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4.2 TRIGONOMETRIC FUNCTIONS: THE UNIT CIRCLE Copyright © Cengage Learning. All rights reserved. What You Should Learn • Identify a unit circle and describe its relationship to real numbers. • Evaluate trigonometric functions using the unit circle. • Use the domain and period to evaluate sine and cosine functions. • Use a calculator to evaluate trigonometric functions. 2 The Unit Circle 3 The Unit Circle The unit circle x2 + y2 = 1 4 The Unit Circle Each real number t also corresponds to a central angle (in standard position) whose radian measure is t. The real number t is the (directional) length of the arc intercepted by the angle , given in radians. 5 The Trigonometric Functions 6 The Trigonometric Functions 7 The Trigonometric Functions The unit circle has been divided into eight equal arcs, corresponding to t-values of 8 The Trigonometric Functions The unit circle has been divided into 12 equal arcs, corresponding to t-values of 9 Example 1 – Evaluating Trigonometric Functions Evaluate the six trigonometric functions at each real number. a. b. c. t = 0 d. t = Solution: a. corresponds to the point (x, y) = . 10 Example 1 – Solution cont’d 11 Example 1 – Solution b. corresponds to the point (x, y) = cont’d . 12 Example 1 – Solution cont’d 13 Example 1 – Solution cont’d c. t = 0 corresponds to the point (x, y) = (1, 0). sin 0 = y = 0 cos 0 = x = 1 tan 0 = = =0 14 Example 1 – Solution cont’d is undefined. is undefined. 15 Example 1 – Solution cont’d d. t = corresponds to the point (x, y) = (–1, 0). sin = y = 0 cos = x = –1 tan = = =0 16 Example 1 – Solution cont’d is undefined. is undefined. 17 Domain and Period of Sine and Cosine 18 Domain and Period of Sine and Cosine The domain of the sine and cosine functions is the set of all real numbers. –1 y 1 –1 Sin t 1 –1 x 1 –1 cos t 1 19 Domain and Period of Sine and Cosine sin (t + 2 n) = sin t cos (t + 2 n) = cos t 20 Domain and Period of Sine and Cosine 21 Example 3 – Using the Period to Evaluate the Sine and Cosine a. Because b. Because , you have , you have 22 Example 3 – Using the Period to Evaluate the Sine and Cosine c. For odd. , cont’d because the sine function is 23