Derivation of the Triple Angle Formulas for Trigonometric Functions Using De Moivre's Theorem
Formulas
$$ \sin 3\theta = 3 \sin \theta - 4 \sin^{3} {\theta} \\ \cos 3\theta = 4 \cos^{3} {\theta} - 3 \cos \theta $$
Explanation
The traditional transformation formulas could usually be obtained by applying the angle addition formulas for trigonometric functions multiple times.
For example, the double angle formula is obtained by substituting $b=a$ into $\sin(a + b ) = \sin {a} \cos {b} + \sin {b} \cos {a}$, yielding $\sin(a+a) = \sin{2a} = 2 \sin{a} \cos{a}$. Of course, there is nothing wrong with deriving the triple angle or quadruple angle formulas in this manner. However, using complex analysis, these formulas can be derived in a smarter way.
Since the derivation process itself is more useful than the transformation formulas themselves, be sure to become familiar with it.
Derivation
De Moivre’s theorem: If $z = r \text{cis} \theta$, then $z^n = r^n \text{cis} n\theta$
By De Moivre’s theorem, letting $r=1$, $$ (\cos{3\theta} + i \sin{3 \theta}) = \text{cis} 3\theta = z^3 = (\cos{\theta} + i \sin{\theta})^3 $$ The above equality holds if and only if the real parts and imaginary parts of both sides are respectively equal. Using the binomial theorem to expand the right-hand side, $$ \begin{align*} (\cos{3\theta} + i \sin{3 \theta}) =& (\cos{\theta} + i \sin{\theta})^3 \\ =& \cos ^3 {\theta} + i 3 \cos ^2 {\theta} \sin{\theta} - 3 \cos {\theta} \sin ^2 {\theta} - i \sin ^3 {\theta} \\ =& ( \cos ^3 {\theta} - 3 \cos {\theta} \sin ^2 {\theta} ) + i ( 3 \cos ^2 {\theta} \sin{\theta} - \sin ^3 {\theta} ) \\ =& (4 \cos^{3} {\theta} - 3 \cos \theta) + i ( 3 \sin \theta - 4 \sin^{3} {\theta} ) \end{align*} $$ That is, $$ \sin 3\theta = 3 \sin \theta - 4 \sin^{3} {\theta} \\ \cos 3\theta = 4 \cos^{3} {\theta} - 3 \cos \theta $$
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Applications
As can be seen from the idea of using the binomial theorem, this derivation is by no means restricted to the triple angle. It can be extended to any natural number, and by expressing it as a series, a general formula could also be obtained. Moreover, tracing the derivation process backwards shows that trigonometric terms of higher degree can be broken up into several terms.