# How do you express sin(pi/ 4 ) * sin( ( 5 pi) / 12 )  without using products of trigonometric functions?

Jun 24, 2016

$\left(\frac{\sqrt{2}}{4}\right) \sqrt{2 + \sqrt{3}}$

#### Explanation:

Product $P = \sin \left(\frac{\pi}{4}\right) . \sin \left(\frac{5 \pi}{12}\right)$
Trig table -->
$\sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$
P can be expressed as:
$\left(\frac{\sqrt{2}}{2}\right) . \sin \left(\frac{5 \pi}{12}\right) .$
We can evaluate $\sin \left(\frac{5 \pi}{12}\right)$ by using the trig identity:
$\cos 2 a = 1 - 2 {\sin}^{2} a$
$\cos \left(\frac{10 \pi}{12}\right) = \cos \left(\frac{5 \pi}{6}\right) = - \frac{\sqrt{3}}{2} = 1 - 2 {\sin}^{2} \left(\frac{5 \pi}{12}\right)$
2sin^2 ((5pi)/12 = 1 + sqrt3/2 = (2 + sqrt3)/2
${\sin}^{2} \left(\frac{5 \pi}{12}\right) = \frac{2 + \sqrt{3}}{4}$
$\sin \left(\frac{5 \pi}{12}\right) = \frac{\sqrt{2 + \sqrt{3}}}{2}$ (note: sin ((5pi)/12) is positive)
Finally,
$P = \left(\frac{\sqrt{2}}{4}\right) \left(\sqrt{2 + \sqrt{3}}\right)$