# Triple angle formulas

## Definition

The expansion of a trigonometric function of triple angle in terms of the same trigonometric ratio of angle is called triple angle identity or triple angle formula.

### List of triple angle identities

The six trigonometric ratios can form six triple angle formulas but triple angle identities of sine, cosine, tangent and cotangent are most useful and there is no priority for the triple angle identities of secant and cosecant.

1

### Sine triple angle formula

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

2

### Cosine triple angle formula

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

3

### Tangent triple angle formula

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

4

### Cotangent triple angle formula

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

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