Stitz-Zeager_College_Algebra_e-book

# We have cos 2 arcsinx cos2t 1 2 sin2 t 1 2x2

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Unformatted text preview: . Cofunction Identities: For all applicable angles θ, π − θ = sin(θ) 2 π • sin − θ = cos(θ) 2 • cos π − θ = csc(θ) 2 π • csc − θ = sec(θ) 2 • sec π − θ = cot(θ) 2 π • cot − θ = tan(θ) 2 • tan With the Cofunction Identities in place, we are now in the position to derive the sum and diﬀerence formulas for sine. To derive the sum formula for sine, we convert to cosines using a cofunction identity, then expand using the diﬀerence formula for cosine π − (α + β ) 2 π = cos −α −β 2 π π = cos − α cos(β ) + sin − α sin(β ) 2 2 = sin(α) cos(β ) + cos(α) sin(β ) sin(α + β ) = cos We can derive the diﬀerence formula for sine by rewriting sin(α − β ) as sin(α + (−β )) and using the sum formula and the Even / Odd Identities. Again, we leave the details to the reader. Theorem 10.15. Sum and Diﬀerence Identities for Sine: For all angles α and β , • sin(α + β ) = sin(α) cos(β ) + cos(α) sin(β ) • sin(α − β ) = sin(α) cos(β ) − cos(α)...
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