534 CHAPTER 7 Analytic Trigonometry 115. Graph ( ) ( ) = = − f x x x sin 1 cos 2 2 2 for π ≤ ≤ x 0 2 by using transformations. 116. Repeat Problem 115 for ( ) = g x x cos . 2 117. Use the fact that π ( ) = + cos 12 1 4 6 2 to find π sin 24 and π cos 24 . 118. Show that π = + cos 8 2 2 2 and use it to find π sin 16 and π cos 16 . 119. Challenge Problem Show that θ θ θ θ ( ) ( ) ( ) + + ° + + ° = − sin sin 120 sin 240 3 4 sin 3 3 3 3 120. Challenge Problem If θ θ = a tan tan 3 , express θ tan 3 in terms of a. 121. Challenge Problem If ( ) ( ) + − + − = x m x m cos 2 2 1 sin 1 0, find m so that there is exactly one real solution for x, π π − ≤ ≤ x 2 2 .† 108. Geometry A rectangle is inscribed in a semicircle of radius 1. See the figure. y x u 1 (a) Express the area A of the rectangle as a function of the angle θ shown in the figure. (b) Show that θ θ ( ) ( ) = A sin 2 . (c) Find the angle θ that results in the largest area A. (d) Find the dimensions of this largest rectangle. 109. If θ = x 2 tan , express θ ( ) sin 2 as a function of x. 110. If θ = x 2 tan , express θ ( ) cos 2 as a function of x. 111. Find the value of the number C: ( ) + = − x C x 1 2 sin 1 4 cos 2 2 112. Find the value of the number C: ( ) + = x C x 1 2 cos 1 4 cos 2 2 113. If α = z tan 2 , show that α = + z z sin 2 1 . 2 114. If α = z tan 2 , show that α = − + z z cos 1 1 . 2 2 †Courtesy of Joliet Junior College Mathematics Department Problems 123–132 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 123. Find an equation of the line that contains the point ( ) − 2, 3 and is perpendicular to the line = − + y x2 9. 124. Graph ( ) = − + + f x x x6 7. 2 Label the vertex and any intercepts. 125. Find the exact value of π π − sin 2 3 cos 4 3 . 126. Graph π( ) = − y x 2 cos 2 . Show at least two periods. 127. Find a polynomial function of degree 3 whose real zeros are − − 5, 2, and 2. Use 1 for the leading coefficient. 128. The function ( ) = − − f x x x 3 2 5 is one-to-one. Find −f 1. 129. Solve: = + + 2 3 x x 7 2 130. Find the distance between the vertices of the parabolas ( ) ( ) = − − = − − − f x x x g x x x 4 1 and 6 2. 2 2 131. Find the average rate of change of ( ) = f x x log2 from 4 to 16. 132. Solve for − − + + − = D x xD y yD D :6 5 5 4 3 4 0 Explaining Concepts 122. Research Chebyshëv polynomials. Write a report on your findings.
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