Triple Angle Formulas

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Theorem

Triple Angle Formula for Sine

$\sin 3 \theta = 3 \sin \theta - 4 \sin^3 \theta$


Triple Angle Formula for Cosine

$\cos 3 \theta = 4 \cos^3 \theta - 3 \cos \theta$


Triple Angle Formula for Tangent

$\map \tan {3 \theta} = \dfrac {3 \tan \theta - \tan^3 \theta} {1 - 3 \tan^2 \theta}$


where $\sin, \cos, \tan$ denote sine, cosine and tangent respectively.


Triple Angle Formula for Hyperbolic Sine

$\sinh 3 x = 3 \sinh x + 4 \sinh^3 x$


Triple Angle Formula for Hyperbolic Cosine

$\cosh 3 x = 4 \cosh^3 x - 3 \cosh x$


Triple Angle Formula for Hyperbolic Tangent

$\tanh \left({3 x}\right) = \dfrac {3 \tanh x + \tanh^3 x} {1 + 3 \tanh^2 x}$


where $\sinh, \cosh, \tanh$ denote hyperbolic sine, hyperbolic cosine and hyperbolic tangent respectively.