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The Unit Circle 4.2 TRIGONOMETRIC FUNCTIONS

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The Unit Circle 4.2 TRIGONOMETRIC FUNCTIONS 292 Chapter 4 Trigonometry 4.2 TRIGONOMETRIC FUNCTIONS : T HE UNIT CIRCLE What you should learn The Unit Circle • Identify a unit circle and describe its relationship to real numbers. The two historical perspectives of trigonometry incorporate different methods for • Evaluate trigonometric functions introducing the trigonometric functions. Our first introduction to these functions is using the unit circle. based on the unit circle. • Use the domain and period to Consider the unit circle given by evaluate sine and cosine functions. x2 ϩ y 2 ϭ 1 Unit circle • Use a calculator to evaluate trigonometric functions. as shown in Figure 4.20. Why you should learn it y Trigonometric functions are used to (0, 1) model the movement of an oscillating weight. For instance, in Exercise 60 on page 298, the displacement from equilibrium of an oscillating weight x suspended by a spring is modeled as (− 1, 0) (1, 0) a function of time. (0,− 1) FIGURE 4.20 Imagine that the real number line is wrapped around this circle, with positive numbers corresponding to a counterclockwise wrapping and negative numbers corresponding to a clockwise wrapping, as shown in Figure 4.21. y y t > 0 (x , y ) t t < 0 t θ (1, 0) Richard Megna/Fundamental Photographs x x (1, 0) θ t (x , y ) t FIGURE 4.21 As the real number line is wrapped around the unit circle, each real number t corresponds to a point ͑x, y͒ on the circle. For example, the real number 0 corresponds to the point ͑1, 0 ͒. Moreover, because the unit circle has a circumference of 2␲, the real number 2␲ also corresponds to the point ͑1, 0 ͒. In general, each real number t also corresponds to a central angle ␪ (in standard position) whose radian measure is t. With this interpretation of t, the arc length formula s ϭ r␪ (with r ϭ 1) indicates that the real number t is the (directional) length of the arc intercepted by the angle ␪, given in radians. Section 4.2 Trigonometric Functions: The Unit Circle 293 The Trigonometric Functions From the preceding discussion, it follows that the coordinates x and y are two functions of the real variable t. You can use these coordinates to define the six trigonometric functions of t. sine cosecant cosine secant tangent cotangent These six functions are normally abbreviated sin, csc, cos, sec, tan, and cot, respectively. Definitions of Trigonometric Functions Let t be a real number and let ͑x, y͒ be the point on the unit circle corresponding Note in the definition at the to t. right that the functions in the second row are the reciprocals y sin t ϭ y cos t ϭ x tan t ϭ , x 0 of the corresponding functions x in the first row. 1 1 x csc t ϭ , y 0 sec t ϭ , x 0 cot t ϭ , y 0 y x y In the definitions of the trigonometric functions, note that the tangent and secant are not defined when x ϭ 0. For instance, because t ϭ ␲͞2 corresponds to y ͑x, y͒ ϭ ͑0, 1 ͒, it follows that tan ͑␲͞2͒ and sec ͑␲͞2͒ are undefined . Similarly, the (0, 1) cotangent and cosecant are not defined when y ϭ 0. For instance, because t ϭ 0 2 2 2 , 2 corresponds to ͑x, y͒ ϭ ͑1, 0 ͒, cot 0 and csc 0 are undefined . (− , ) ( 2 2 ) 2 2 In Figure 4.22, the unit circle has been divided into eight equal arcs, corresponding to t-values of x ␲ ␲ 3␲ 5␲ 3␲ 7␲ (− 1, 0) (1, 0) 0, , , , ␲, , , , and 2␲. 4 2 4 4 2 4 Similarly, in Figure 4.23, the unit circle has been divided into 12 equal arcs, 2 2 corresponding to t-values of − , − 2, − 2 (2 2 ) − ( 2 2 ) (0, 1) ␲ ␲ ␲ 2␲ 5␲ 7␲ 4␲ 3␲ 5␲ 11 ␲ 0, , , , , , ␲, , , , , , and 2␲. FIGURE 4.22 6 3 2 3 6 6 3 2 3 6 Ί2 Ί2 To verify the points on the unit circle in Figure 4.22, note that , also lies ΂ 2 2 ΃ y 1 , 3 (2 2 ) on the line y ϭ x. So, substituting x for y in the equation of the unit circle produces − 1 , 3 (0, 1) (2 2 ) the following. 3 , 1 − 3 , 1 (2 2 ) 1 Ί2 (2 2 ) x2 ϩ x2 ϭ 1 2x2 ϭ 1 x2 ϭ x ϭ ± 2 2 x Ί2 (− 1, 0) (1, 0) Because the point is in the first quadrant,x ϭ and because y ϭ x, you also 2 Ί2 have y ϭ . You can use similar reasoning to verify the rest of the points in (− 3 , − 1 ) 2 2 2 Figure 4.22 and the points in Figure 4.23. 1 , − 3 (2 2 ) Using the ͑x, y͒ coordinates in Figures 4.22 and 4.23, you can evaluate the trigono- − 1 , − 3 (0,− 1) metric functions for common t-values. This procedure is demonstrated in Examples 1, (2 2 ) 3 , − 1 (2 2 ) 2, and 3. You should study and learn these exact function values for common t-values FIGURE 4.23 because they will help you in later sections to perform calculations. 294 Chapter 4 Trigonometry Example 1 Evaluating Trigonometric Functions You can review dividing Evaluate the six trigonometric functions at each real number. fractions and rationalizing ␲ 5␲ a.t ϭ b.t ϭ c.t ϭ 0 d. t ϭ ␲ denominators in Appendix A.1 6 4 and Appendix A.2, respectively. Solution For each t-value, begin by finding the corresponding point ͑x, y͒ on the unit circle. Then use the definitions of trigonometric functions listed on page 293. ␲ Ί3 1 a. t ϭ corresponds to the point ͑x, y͒ ϭ , . 6 ΂ 2 2΃ ␲ 1 ␲ 1 1 sin ϭ y ϭ csc ϭ ϭ ϭ 2 6 2 6 y 1͞2 ␲ Ί3 ␲ 1 2 2Ί3 cos ϭ x ϭ sec ϭ ϭ ϭ 6 2 6 x Ί3 3 ␲ y 1͞2 1 Ί3 ␲ x Ί3͞2 tan ϭ ϭ ϭ ϭ cot ϭ ϭ ϭ Ί3 6 x Ί3͞2 Ί3 3 6 y 1͞2 5␲ Ί2 Ί2 b. t ϭ corresponds to the point ͑x, y͒ ϭ Ϫ , Ϫ . 4 ΂ 2 2 ΃ 5␲ Ί2 5␲ 1 2 sin ϭ y ϭ Ϫ csc ϭ ϭ Ϫ ϭ Ϫ Ί2 4 2 4 y Ί2 5␲ Ί2 5␲ 1 2 cos ϭ x ϭ Ϫ sec ϭ ϭ Ϫ ϭ Ϫ Ί2 4 2 4 x Ί2 5␲ y ϪΊ2͞2 5␲ x ϪΊ2͞2 tan ϭ ϭ ϭ 1 cot ϭ ϭ ϭ 1 4 x ϪΊ2͞2 4 y ϪΊ2͞2 c. t ϭ 0 corresponds to the point ͑x, y͒ ϭ ͑1, 0 ͒. 1 sin 0 ϭ y ϭ 0 csc 0 ϭ is undefined. y 1 1 cos 0 ϭ x ϭ 1 sec 0 ϭ ϭ ϭ 1 x 1 y 0 x tan 0 ϭ ϭ ϭ 0 cot 0 ϭ is undefined. x 1 y d. t ϭ ␲ corresponds to the point ͑x, y͒ ϭ ͑Ϫ1, 0 ͒. 1 sin ␲ ϭ y ϭ 0 csc ␲ ϭ is undefined. y 1 1 cos ␲ ϭ x ϭ Ϫ 1 sec ␲ ϭ ϭ ϭ Ϫ 1 x Ϫ1 y 0 x tan ␲ ϭ ϭ ϭ 0 cot ␲ ϭ is undefined. x Ϫ1 y Now try Exercise 23. Section 4.2 Trigonometric Functions: The Unit Circle 295 Example 2 Evaluating Trigonometric Functions ␲ Evaluate the six trigonometric functions at t ϭ Ϫ . 3 Solution Moving clockwise around the unit circle, it follows that t ϭ Ϫ ␲͞3 corresponds to the point ͑x, y͒ ϭ ͑1͞2, ϪΊ3͞2͒. ␲ Ί3 ␲ 2 2Ί3 sin Ϫ ϭ Ϫ csc Ϫ ϭ Ϫ ϭ Ϫ ΂ 3΃ 2 ΂ 3΃ Ί3 3 ␲ 1 ␲ cos Ϫ ϭ sec Ϫ ϭ 2 ΂ 3΃ 2 ΂ 3΃ ␲ ϪΊ3͞2 ␲ 1͞2 1 Ί3 tan Ϫ ϭ ϭ Ϫ Ί3 cot Ϫ ϭ ϭ Ϫ ϭ Ϫ ΂ 3΃ 1͞2 ΂ 3΃ ϪΊ3͞2 Ί3 3 Now try Exercise 33. Domain and Period of Sine and Cosine y The domain of the sine and cosine functions is the set of all real numbers. To determine (0, 1) the range of these two functions, consider the unit circle shown in Figure 4.24. By definition,sin t ϭ y and cos t ϭ x. Because ͑x, y͒ is on the unit circle, you know that Ϫ1 y 1 and Ϫ1 x 1. So, the values of sine and cosine also range between (−1, 0) (1, 0) Յ Յ Յ Յ x −1 ≤ y ≤ 1 Ϫ1 and 1. Ϫ1 Յ y Յ 1 Ϫ1 Յ x Յ 1 and (0, −1) Ϫ1 Յ sin t Յ 1 Ϫ1 Յ cos t Յ 1 Adding 2␲ to each value of t in the interval ͓0, 2 ␲͔ completes a second revolution −1 ≤ x ≤ 1 around the unit circle, as shown in Figure 4.25. The values of sin ͑t ϩ 2␲͒ and FIGURE 4.24 cos ͑t ϩ 2␲͒ correspond to those of sin t and cos t. Similar results can be obtained for repeated revolutions (positive or negative) on the unit circle. This leads to the general result sin ͑t ϩ 2␲n͒ ϭ sin t and π π π t = , + 2 π, + 4π , ... 2 2 y 2 cos ͑t ϩ 2␲ n͒ ϭ cos t π π π π t = 3 3 π t π 4 , 4 + 2 , ... = 4 , 4 + 2 , ... for any integer n and real number t. Functions that behave in such a repetitive (or cyclic) manner are called periodic. t = π , 3 π , ... x Definition of Periodic Function t = 0, 2π , ..
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