Pre-Calc Exam Notes 79

Pre-Calc Exam Notes 79 - Double-Angle and Half-Angle...

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Double-Angle and Half-Angle Formulas Section 3.3 79 Example 3.14 Prove that sin 4 z = 4 tan z (1 tan 2 z ) (1 + tan 2 z ) 2 . Solution: Expand the right side and use 1 + tan 2 z = sec 2 z : 4 tan z (1 tan 2 z ) (1 + tan 2 z ) 2 = 4 · sin z cos z · p cos 2 z cos 2 z sin 2 z cos 2 z P (sec 2 z ) 2 = 4 · sin z cos z · cos 2 z cos 2 z p 1 cos 2 z P 2 (by formula (3.24)) = (4 sin z cos 2 z ) cos z = 2 (2 sin z cos z ) cos 2 z = 2 sin 2 z cos 2 z (by formula (3.23)) = sin 4 z (by formula (3.23) with θ replaced by 2 z ) Note: Perhaps surprisingly, this seemingly obscure identity has found a use in physics, in the deriva- tion of a solution of the sine-Gordon equation in the theory of nonlinear waves. 3 Closely related to the double-angle formulas are the half-angle formulas : sin 2 1 2 θ = 1 cos θ 2 (3.28) cos 2 1 2 θ = 1 + cos θ 2 (3.29) tan 2 1 2 θ = 1 cos θ 1 + cos θ (3.30) These formulas are just the double-angle formulas rewritten with θ replaced by 1 2 θ : cos 2 θ = 1 2 sin 2 θ sin 2 θ = 1 cos 2 θ
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This note was uploaded on 01/21/2012 for the course MAC 1130 taught by Professor Dr.cheun during the Fall '11 term at FSU.

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