Trigonometric additions and subtractions

Since the coordinates of $ A'$ and $ B'$ in the figure below are given respectively by

$\displaystyle (\overline{OA}\cos v,  \overline{OA}\sin v)$    and $\displaystyle \quad
(-\overline{OB}\sin v,  \overline{OB}\cos v),

the coordinate $ (\cos(u+v),\sin(u+v))$ on the unit circle can be expressed as

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


Together with $ \overline{OA} = \cos u$ and $ \overline{OB} = \sin u$, we can find

$\displaystyle \cos(u + v)$ $\displaystyle = \cos u \cos v - \sin u \sin v;$ (3-1)
$\displaystyle \sin(u + v)$ $\displaystyle = \cos u \sin v + \sin u \cos v$    
  $\displaystyle = \sin u \cos v + \cos u \sin v.$ (3-2)

By replacing $ v$ by $ (-v)$ in both (3-1) and (3-2), we obtain

$\displaystyle \cos(u - v)$ $\displaystyle = \cos u \cos v + \sin u \sin v;$ (3-3)
$\displaystyle \sin(u - v)$ $\displaystyle = \sin u \cos v - \cos u \sin v.$ (3-4)

Department of Mathematics
Last modified: 2005-09-29