Section 1.2 Trigonometric Functions: The Unit Circle 37 Course Number Section 1.2 Trigonometric Functions: The Unit Circle Instructor Objective: In this lesson you learned how to identify a unit circle and its relationship to real numbers. Date Important Vocabulary Define each term or concept. Unit circle A circle of radius 1 centered at the origin and given by the equation x2 + y2 = 1. The basis for one of the perspectives of trigonometry. Period A function f is periodic if there exists a positive real number c such that f(t + c) = f(t) for all t in the domain of f. The smallest number c for which f is periodic is called the period of f. I. The Unit Circle (Page 137) What you should learn How to identify a unit As the real number line is wrapped around the unit circle, each circle and its relationship real number t corresponds to . a point (x, y) on the circle. to real numbers The real number 2p corresponds to the point (1, 0) on the unit circle. Each real number t also corresponds to a central angle q (in standard position) whose radian measure is t. With this interpretation of t, the arc length formula s = rq (with r = 1) indicates that . the real number t is the length of the arc intercepted by the angle q, given in radians. II. The Trigonometric Functions (Pages 138-140) What you should learn How to evaluate The coordinates x and y are two functions of the real variable t. trigonometric functions These coordinates can be used to define six trigonometric using the unit circle functions of t. List the abbreviation for each trigonometric function. Sine sin Cosecant csc Cosine cos Secant sec Tangent tan Cotangent cot Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company. All rights reserved. 38 Chapter 1 Trigonometry Let t be a real number and let (x, y) be the point on the unit circle corresponding to t. Complete the following definitions of the trigonometric functions: sin t = y cos t = x tan t = y/x, x ¹ 0 cot t = x/y, y ¹ 0 sec t = 1/x, x ¹ 0 csc t = 1/y, y ¹ 0 The cosecant function is the reciprocal of the sine function. The cotangent function is the reciprocal of the tangent function. The secant function is the reciprocal of the cosine function. Complete the following table showing the correspondence between the real number t and the point (x, y) on the unit circle when the unit circle is divided into eight equal arcs. t 0 p/4 p/2 3p/4 p 5p/4 3p/2 7p/4 x 1 Ö2 /2 0 - Ö2 /2 - 1 - Ö2 /2 0 Ö2 /2 y 0 Ö2 /2 1 Ö2 /2 0 - Ö2 /2 - 1 - Ö2 /2 Complete the following table showing the correspondence between the real number t and the point (x, y) on the unit circle when the unit circle is divided into 12 equal arcs. t 0 p/6 p/3 p/2 2p/3 5p/6 p 7p/6 4p/3 3p/2 5p/3 11p/6 x 1 Ö3 /2 1/2 0 - 1/2 - Ö3 /2 - 1 - Ö3 /2 - 1/2 0 1/2 Ö3 /2 y 0 1/2 Ö3 /2 1 Ö3 /2 1/2 0 - 1/2 - Ö3 /2 - 1 - Ö3 /2 - 1/2 Example 1: Find the following: p 3p 7p (a) cos (b) tan (c) csc 3 4 6 (a) 1/2 (b) - 1 (c) - 2 III. Domain and Period of Sine and Cosine (Pages 140-141) What you should learn How to use the domain The sine function’s domain is the set of all real numbers , and period to evaluate and its range is [- 1, 1] . sine and cosine functions Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company. All rights reserved. Section 1.2 Trigonometric Functions: The Unit Circle 39 The cosine function’s domain is the set of all real numbers , and its range is [- 1, 1] . The period of the sine function is 2p . The period of the cosine function is 2p . Which trigonometric functions are even functions? cosine and secant Which trigonometric functions are odd functions? Sine, cosecant, tangent, and cotangent 31p Example 2: Evaluate sin 6 - 1/2 IV. Evaluating Trigonometric Functions with a Calculator What you should learn (Page 141) How to use a calculator to evaluate trigonometric To evaluate the secant function with a calculator, . functions evaluate its reciprocal function, cosine, and then use the x-1 key. Example 3: Use a calculator to evaluate (a) tan 4p/3, and (b) cos 3. (a) 1.732050808 (b) - 0.9899924966 y y y x x x Homework Assignment Page(s) Exercises Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company. All rights reserved. .