Stitz-Zeager_College_Algebra_e-book

Provided 0 x 2 or x 3 2 a assuming the calculus

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Unformatted text preview: Sum Formulas: For all angles α and β , • cos(α) cos(β ) = 1 2 [cos(α − β ) + cos(α + β )] • sin(α) sin(β ) = 1 2 [cos(α − β ) − cos(α + β )] • sin(α) cos(β ) = 1 2 [sin(α − β ) + sin(α + β )] Related to the Product to Sum Formulas are the Sum to Product Formulas, which we will have need of in Section 10.7. These are easily verified using the Sum to Product Formulas, and as such, their proofs are left as exercises. Theorem 10.21. Sum to Product Formulas: For all angles α and β , • cos(α) + cos(β ) = 2 cos • cos(α) − cos(β ) = −2 sin • sin(α) ± sin(β ) = 2 sin α+β 2 α+β 2 α±β 2 α−β 2 cos α−β 2 sin cos α β 2 Example 10.4.6. 1. Write cos(2θ) cos(6θ) as a sum. 2. Write sin(θ) − sin(3θ) as a product. Solution. 1. Identifying α = 2θ and β = 6θ, we find cos(2θ) cos(6θ) = = = 1 2 1 2 1 2 [cos(2θ − 6θ) + cos(2θ + 6θ)] cos(−4θ) + 1 cos(8θ) 2 cos(4θ) + 1 cos(8θ), 2 where the last equality is courtesy of the even identity for cosine, cos(−4θ...
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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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