Triple Angle Identities

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

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

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:

$$\begin{aligned} \sin 3 \theta & = \sin (2 \theta + \theta)\\ & = \sin 2 \theta \cos \theta + \cos 2 \theta \sin \theta \\ & = (2 \sin \theta \cos \theta) \cos \theta + \big(1 - 2 \sin^2 \theta\big) \sin \theta \\ & = 2 \sin \theta \cos^2 \theta + \sin \theta - 2 \sin^3 \theta \\ & = 2 \sin \theta \big(1 - \sin^2 \theta\big) + \sin \theta - 2 \sin^3 \theta \\ & = 2 \sin \theta - 2 \sin^3 \theta + \sin \theta - 2 \sin^3 \theta \\ & = 3 \sin \theta - 4 \sin^3 \theta.\ _\square \end{aligned}$$

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:

$$\sin^3(\theta) = \frac{3 \sin (\theta) - \sin \left( 3 \theta \right) }{4},\quad \cos^3 (\theta) = \frac{\cos(3\theta) + 3 \cos (\theta)}{4}.$$

To remember the cosine formula, the trick that I like to use is to read cosine as "dollar." Then, we say

"Dollar thirty is equal to four dollar thirty minus three dollar."

$$\begin{array} {l l l l l } \$1. 30 & = \$ 4.30 & - \$ 3 \\ 1 \cos 3 \theta & = 4 \cos ^3 \theta & - 3 \cos \theta \\ \end{array}$$

Just leaving a mark here,

$$\begin{aligned} \tan x \tan(60^\circ - x) \tan (60^\circ + x) &=& \tan(3x) \\ 4\sin x \sin(60^\circ - x) \sin (60^\circ + x) &=&\sin(3x) \\ 4\cos x \cos(60^\circ - x) \cos(60^\circ + x) &=& \cos(3x) \\ \tan(3x) &=& \frac {3\tan x - \tan^3 x}{1 - 3\tan^2 x}. \end{aligned}$$

Prove that

$$\tan (6^\circ) = \tan (12^\circ) \tan(24^\circ) \tan(48^\circ ).$$

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