Multiple Angle Formulas

Trigonometric Identities

Double Angle Formulas

Double Angle Formula for Sine

$\sin 2 \theta = 2 \sin \theta \cos \theta$

Double Angle Formula for Cosine

$\cos 2 \theta = \cos^2 \theta - \sin^2 \theta$

Double Angle Formula for Tangent

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

Triple Angle Formulas

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

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

Quadruple Angle Formulas

Quadruple Angle Formula for Sine

$\sin 4 \theta = 4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta$

Quadruple Angle Formula for Cosine

$\cos 4 \theta = 8 \cos^4 \theta - 8 \cos^2 \theta + 1$

Quadruple Angle Formula for Tangent

$\tan 4 \theta = \dfrac {4 \tan \theta - 4 \tan^3 \theta} {1 - 6 \tan^2 \theta + \tan^4 \theta}$

Quintuple Angle Formulas

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

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

Sextuple Angle Formulas

Sextuple Angle Formula for Sine

$\dfrac {\sin 6 \theta} {\sin \theta} = 32 \cos^5 \theta - 32 \cos^3 \theta + 6 \cos \theta$

Sextuple Angle Formula for Cosine

$\cos 6 \theta = 32 \cos^6 \theta - 48 \cos^4 \theta + 18 \cos^2 \theta - 1$

Sextuple Angle Formula for Tangent

$\tan 6 \theta = \dfrac { 6 \tan \theta - 20 \tan^3 \theta + 6 \tan^5 \theta } { 1 - 15 \tan^2 \theta + 15 \tan^4 \theta - \tan^6 \theta }$

Septuple Angle Formulas

Septuple Angle Formula for Cosine

$\cos 7 \theta = 64 \cos^7 \theta - 112 \cos^5 \theta + 56 \cos^3 \theta - 7 \cos \theta$

Also known as

The multiple angle formulas are often presented hyphenated, and the older plural formulae can also be found, that is: multiple-angle formulae.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): multiple-angle formulae
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): multiple-angle formulae