SECTION 7.5 Sum and Difference Formulas 511 ‘Are You Prepared?’ Answers 1. True 2. True 117. Find the exact values of the six trigonometric functions of an angle θ in standard position if 12, 5 ( ) − is a point on its terminal side. 118. Find the average rate of change of f x x cos ( ) = from 0 to 2 . π 119. Find the length of a line segment with endpoints 3, 4 ( ) − − and 5,8 . ( ) 120. Find the area of the sector of a circle of radius 8 meters formed by an angle of 54 .° 121. Kayaking Ben paddled his kayak 8 miles upstream against a 1 mile per hour current and back again in 6 hours. How far could Ben have paddled in that time if there had been no current? 122. If an angle θ lies in quadrant III and cot 8 5 , θ = find sec .θ 123. Write the equation of the circle in standard form: x y x y 12 4 31 0 2 2 + − + + = 124. If f x x 4 ( ) = − and g x x x 3 6 , ( ) = + − find the domain of f g x . ( )( ) This section continues the derivation of trigonometric identities by obtaining formulas that involve the sum or the difference of two angles, such as α β ( ) + cos , α β ( ) − cos , and α β ( ) + sin . These formulas are referred to as the Sum and Difference Formulas . We begin with the formulas for α β ( ) + cos and α β ( ) − cos . OBJECTIVES 1 Use Sum and Difference Formulas to Find Exact Values (p. 512) 2 Use Sum and Difference Formulas to Establish Identities (p. 516) 3 Use Sum and Difference Formulas Involving Inverse Trigonometric Functions (p. 517) 4 Solve Trigonometric Equations Linear in Sine and Cosine (p. 518) 7.5 Sum and Difference Formulas Now Work the ‘Are You Prepared?’ problems on page 520. PREPARING FOR THIS SECTION Before getting started, review the following: • Distance Formula (Section 1.2, pp. 13–16) • Values of the Trigonometric Functions (Section 6.2, pp. 397–405) • Congruent Triangles (Section A.2, pp. A16–A19) • Finding Exact Values of the Trigonometric Functions, Given the Value of a Trigonometric Function and the Quadrant of the Angle (Section 6.3, pp. 419–422) Proof We prove formula (2) first. Although the formula is true for all numbers α and β, we assume in our proof that β α π < < < 0 2 . Begin with the unit circle and place the angles α and β in standard position, as shown in Figure 32(a) on the next page. The point P1 lies on the terminal side of β, so its coordinates are β β ( ) cos , sin ; and the point P2 lies on the terminal side of α, so its coordinates are α α ( ) cos , sin . THEOREM Sum and Difference Formulas for the Cosine Function i α β α β α β ( ) + = − cos cos cos sin sin (1) i α β α β α β ( ) − = + cos cos cos sin sin (2) (continued) In Words Formula (1) states that the cosine of the sum of two angles equals the cosine of the first angle times the cosine of the second angle minus the sine of the first angle times the sine of the second angle.
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