Quintuple Angle Formulas

Theorem

Quintuple Angle Formula for Sine

$\sin 5 \theta = 5 \sin \theta - 20 \sin^3 \theta + 16 \sin^5 \theta$

Quintuple Angle Formula for Cosine

$\cos 5 \theta = 16 \cos^5 \theta - 20 \cos^3 \theta + 5 \cos \theta$

Quintuple Angle Formula for Tangent

$\tan 5 \theta = \dfrac {\tan^5 \theta - 10 \tan^3 \theta + 5 \tan \theta} {1 - 10 \tan^2 \theta + 5 \tan^4 \theta}$

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

Also see

• Quadruple Angle Formulas
• Sextuple Angle Formulas