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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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