Powers of sine and cosine
and multiple angle sine and cosine functions

Branko Ćurgus


By \(\mathbb{R}\) we denote the set of real numbers and by \(\mathbb{N}\) the set of positive integers. By trigonometric functions here we mean the functions cosine and sine; that is with \(\theta \in \mathbb{R}\), \[ \theta \mapsto \cos \theta \quad \text{and} \quad \theta \mapsto \sin \theta. \] By Multiple-Angle Functions we mean the trigonometric functions in which the variable is multiplied by a positive integer; that is with \(n\in\mathbb{N}\) and \(\theta \in \mathbb{R}\). \[ \theta \mapsto \cos(n\theta) \quad \text{and} \quad \theta \mapsto \sin(n\theta). \]

Powers of Cosine and Sine as linear combinations of Multiple-Angle Functions

Below are my versions of the Power-reduction formulae: \begin{align*} (\cos \theta)^{2m} & = \frac{1}{4^{m}} \binom{2m}{m} + \frac{2}{4^{m}} \sum_{k=1}^{m} \binom{2m}{m-k} \cos(2k\theta), \ \ m\in\mathbb{N}, \\[6pt] (\cos \theta)^{2m+1} & =\frac{1}{4^{m}} \sum_{k=0}^{m} \binom{2m+1}{m-k} \cos\bigl((2k+1)\theta\bigr), \ \ m\in\mathbb{N}\cup\{0\}, \\[6pt] \bigl(\sin \theta\bigr)^{2m} & = \frac{1}{4^{m}} \binom{2m}{m} + \frac{2}{4^{m}} \sum_{k=1}^{m} (-1)^{k} \binom{2m}{m-k} \cos (2k \theta ), \ \ m\in\mathbb{N}, \\[6pt] \bigl(\sin \theta\bigr)^{2m+1} & = \frac{1}{4^{m}} \sum_{k=0}^{m} (-1)^{k} \binom{2m + 1}{m - k} \sin\bigl((2k + 1)\theta\bigr), \ \ m\in\mathbb{N}\cup\{0\}. \end{align*}

The special cases for the first four instances of each of the above formulas are given below.

The first four even powers of \(\cos\theta\): \begin{align*} \bigl(\cos \theta\bigr)^{2} & = \frac{1}{2} + \frac{1}{2} \cos(2\theta),\\[6pt] \bigl(\cos \theta\bigr)^{4} & = \frac{3}{8} + \frac{1}{2} \cos(2\theta) + \frac{1}{8} \cos(4\theta),\\[6pt] \bigl(\cos \theta\bigr)^{6} & = \frac{5}{16} + \frac{15}{32} \cos(2\theta) + \frac{3}{16} \cos(4\theta) + \frac{1}{32} \cos(6\theta), \\[6pt] \bigl(\cos \theta\bigr)^{8} & = \frac{35}{128} + \frac{7}{16} \cos(2\theta) + \frac{7}{32} \cos(4\theta) + \frac{1}{16} \cos(6\theta) + \frac{1}{128} \cos(8\theta). \end{align*}

The first four odd powers of \(\cos\theta\): \begin{align*} \bigl(\cos \theta\bigr)^{3} & = \frac{3}{4} \cos(\theta) + \frac{1}{4} \cos(3\theta),\\[6pt] \bigl(\cos \theta\bigr)^{5} & = \frac{5}{8} \cos(\theta) + \frac{5}{16} \cos(3\theta) + \frac{1}{16} \cos(5\theta),\\[6pt] \bigl(\cos \theta\bigr)^{7} & = \frac{35}{64} \cos(\theta) + \frac{21}{64} \cos(3\theta) + \frac{7}{64} \cos(5\theta) + \frac{1}{64} \cos(7\theta), \\[6pt] \bigl(\cos \theta\bigr)^{9} & = \frac{63}{128} \cos(\theta) + \frac{21}{64} \cos(3\theta) + \frac{9}{64} \cos(5\theta) + \frac{9}{256} \cos(7\theta) + \frac{1}{256} \cos(9\theta). \end{align*}

The first four even powers of \(\sin\theta\): \begin{align*} \bigl(\sin \theta\bigr)^{2} & = \frac{1}{2} - \frac{1}{2} \cos(2\theta),\\[6pt] \bigl(\sin \theta\bigr)^{4} & = \frac{3}{8} - \frac{1}{2} \cos(2\theta) + \frac{1}{8} \cos(4\theta),\\[6pt] \bigl(\sin \theta\bigr)^{6} & = \frac{5}{16} - \frac{15}{32} \cos(2\theta) + \frac{3}{16} \cos(4\theta) - \frac{1}{32} \cos(6\theta), \\[6pt] \bigl(\sin \theta\bigr)^{8} & = \frac{35}{128} - \frac{7}{16} \cos(2\theta) + \frac{7}{32} \cos(4\theta) - \frac{1}{16} \cos(6\theta) + \frac{1}{128} \cos(8\theta). \end{align*}

The first four odd powers of \(\sin\theta\): \begin{align*} \bigl(\sin \theta\bigr)^{3} & = \frac{3}{4}\sin(\theta) - \frac{1}{4} \sin(3\theta),\\[6pt] \bigl(\sin \theta\bigr)^{5} & = \frac{5}{8}\sin(\theta) - \frac{5}{16}\sin(3\theta) + \frac{1}{16} \sin(5\theta) ,\\[6pt] \bigl(\sin \theta\bigr)^{7} & = \frac{35}{64}\sin(\theta) - \frac{21}{64}\sin(3\theta) + \frac{7}{64}\sin(5\theta) - \frac{1}{64} \sin(7\theta), \\[6pt] (\sin \theta)^9 &= \frac{63}{128} \sin \theta - \frac{21}{64} \sin(3\theta) + \frac{9}{64} \sin(5\theta) - \frac{9}{256} \sin(7\theta) + \frac{1}{256} \sin(9\theta). \end{align*}


Multiple-Angle Functions as linear combinations of Powers of Cosine and Sine


Powers of Cosine and Sine as linear combinations of their Multiple-angles