524 CHAPTER 7 Analytic Trigonometry Problems 116–125 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 116. Determine the points of intersection of the graphs of = + + f x x x ( ) 5 1 2 and = − − − g x x x ( ) 2 11 4 2 by solving = f x g x ( ) ( ). 117. Convert π 17 6 to degrees. 118. Find the area of the sector of a circle of radius 6 meters formed by an angle of ° 45 . Give both the exact area and an approximation rounded to two decimal places. 119. Given tan 2, 270 360 , θ θ = − ° < < ° find the exact value of the remaining five trigonometric functions. 120. Write = + − f x x x ( ) 1 4 2 2 in vertex form. 121. Solve: = − − 8 4 x x 4 2 9 122. Write as a single logarithm: + − x y z 3 log 2 log 5 log 7 7 7 123. Simplify: ( ) ( ) x y x y 2 3 2 3 4 5 2 124. Solve: − − − = x x 3 2 2 3 1 125. Write ( ) ( ) + + + >− x x x x 6 3 8 3 , 3, 1/4 3/4 as a single quotient with only positive exponents. ‘Are You Prepared?’ Answers 1. 5 2. − 3 5 3. (a) 2 4 (b) 1 2 4. − 3 5 5. congruent 6. 2 2 3 ; 1 3 ; 22 − − 7.6 Double-angle and Half-angle Formulas In this section, formulas for θ θ θ ( ) ( ) ( ) sin 2 , cos 2 , sin 1 2 , and θ ( ) cos 1 2 are established in terms of θ sin and θ cos . They are derived using the Sum Formulas. In the Sum Formulas for α β ( ) + sin and α β ( ) + cos , let α β θ = = . Then α β α β α β θ θ θ θ θ θ θ θ θ ( ) ( ) ( ) + = + + = + = sin sin cos cos sin sin sin cos cos sin sin 2 2 sin cos and α β α β α β θ θ θ θ θ θ θ θ θ ( ) ( ) ( ) + = − + = − = − cos cos cos sin sin cos cos cos sin sin cos 2 cos sin 2 2 An application of the Pythagorean Identity θ θ + = sin cos 1 2 2 results in two other ways to express θ ( ) cos 2 . θ θ θ θ θ θ ( ) ( ) = − = − − = − cos 2 cos sin 1 sin sin 1 2 sin 2 2 2 2 2 and θ θ θ θ θ θ ( ) ( ) = − = − − = − cos 2 cos sin cos 1 cos 2 cos 1 2 2 2 2 2 OBJECTIVES 1 Use Double-angle Formulas to Find Exact Values (p. 525) 2 Use Double-angle Formulas to Establish Identities (p. 525) 3 Use Half-angle Formulas to Find Exact Values (p. 528)
RkJQdWJsaXNoZXIy NjM5ODQ=