculator to evaluatetrigonometric functions.The Unit CircleThe two historical perspectives of trigonometry incorporate different methods forintroducing the trigonometric functions. Our first introduction to these functions isbased on the unit circle.Consider the unit circle given byx2 y 2 1Unit circleas shown in Figure 4.20.Why you should learn ity(0, 1)Trigonometric functions are used tomodel the movement of an oscillatingweight. For instance, in Exercise 60on page 298, the displacement fromequilibrium of an oscillating weightsuspended by a spring is modeled asa function of time.( 1, 0)(1, 0)x(0, 1)Richard Megna/Fundamental PhotographsFIGURE4.20Imagine that the real number line is wrapped around this circle, with positive numberscorresponding to a counterclockwise wrapping and negative numbers corresponding toa clockwise wrapping, as shown in Figure 4.21.yy(x , y)tt 0t 0t(1, 0)θ(1, 0)xt(x , y)FIGURExθt4.21As the real number line is wrapped around the unit circle, each real number tcorresponds to a point x, y on the circle. For example, the real number 0 correspondsto the point 1, 0 . Moreover, because the unit circle has a circumference of 2 , the realnumber 2 also corresponds to the point 1, 0 .In general, each real number t also corresponds to a central angle (in standardposition) whose radian measure is t. With this interpretation of t, the arc lengthformula s r (with r 1) indicates that the real number t is the (directional) lengthof the arc intercepted by the angle , given in radians.

Section 4.2Trigonometric Functions: The Unit Circle293The Trigonometric FunctionsFrom the preceding discussion, it follows that the coordinates x and y are two functionsof the real variable t. You can use these coordinates to define the six trigonometricfunctions of t.sinecosecantcosinesecanttangentcotangentThese six functions are normally abbreviated sin, csc, cos, sec, tan, and cot, respectively.Definitions of Trigonometric FunctionsLet t be a real number and let x, y be the point on the unit circle correspondingto t.Note in the definition at theright that the functions in thesecond row are the reciprocalsof the corresponding functionsin the first row.sin t y1csc t ,yy(0, 1)( 2,222( 1, 0)( 2,222 ())2,222x(1, 0)((0, 1)2,2) 22)4.22FIGUREy 0cos t xytan t ,x1sec t , x 0xxcot t , y 0yx 0In the definitions of the trigonometric functions, note that the tangent and secantare not defined when x 0. For instance, because t 2 corresponds to x, y 0, 1 , it follows that tan 2 and sec 2 are undefined. Similarly, thecotangent and cosecant are not defined when y 0. For instance, because t 0corresponds to x, y 1, 0 , cot 0 and csc 0 are undefined.In Figure 4.22, the unit circle has been divided into eight equal arcs, correspondingto t-values of 3 5 3 7 0, , , , , , , , and 2 .4 2 44 2 4Similarly, in Figure 4.23, the unit circle has been divided into 12 equal arcs,corresponding to t-values of 2 5 7 4 3 5 11 , and 2 .0, , , , , , , , , , ,6 3 2 3 66 3 2 3 622,also lies22on the line y x. So, substituting x for y in the equation of the unit circle producesthe following.To verify the points on the unit circle in Figure 4.22, note thaty( 21 , 23 )( 23 , 21 )( 1, 0)( 3,2( 21 21 ,FIGURE(0, 1)( 21 , 23 )( 23 , 21 )(1, 0)x4.2332( 21 , 23 )) (0, 1) ( 3 , 1 )222x2 1x2 Because the point is in the first quadrant, x 12x 222and because y x, you also22. You can use similar reasoning to verify the rest of the points in2Figure 4.22 and the points in Figure 4.23.Using the x, y coordinates in Figures 4.22 and 4.23, you can evaluate the trigonometric functions for common t-values. This procedure is demonstrated in Examples 1,2, and 3. You should study and learn these exact function values for common t-valuesbecause they will help you in later sections to perform calculations.have y ) x2 x2 1

294Chapter 4TrigonometryExample 1You can review dividingfractions and rationalizingdenominators in Appendix A.1and Appendix A.2, respectively.Evaluating Trigonometric FunctionsEvaluate the six trigonometric functions at each real number. 5 a. t b. t c. t 0d. t 64SolutionFor each t-value, begin by finding the corresponding point x, y on the unit circle. Thenuse the definitions of trigonometric functions listed on page 293.a. t b. t corresponds to the point x, y 63 1, .2 2sin1 y 62csc1 1 26y1 2cos 3 x 62sec2 3 12 6x33tan1 21 y3 6x3 233cot x3 2 6y1 235 22.corresponds to the point x, y , 422sin25 y 42csc5 12 4y22cos5 2 x 42sec25 1 4x22tan5 y 4x cot5 x 4y 2 2 12 22 2 12 2c. t 0 corresponds to the point x, y 1, 0 .sin 0 y 0csc 0 1is undefined.ycos 0 x 1sec 0 1 1 1x1cot 0 xis undefined.yy 0 0x 1tan 0 d. t corresponds to the point x, y 1, 0 .sin y 0csc 1is undefined.ycos x 1sec 11 1x 1cot xis undefined.ytan y0 0x 1Now try Exercise 23.

Section 4.2Example 2Trigonometric Functions: The Unit Circle295Evaluating Trigonometric Functions Evaluate the six trigonometric functions at t .3SolutionMoving clockwise around the unit circle, it follows that t 3 corresponds to thepoint x, y 1 2, 3 2 .sin 3 32csc 22 3 3331 32sec 23cot 1 2 31 3 3 233cos tan 3 2 31 23Now try Exercise 33.Domain and Period of Sine and Cosiney(0, 1)(1, 0)( 1, 0)x 1 y 1The domain of the sine and cosine functions is the set of all real numbers. To determinethe range of these two functions, consider the unit circle shown in Figure 4.24. Bydefinition, sin t y and cos t x. Because x, y is on the unit circle, you know that 1 " y " 1 and 1 " x " 1. So, the values of sine and cosine also range between 1 and 1. 1 ""1 1 " sin t " 1(0, 1)and 1 "x"1 1 " cos t " 1Adding 2 to each value of t in the interval 0, 2 completes a second revolutionaround the unit circle, as shown in Figure 4.25. The values of sin t 2 andcos t 2 correspond to those of sin t and cos t. Similar results can be obtained forrepeated revolutions (positive or negative) on the unit circle. This leads to the generalresult 1 x 1FIGUREy4.24sin t 2 n sin tandt t 3π 3π4, 4π π,2 2 2π , π2 4π, .y 2π , .t π π,4 4 2π , .cos t 2 n cos tfor any integer n and real number t. Functions that behave in such a repetitive (or cyclic)manner are called periodic.t π, 3π, .xt 0, 2π, .t 5π 5π4, 4 2π , .t FIGURE4.253π 3π,2 2t 74π , 74π 2π , . 2π , 32π 4π, .Definition of Periodic FunctionA function f is periodic if there exists a positive real number c such thatf t c f t for all t in the domain of f. The smallest number c for which f is periodic iscalled the period of f.

296Chapter 4TrigonometryRecall from Section 1.5 that a function f is even if f t f t , and is odd iff t f t .Even and Odd Trigonometric FunctionsThe cosine and secant functions are even.cos t cos tsec t sec tThe sine, cosecant, tangent, and cotangent functions are odd.sin t sin tcsc t csc ttan t tan tcot t cot tExample 3Using the Period to Evaluate the Sine and Cosine13 113 2 , you have sin sin 2 sin .6666627 b. Because 4 , you have22a. BecauseFrom the definition of periodicfunction, it follows that thesine and cosine functions areperiodic and have a period of2 . The other four trigonometricfunctions are also periodic, andwill be discussed further inSection 4.6.cos 7 cos 4 cos 0.22244c. For sin t , sin t because the sine function is odd.55Now try Exercise 37.Evaluating Trigonometric Functions with a CalculatorT E C H N O LO G YWhen evaluating trigonometricfunctions with a calculator,remember to enclose all fractionalangle measures in parentheses.For instance, if you want toevaluate sin t for t !/6, youshould enterSIN #6 ENTER.These keystrokes yield the correctvalue of 0.5. Note that somecalculators automatically place aleft parenthesis after trigonometricfunctions. Check the user’s guidefor your calculator for specifickeystrokes on how to evaluatetrigonometric functions.When evaluating a trigonometric function with a calculator, you need to set thecalculator to the desired mode of measurement (degree or radian).Most calculators do not have keys for the cosecant, secant, and cotangentfunctions. To evaluate these functions, you can use the x 1 key with their respectivereciprocal functions sine, cosine, and tangent. For instance, to evaluate csc 8 , usethe fact thatcsc 1 8sin 8 and enter the following keystroke sequence in radian mode. SINExample 4Function2 a. sin3b. cot 1.5 #8 x 1ENTERDisplay 2.6131259Using a CalculatorModeCalculator KeystrokesRadianSINRadian TANNow try Exercise 55.2 3#1.5 Display ENTER0.8660254 x 10.0709148ENTER

Section 4.2EXERCISES4.2Trigonometric Functions: The Unit Circle297See www.CalcChat.com for worked-out solutions to odd-numbered exercises.VOCABULARY: Fill in the blanks.1. Each real number t corresponds to a point x, y on the .2. A function f is if there exists a positive real number c such that f t c f t for all t in the domain of f.3. The smallest number c for which a function f is periodic is called the of f.4. A function f is if f t f t and if f t f t .SKILLS AND APPLICATIONSIn Exercises 5–8, determine the exact values of the sixtrigonometric functions of the real number t.y5.6.(12 5,13 13(y( 178 , 1715 (tθθ t x11 63 25. t 223. t y2 34 29. t 3y8.ttθθ31. t xx(( 45 , 35(12,135 13(In Exercises 9–16, find the point x, y on the unit circle thatcorresponds to the real number t. 2 11. t 45 13. t 64 15. t 39. t 10. t 33 14. t 45 16. t 312. t In Exercises 17–26, evaluate (if possible) the sine, cosine, andtangent of the real number.17. t 4 67 21. t 419. t 18. t 3 44 22. t 320. t 5 326. t 2 In Exercises 27–34, evaluate (if possible) the six trigonometricfunctions of the real number.x5 67 30. t 428. t 27. t 7.24. t 3 433. t 32. t 23 234. t In Exercises 35– 42, evaluate the trigonometric functionusing its period as an aid.35. sin 4 36. cos 3 7 37. cos338. sin9 440. sin19 639. cos17 441. sin 8 342. cos 9 4In Exercises 43–48, use the value of the trigonometricfunction to evaluate the indicated functions.143. sin t 2(a) sin t 344. sin t 8(a) sin t(b) csc t 145. cos t 5(a) cos t(b) sec t (b) csc t346. cos t 4(a) cos t (b) sec t 447. sin t 5(a) sin t (b) sin t 448. cos t 5(a) cos t (b) cos t

298Chapter 4TrigonometryIn Exercises 49–58, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places.(Be sure the calculator is set in the correct angle mode.)49. sin 450. tan 351. cot 452. csc2 353. cos 1.7 55. csc 0.857. sec 22.8 54. cos 2.5 56. sec 1.858. cot 0.9 59. HARMONIC MOTION The displacement fromequilibrium of an oscillating weight suspended by aspring is given by y t 41 cos 6t, where y is thedisplacement (in feet) and t is the time (in seconds). Findthe displacements when (a) t 0, (b) t 14, and (c) t 21.60. HARMONIC MOTION The displacement fromequilibrium of an oscillating weight suspended by aspring and subject to the damping effect of friction isgiven by y t 41e t cos 6t, where y is the displacement(in feet) and t is the time (in seconds).067. Verify that cos 2t 2 cos t by approximating cos 1.5and 2 cos 0.75.68. Verify that sin t1 t2 sin t1 sin t2 by approximatingsin 0.25, sin 0.75, and sin 1.69. THINK ABOUT IT Because f t sin t is an oddfunction and g t cos t is an even function, what canbe said about the function h t f t g t ?70. THINK ABOUT IT Because f t sin t andg t tan t are odd functions, what can be said aboutthe function h t f t g t ?71. GRAPHICAL ANALYSIS With your graphing utility inradian and parametric modes, enter the equationsX1T cos T and Y1T sin Tand use the following settings.(a) Complete the table.t(b) Make a conjecture about any relationship betweensin t1 and sin t1 .(c) Make a conjecture about any relationship betweencos t1 and cos t1 .66. Use the unit circle to verify that the cosine and secantfunctions are even and that the sine, cosecant, tangent,and cotangent functions are odd.1412341yTmin 0, Tmax 6.3, Tstep 0.1Xmin 1.5, Xmax 1.5, Xscl 1Ymin 1, Ymax 1, Yscl 1(a) Graph the entered equations and describe the graph.(b) Use the table feature of a graphing utility to approximate the time when the weight reaches equilibrium.(c) What appears to happen to the displacement as tincreases?(b) Use the trace feature to move the cursor around thegraph. What do the t-values represent? What do thex- and y-values represent?(c) What are the least and greatest values of x and y?EXPLORATIONTRUE OR FALSE? In Exercises 61– 64, determine whetherthe statement is true or false. Justify your answer.61. Because sin t sin t, it can be said that the sine ofa negative angle is a negative number.62. tan a tan a 6 72. CAPSTONE A student you are tutoring has used aunit circle divided into 8 equal parts to complete thetable for selected values of t. What is wrong?t0 4 2x1220y022165. Let x1, y1 and x2, y2 be points on the unit circlecorresponding to t t1 and t t1, respectively.sin t1220(a) Identify the symmetry of the points x1, y1 and x2, y2 .cos t0221220tan tUndef.10 1Undef.63. The real number 0 corresponds to the point 0, 1 on theunit circle.64. cos 7 cos 223 4 2222 22 10 1

Section 4.3Right Triangle Trigonometry2994.3 RIGHT TRIANGLE TRIGONOMETRYWhat you should learn Evaluate trigonometric functionsof acute angles. Use fundamental trigonometricidentities. Use a calculator to evaluatetrigonometric functions. Use trigonometric functions tomodel and solve real-life problems.The Six Trigonometric FunctionsOur second look at the trigonometric functions is from a right triangle perspective.Consider a right triangle, with one acute angle labeled , as shown in Figure 4.26.Relative to the angle , the three sides of the triangle are the hypotenuse, theopposite side (the side opposite the angle ), and the adjacent side (the side adjacent tothe angle ).HypotenuseSide opposite θWhy you should learn itTrigonometric functions are often usedto analyze real-life situations. Forinstance, in Exercise 76 on page 309,you can use trigonometric functionsto find the height of a helium-filledballoon.θSide adjacent to θFIGURE4.26Using the lengths of these three sides, you can form six ratios that define the sixtrigonometric functions of the acute angle .sine cosecant cosine secant tangent cotangentJoseph Sohm/Visions of America/CorbisIn the following definitions, it is important to see that 0 90 lies in the firstquadrant) and that for such angles the value of each trigonometric function is positive.Right Triangle Definitions of Trigonometric FunctionsLet be an acute angle of a right triangle. The six trigonometric functions of theangle are defined as follows. (Note that the functions in the second row are thereciprocals of the corresponding functions in the first row.)sin opphypcos adjhyptan oppadjcsc hypoppsec hypadjcot adjoppThe abbreviations opp, adj, and hyp represent the lengths of the three sides of aright triangle.opp the length of the side oppositeadj the length of the side adjacent tohyp the length of the hypotenuse

300Chapter 4TrigonometryExample 1tenuseUse the triangle in Figure 4.27 to find the values of the six trigonometric functions of .Solutionpo4HyBy the Pythagorean Theorem, hyp 2 opp 2 adj 2, it follows that42 32hyp θ 3FIGUREEvaluating Trigonometric Functions25 5.4.27So, the six trigonometric functions of opp 4 hyp 5csc hyp 5 opp 4cos adj3 hyp 5sec hyp 5 adj3tan opp 4 adj3cot adj3 .opp 4sinYou can review the PythagoreanTheorem in Section 1.1.HISTORICAL NOTEGeorg Joachim Rhaeticus(1514–1574) was the leadingTeutonic mathematicalastronomer of the 16th century.He was the first to define thetrigonometric functions as ratiosof the sides of a right triangle.areNow try Exercise 7.In Example 1, you were given the lengths of two sides of the right triangle, but notthe angle . Often, you will be asked to find the trigonometric functions of a given acuteangle . To do this, construct a right triangle having as one of its angles.Example 2Evaluating Trigonometric Functions of 45"Find the values of sin 45 , cos 45 , and tan 45 .Solution45 21Construct a right triangle having 45 as one of its acute angles, as shown in Figure 4.28.Choose the length of the adjacent side to be 1. From geometry, you know that the otheracute angle is also 45 . So, the triangle is isosceles and the length of the opposite sideis also 1. Using the Pythagorean Theorem, you find the length of the hypotenuse tobe 2.sin 45 opp21 hyp22cos 45 1adj2 hyp22tan 45 opp 1 1adj145 1FIGURE4.28Now try Exercise 23.

Section 4.3Example 3Because the angles 30 , 45 , and60 6, 4, and 3 occurfrequently in trigonometry, youshould learn to construct thetriangles shown in Figures 4.28and 4.29.301Right Triangle TrigonometryEvaluating Trigonometric Functions of 30" and 60"Use the equilateral triangle shown in Figure 4.29 to find the values of sin 60 ,cos 60 , sin 30 , and cos 30 .30 22360 1FIGURE14.29SolutionT E C H N O LO G YYou can use a calculator toconvert the answers in Example 3to decimals. However, theradical form is the exact valueand in most cases, the exactvalue is preferred.Use the Pythagorean Theorem and the equilateral triangle in Figure 4.29 to verify thelengths of the sides shown in the figure. For 60 , you have adj 1, opp 3, andhyp 2. So,sin 60 For3opp hyp2 30 , adj sin 30 cos 60 and1adj .hyp 23, opp 1, and hyp 2. So,opp 1 hyp 2cos 30 and3adj .hyp2Now try Exercise 27.Sines, Cosines, and Tangents of Special Anglessin 30 sin 1 62cos 30 cos 3 62tan 30 tan 3 63sin 45 sin 2 42cos 45 cos 2 42tan 45 tan 14sin 60 sin 3 32cos 60 cos 1 32tan 60 tan 33In the box, note that sin 30 12 cos 60 . This occurs because 30 and 60 arecomplementary angles. In general, it can be shown in 30 (b) cos 30 (c) tan 60 (d) cot 60 31. sin 60 132. sin 30 ,2(a) csc 30 3tan 30 3(b) cot 60 (c) cos 30 33. cos 13(a) sin(c) sec34. sec 5(a) cos(c) cot 90 35. cot ) 5(a) tan )(c) cot 90 ) 736. cos ( 4(a) sec ((c) cot ((d) cot 30 (b) tan(d) csc 90 (b) cot(d) sin(b) csc )(d) cos )(b) sin ((d) sin 90 ( In Exercises 37– 46, use trigonometric identities to transformthe left side of the equation into the right side!0 # ! / 2".37.38.39.40.41.42.43.44.tan cot 1cos sec 148. (a) tan 23.5 49. (a) sin 16.35 50. (a) cot 79.56 (b) cot 66.5 (b) csc 16.35 (b) sec 79.56 51. (a) cos 4 50% 15&52. (a) sec 42 12%53. (a) cot 11 15%(b) sec 4 50% 15&(b) csc 48 7%(b) tan 11 15%54. (a) sec 56 8% 10&55. (a) csc 32 40% 3&56. (a) sec 95 ' 20 32 (b) cos 56 8% 10&(b) tan 44 28% 16&(b) cot 95 ' 30 32 In Exercises 57– 62, find the values of # in degrees!0" # 90"" and radians !0 # ! / 2" without the aidof a calculator.57. (a) sin 2158. (a) cos 2259. (a) sec 260. (a) tan 32 361. (a) csc 3362. (a) cot 3(b) csc 2(b) tan 1(b) cot 1(b) cos 21(b) sin 22(b) sec 2In Exercises 63–66, solve for x, y, or r as indicated.63. Solve for y.64. Solve for x.3018y30 x60 tan ) cos ) sin )cot ) sin ) cos ) 1 sin 1 sin cos2 1 cos 1 cos sin2 sec tan sec tan 1sin2 cos2 2 sin2 1sincos45. csc seccossintan ( cot (46. csc2 (tan (In Exercises 47–56, use a calculator to evaluate each function.Round your answers to four decimal places. (Be sure thecalculator is in the correct angle mode.)47. (a) sin 10 307Right Triangle Trigonometry(b) cos 80 65. Solve for x.66. Solve for r.r3260 x2045 67. EMPIRE STATE BUILDING You are standing 45 metersfrom the base of the Empire State Building. Youestimate that the angle of elevation to the top of the 86thfloor (the observatory) is 82 . If the total height of thebuilding is another 123 meters above the 86th floor,what is the approximate height of the building? One ofyour friends is on the 86th floor. What is the distancebetween you and your friend?

308Chapter 4Trigonometry68. HEIGHT A six-foot person walks from the base of abroadcasting tower directly toward the tip of theshadow cast by the tower. When the person is 132 feetfrom the tower and 3 feet from the tip of the shadow, theperson’s shadow starts to appear beyond the tower’sshadow.72. HEIGHT OF A MOUNTAIN In traveling across flatland, you notice a mountain directly in front of you. Itsangle of elevation (to the peak) is 3.5 . After you drive13 miles closer to the mountain, the angle of elevationis 9 . Approximate the height of the mountain.(a) Draw a right triangle that gives a visualrepresentation of the problem. Show the knownquantities of the triangle and use a variable toindicate the height of the tower.3.5 13 mi(b) Use a trigonometric function to write an equationinvolving the unknown quantity.(c) What is the height of the tower?69. ANGLE OF ELEVATION You are skiing down amountain with a vertical height of 1500 feet. Thedistance from the top of the mountain to the base is3000 feet. What is the angle of elevation from the baseto the top of the mountain?70. WIDTH OF A RIVER A biologist wants to know thewidth w of a river so that instrume

Section 4.2 Trigonometric Functions: The Unit Circle 295 Evaluating Trigonometric Functions Evaluate the six trigonometric functions at Solution Moving clockwise around the unit circle,it follows that corresponds to the point Now try Exercise 33. Domain and Period of Sine and Cosine The domain of the sine an