The rotation matrix 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 . Because rotating by $\alpha+\beta$ radians is the same as rotating by $\beta$ radians followed by rotating by $\alpha$ radians, we obtain:

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. $\sin(-x)=-\sin x$ and $\cos(-x)=-\cos x$ . These facts enable us to obtain the subtraction formulas for sine and cosine.

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

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