412 CHAPTER 6 Trigonometric Functions the answer as a single fraction, completely simplified, with rationalized denominator. 141. Challenge Problem If the terminal side of an angle contains the point n n 5, 12 ( ) − with n 0, > find sin .θ 140. Challenge Problem Let θ be the measure of an angle, in radians, in standard position with 2 . π θ π < < Find the exact x -coordinate of the intersection of the terminal side of θ with the unit circle, given cos sin 1 9 . 2 θ θ − = − State Explaining Concepts 142. Write a brief paragraph that explains how to quickly compute the trigonometric functions of 30 , 45 , ° ° and 60 .° 143. Write a brief paragraph that explains how to quickly compute the trigonometric functions of 0 , 90 , 180 , ° ° ° and 270 .° 144. How would you explain the meaning of the sine function to a student who has just completed college algebra? 145. Draw a unit circle. Label the angles 0, π π π π 6 , 4 , 3 , . . . , 7 4 , π π 11 6 , 2 and the coordinates of the points on the unit circle that correspond to each of these angles. Explain how symmetry can be used to find the coordinates of points on the unit circle for angles whose terminal sides are in quadrants II, III, and IV. ‘Are You Prepared?’ Answers 1. = + c a b 2 2 2 2. 8 3. True 4. equal; proportional 5. − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ 1 2 , 3 2 6. − 1 2 Problems 146–155 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. Retain Your Knowledge 146. Find the domain of f x x ln 5 2 . ( ) ( ) = + 147. If the polynomial function P x x x x x 5 9 155 250 4 3 2 ( ) = − − + − has zeros of i 4 3 + and 2, find the remaining zeros of the function. 148. Find the remainder when P x x x x 8 2 8 4 3 ( ) = − + − is divided by x 2. + 149. Sidewalk Area A sidewalk with a uniform width of 3 feet is to be placed around a circular garden with a diameter of 24 feet. Find the exact area of the sidewalk. 150. Find the real zeros of f x x x 3 7 9. 2 ( ) = − − 151. If g x x 1 1 , 2 ( ) = + find f x( ) so that f g x x 1 2 . 2 ( ) ( ) = + 152. If f x x 3 2 ( ) = − and g x x 3, ( ) = − + determine where g x f x . ( ) ( ) ≥ 153. Solve x int 3 2. ( ) + = − 154. If the point 3, 4 ( ) − is on the graph of y f x , ( ) = what corresponding point must be on the graph of y f x 1 2 3 ? ( ) = − 155. If g x x x 4 1 , 2 2 ( ) = − simplify g x 1 . 2 [ ] ( ) + 6.3 Properties of the Trigonometric Functions Now Work the ‘Are You Prepared?’ problems on page 423. • Functions (Section 2.1, pp. 61–73) • Identity (Appendix A, Section A.6, p. A44) • Even and Odd Functions (Section 2.3, pp. 87–89) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Determine the Domain and the Range of the Trigonometric Functions (p. 413) 2 Determine the Period of the Trigonometric Functions (p. 415) 3 Determine the Signs of the Trigonometric Functions in a Given Quadrant (p. 416) 4 Find the Values of the Trigonometric Functions Using Fundamental Identities (p. 417) 5 Find the Exact Values of the Trigonometric Functions of an Angle Given One of the Functions and the Quadrant of the Angle (p. 419) 6 Use Even–Odd Properties to Find the Exact Values of the Trigonometric Functions (p. 422)
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