SECTION 7.4 Trigonometric Identities 503 Explaining Concepts 125. Explain in your own words how you would use your calculator to solve the equation π = − ≤ < x x cos 0.6, 0 2 . How would you modify your approach to solve the equation π = < < x x cot 5, 0 2 ? 126. Explain why no further points of intersection (and therefore no further solutions) exist in Figure 30 for π <− x or π > x 4 . Retain Your Knowledge Problems 127–136 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. 127. Convert = y 6x to an equivalent statement involving a logarithm. 128. Find the real zeros of ( ) = − + f x x x 2 9 8. 2 129. If θ θ = − = sin 10 10 and cos 3 10 10 , find the exact value of each of the four remaining trigonometric functions. 130. Find the amplitude, period, and phase shift of the function π ( ) = − y x 2 sin 2 . Graph the function. Show at least two periods. 131. If ( ) = + − f x e 1 2 3, x 1 find the domain of ( ) −f x . 1 132. Find the length of the arc of a circle of radius 15 centimeters subtended by a central angle of ° 36 . 133. Find the value of a so that the line − = ax y3 10 has slope 2. 134. Is the function ( ) = − f x x x 3 5 2 even, odd, or neither? 135. If ( ) = f x x 8 , 2 find an equation of the secant line containing the points ( ) ( ) f 1, 1 and ( ) ( ) f 4, 4. 136. Find the average rate of change of ( ) = − f x x cos 1 from 1 2 to 1. ‘Are You Prepared?’ Answers 1. 1 2. − 2 2 ; 1 2 3. { } −1, 5 4 4. { } − + 1 5 2 , 1 5 2 5. { } 0, 5 2 6. { } 0.76 OBJECTIVES 1 Use Algebra to Simplify Trigonometric Expressions (p. 504) 2 Establish Identities (p. 505) 7.4 Trigonometric Identities Now Work the ‘Are You Prepared?’ problems on page 508. • Fundamental Identities (Section 6.3, pp. 417–419) • Even–Odd Properties (Section 6.3, pp. 422–423) PREPARING FOR THIS SECTION Before getting started, review the following: This section establishes additional identities involving trigonometric functions. First, let’s define an identity. DEFINITION Identically Equal, Identity, and Conditional Equation Two functions f and g are identically equal if f x g x ( ) ( ) = for every value of x for which both functions are defined. Such an equation is referred to as an identity . An equation that is not an identity is called a conditional equation .
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