SECTION 7.7 Product-to-Sum and Sum-to-Product Formulas 535 1 Express Products as Sums Sum and Difference Formulas can be used to derive formulas for writing the products of sines and/or cosines as sums or differences. These identities are usually called the Product-to-Sum Formulas. 7.7 Product-to-Sum and Sum-to-Product Formulas OBJECTIVES 1 Express Products as Sums (p. 535) 2 Express Sums as Products (p. 536) THEOREM Product-to-Sum Formulas • α β α β α β ( ) ( ) [ ] = − − + sin sin 1 2 cos cos (1) • α β α β α β ( ) ( ) [ ] = − + + cos cos 1 2 cos cos (2) • α β α β α β ( ) ( ) [ ] = + + − sin cos 1 2 sin sin (3) These formulas do not have to be memorized. Instead, remember how they are derived. Then, when you want to use them, either look them up or derive them, as needed. To derive Product-to-Sum Formulas (1) and (2), write down the Sum and Difference Formulas for cosine: cos cos cos sin sin α β α β α β ( ) − = + (4) cos cos cos sin sin α β α β α β ( ) + = − (5) To derive formula (1), subtract equation (5) from equation (4) cos cos 2 sin sin α β α β α β ( ) ( ) − − + = from which sin sin 1 2 cos cos α β α β α β ( ) ( ) [ ] = − − + To derive formula (2), add equations (4) and (5) cos cos 2cos cos α β α β α β ( ) ( ) − + + = from which α β α β α β ( ) ( ) [ ] = − + + cos cos 1 2 cos cos To derive Product-to-Sum Formula (3), use the Sum and Difference Formulas for sine in a similar way. (You are asked to do this in Problem 55.) Expressing Products as Sums Express each of the following products as a sum containing only sines or only cosines. (a) sin 6 sin 4 θ θ ( ) ( ) (b) cos 3 cos θ θ ( ) (c) sin 3 cos 5 θ θ ( ) ( ) EXAMPLE 1 (continued)
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