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Pre-Calc Exam Notes 78

Pre-Calc Exam Notes 78 - cos 2 θ = 2 cos 2 θ − 1(3.26 =...

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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 θ = 2 cos 2 θ − 1 (3.26) = 1 − 2 sin 2 θ (3.27) Example 3.13 Prove that sin 3 θ = 3 sin θ − 4 sin 3 θ . Solution: Using 3 θ = 2 θ + θ , the addition formula for sine, and the double-angle formulas (3.23) and (3.27), we get: sin 3 θ = sin (2 θ + θ ) = sin 2 θ cos θ + cos 2 θ sin θ = (2 sin θ cos θ ) cos θ + (1 − 2 sin 2 θ ) sin θ = 2 sin θ cos 2 θ + sin θ − 2 sin 3 θ = 2 sin θ (1 − sin 2 θ ) + sin θ − 2 sin 3 θ = 3 sin θ − 4 sin 3 θ...
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