# Trigonometric Equations - Triple Angle Formula

The trigonometric **triple angle identities** give a relationship between the basic trigonometric functions applied to three times an angle in terms of trigonometric functions of the angle itself.

Triple angle Identities\[ \sin 3 \theta = 3 \sin \theta - 4 \sin ^3 \theta \] \[ \cos 3\theta = 4 \cos ^ 3 \theta - 3 \cos \theta \]

Proof:To prove the triple angle identities, we can write \(\sin 3 \theta\) as \(\sin(2 \theta + \theta)\). Then we can use the sum formula and the double angle identities to get the desired form.\[\begin{align} \sin 3 \theta & = \sin (2 \theta + \theta)\\ & = \sin 2 \theta \cos \theta + \cos 2 \theta \sin \theta \\ & = (2 \sin \theta \cos \theta) \cos \theta + (1 - 2 \sin^2 \theta ) \sin \theta \\ & = 2 \sin \theta \cos^2 \theta + \sin \theta - 2 \sin^3 \theta \\ & = 2 \sin \theta (1 - \sin^2 \theta) + \sin \theta - 2 \sin^3 \theta \\ & = 2 \sin \theta - 2 \sin^3 \theta + \sin \theta - 2 \sin^3 \theta \\ & = 3 \sin \theta - 4 \sin^3 \theta & _\square \end{align} \]

The triple angle identity of \(\cos 3 \theta\) can be proved in a very similar manner.

From these formulas, we also have the following identities for \(\sin^3(\theta)\) and \(\cos^3 (\theta) \) in terms of lower powers:

\[\begin{align} \sin^3(\theta) &= \frac{3 \sin (\theta) - \sin \left( 3 \theta \right) }{4}\\ \cos^3 (\theta) & = \frac{\cos(3\theta) + 3 \cos (\theta)}{4} \end{align} \]

**Cite as:**Trigonometric Equations - Triple Angle Formula.

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