Precalculus Review

Trigonometric additions and subtractions

Let $\overline{OA} = \cos u$ on the -axis, and $\overline{OB} = \sin u$ on the -axis. By rotating the point and counterclockwise about the origin by the angle , we obtain the point and as in the figure below. Their respective coordinates are given by $(\overline{OA}\cos v, \overline{OA}\sin v)$ and $(-\overline{OB}\sin v, \overline{OB}\cos v)$ . By adding these two coordinates together, we can find the the coordinate on the unit circle

$\displaystyle (\cos(u+v),\sin(u+v)) = (\overline{OA}\cos v - \overline{OB}\sin v, \overline{OA}\sin v + \overline{OB}\cos v).$

$\includegraphics{lec13a.ps}$

By setting $\overline{OA} = \cos u$ and $\overline{OB} = \sin u$ , we can observe

$\displaystyle \cos(u + v) & = \cos u \cos v - \sin u \sin v$

and

$\displaystyle \sin(u + v) & = \cos u \sin v + \sin u \cos v = \sin u \cos v + \cos u \sin v.$

By replacing

in the above identities, we can find $\cos(u - v) & = \cos u \cos v + \sin u \sin v$ and $\sin(u - v) & = \sin u \cos v - \cos u \sin v$ .