# addition and subtraction formulas for sine and cosine

The rotation matrix  $\displaystyle\left(\begin{array}[]{lr}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{array}\right)$ will be used to obtain the addition formulas for sine and cosine.

Recall that a vector in $\mathbb{R}^{2}$ can be rotated $\theta$ radians in the counterclockwise direction by multiplying on the left by the rotation matrix $\displaystyle\left(\begin{array}[]{lr}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{array}\right)$. Because rotating by $\alpha+\beta$ radians is the same as rotating by $\beta$ radians followed by rotating by $\alpha$ radians, we obtain:

$\begin{array}[]{rl}\displaystyle\left(\begin{array}[]{lr}\cos(\alpha+\beta)&-% \sin(\alpha+\beta)\\ \sin(\alpha+\beta)&\cos(\alpha+\beta)\end{array}\right)&=\displaystyle\left(% \begin{array}[]{lr}\cos\alpha&-\sin\alpha\\ \sin\alpha&\cos\alpha\end{array}\right)\left(\begin{array}[]{lr}\cos\beta&-% \sin\beta\\ \sin\beta&\cos\beta\end{array}\right)\\ &\\ &=\displaystyle\left(\begin{array}[]{lr}\cos\alpha\cos\beta-\sin\alpha\sin% \beta&-\cos\alpha\sin\beta-\sin\alpha\cos\beta\\ \sin\alpha\cos\beta+\cos\alpha\sin\beta&-\sin\alpha\sin\beta+\cos\alpha\cos% \beta\end{array}\right)\end{array}$

Hence, $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$ and $\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$.

Note that sine is an even function  and that cosine is an odd function, i.e. (http://planetmath.org/Ie) $\sin(-x)=-\sin x$ and $\cos(-x)=-\cos x$. These facts enable us to obtain the subtraction formulas for sine and cosine.

 $\sin(\alpha-\beta)=\sin(\alpha+(-\beta))=\sin(\alpha)\cos(-\beta)+\cos(\alpha)% \sin(-\beta)=\sin(\alpha)\cos(\beta)-\cos(\alpha)\sin(\beta)$
 $\cos(\alpha-\beta)=\cos(\alpha+(-\beta))=\cos(\alpha)\cos(-\beta)-\sin(\alpha)% \sin(-\beta)=\cos(\alpha)\cos(\beta)+\sin(\alpha)\sin(\beta)$