# How do you express cos(pi/ 3 ) * sin( ( 9 pi) / 8 )  without using products of trigonometric functions?

Jun 24, 2016

$P = - \left(\frac{1}{2}\right) \sin \left(\frac{\pi}{8}\right)$

#### Explanation:

Product $P = \cos \left(\frac{\pi}{3}\right) . \sin \left(\frac{9 \pi}{8}\right)$
Trig table --> $\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}$
$\sin \left(\frac{9 \pi}{8}\right) = \sin \left(\frac{\pi}{8} + \pi\right) = - \sin \left(\frac{\pi}{8}\right)$
P can be expressed as:
$P = - \left(\frac{1}{2}\right) \sin \left(\frac{\pi}{8}\right) .$
Note. We can evaluate $\sin \left(\frac{\pi}{8}\right)$ by using the trig identity:
$\cos 2 a = 1 - 2 {\sin}^{2} a .$
$\cos \left(\frac{2 \pi}{8}\right) = \cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} = 1 - 2 {\sin}^{2} \left(\frac{\pi}{8}\right)$
${\sin}^{2} \left(\frac{\pi}{8}\right) = 1 - \sqrt{2} = \frac{2 - \sqrt{2}}{4}$
$\sin \left(\frac{\pi}{8}\right) = \frac{\sqrt{2 - \sqrt{2}}}{2}$.
Take the positive value since sin (pi/8) is positive.
$P = - \left(\frac{1}{4}\right) \sqrt{2 - \sqrt{2}}$