# 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