Pre-Calc Exam Notes 78

Pre-Calc Exam Notes 78 - cos 2 = 2 cos 2 1 (3.26) = 1 2 sin...

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78 Chapter 3 Identities §3.3 3.3 Double-Angle and Half-Angle Formulas A special case of the addition formulas is when the two angles being added are equal, result- ing in the double-angle formulas : sin 2 θ = 2 sin θ cos θ (3.23) cos 2 θ = cos 2 θ sin 2 θ (3.24) tan 2 θ = 2 tan θ 1 tan 2 θ (3.25) To derive the sine double-angle formula, we see that sin 2 θ = sin ( θ + θ ) = sin θ cos θ + cos θ sin θ = 2 sin θ cos θ . Likewise, for the cosine double-angle formula, we have cos 2 θ = cos ( θ + θ ) = cos θ cos θ sin θ sin θ = cos 2 θ sin 2 θ , and for the tangent we get tan 2 θ = tan ( θ + θ ) = tan θ + tan θ 1 tan θ tan θ = 2 tan θ 1 tan 2 θ Using the identities sin 2 θ = 1 cos 2 θ and cos 2 θ = 1 sin 2 θ , we get the following useful alternate forms for the cosine double-angle formula:
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Unformatted text preview: cos 2 = 2 cos 2 1 (3.26) = 1 2 sin 2 (3.27) Example 3.13 Prove that sin 3 = 3 sin 4 sin 3 . Solution: Using 3 = 2 + , the addition formula for sine, and the double-angle formulas (3.23) and (3.27), we get: sin 3 = sin (2 + ) = sin 2 cos + cos 2 sin = (2 sin cos ) cos + (1 2 sin 2 ) sin = 2 sin cos 2 + sin 2 sin 3 = 2 sin (1 sin 2 ) + sin 2 sin 3 = 3 sin 4 sin 3...
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