advertisement

4.2 – Trigonometric Functions: The Unit Circle Learning Target: be able to evaluate trigonometric functions using the unit circle. The Unit Circle – is a circle centered at the origin with a radius of 1. 𝑥2 + 𝑦2 = 1 Memorize the Unit Circle Shown Below: A Trick for memorizing the unit circle: https://www.youtube.com/watch?v=LE6dmczMc68 http://www.youtube.com/watch?v=1CiX AP8XaBg The Six Trigonometric Functions: Common Trig. And their inverses Values Sine Cosecant Cosine Secant Tangent Cotangent Definitions of Trigonometric Functions: 𝑦 sin 𝑡 = 𝑦 cos 𝑡 = 𝑥 tan 𝑡 = 𝑥 𝑥 1 1 cot 𝑡 = csc 𝑡 = sec 𝑡 = 𝑦 𝑦 𝑥 EX: Evaluate the six trigonometric functions at each real number. a.) 𝑡 = 2𝜋 3 b.) 𝑡 = 4𝜋 3 c.) 𝑡 = 2𝜋 a.) 𝑡 = 2𝜋 corresponds to the point 3 −1 √3 ( , ). 2 2 √3 sin 𝑡 = 𝑦 = 2 −1 cos 𝑡 = 𝑥 = 2 1 2 csc 𝑡 = = 𝑦 √3 2√3 = 3 1 2 sec 𝑡 = = 𝑥 −1 = −2 3⁄ √ 𝑦 2 tan 𝑡 = = 𝑥 −1⁄ 2 𝑥 1 cot 𝑡 = = 𝑦 −√3 √3 =− 3 √3 2 = ∙ 2 −1 = −√3 4𝜋 b.) 𝑡 = corresponds to the point 3 1 √3 (− , − ). 2 2 −√3 sin 𝑡 = 𝑦 = 2 1 cos 𝑡 = 𝑥 = − 2 1 −2 csc 𝑡 = = 𝑦 √3 −2√3 = 3 1 sec 𝑡 = = −2 𝑥 𝑦 tan 𝑡 = 𝑥 𝑥 1 cot 𝑡 = = 𝑦 √3 √3 = 3 2 √3 =− ∙− 2 1 = √3 c.) 𝑡 = 2𝜋 corresponds to the point (1,0). 1 1 csc 𝑡 = = 𝑦 0 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 1 1 cos 𝑡 = 𝑥 = 1 sec 𝑡 = = = 1 𝑥 1 𝑥 1 𝑦 0 cot 𝑡 = = tan 𝑡 = = = 0 𝑦 0 𝑥 1 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 EX: Evaluate the six trigonometric sin 𝑡 = 𝑦 = 0 functions of 𝑡 = − 2𝜋 3 . Since t is negative, we must find its positive coterminal angle. 2𝜋 2𝜋 6𝜋 4𝜋 𝑡=− + 2𝜋 = − + = 3 3 3 3 t now corresponds to the point −1 −√3 ( 2 , 2 ). This problem is now identical to part (b) in the previous example. Facts about Sine and Cosine Curves: - The domain of both sine and cosine functions is all real numbers. - The range of both sine and cosine is between -1 and 1. We can graph the curves to confirm this. - Both curves are periodic (repetitive in nature). Their period is 2𝜋. - The cosine and secant functions are even (symmetric to the y-axis). - The cosecant, tangent, and cotangent functions are odd (symmetric to the origin). EX: Find the following: a.) cos 9𝜋 3 b.) sin (− 11𝜋 2 ) 2 c.) If tan(𝑡) = , find tan(−𝑡) 3 a.) We must find a coterminal angle for 9𝜋 3 9𝜋 3 that is between 0 and 2𝜋. 𝜃 = − 2𝜋 = 9𝜋 3 − 6𝜋 3 = 𝜋. Theta is now equivalent to the point 1,0). cos 𝑡 = 𝑥 = −1. (- b.) −11𝜋 2 −11𝜋 + 2𝜋 = 2𝜋 = −7𝜋 2 + 2 4𝜋 2 = + 4𝜋 −3𝜋 2 2 = −7𝜋 2 + 𝜋 + 2𝜋 = . 2 This corresponds to the point (0,1). Sin 𝑡 = 𝑦 = 1. c.) Since tangent is an odd function, we know that tan(-t)= - tan(t). Thus, tan(-t)= - tan(t) = - 2/3. EX: Use a calculator to evaluate: a.) sin 5𝜋 7 b.) csc 2 *We must make sure that our calculator is in radian mode! a.) b.) csc 2 = 1 sin 2 Upon completion of this lesson, you should be able to: 1. evaluate trig functions using the unit circle. 2. Identify the facts about sine and cosine curves and explain their differences. For more information, visit http://www.mathsisfun.com/geometry/unit-circle.html HW Pg.299 3-51 3rds, 59, 63-66