# How do you express cos(pi/ 3 ) * sin( ( 11 pi) / 8 )  without using products of trigonometric functions?

Apr 15, 2016

$= \frac{1}{2} \left[\sin \left(\frac{41 \pi}{24}\right) + \sin \left(\frac{25 \pi}{24}\right)\right]$

#### Explanation:

Use formula

$\sin \left(A + B\right) - \sin \left(A - B\right) = 2 \cos A \sin B$

$\cos A \sin B = \frac{1}{2} \left[\sin \left(A + B\right) - \sin \left(A - B\right)\right]$

$A = \frac{\pi}{3} \mathmr{and} B = \frac{11 \pi}{8}$

$\cos \left(\frac{\pi}{3}\right) \sin \left(\frac{11 \pi}{8}\right) = \frac{1}{2} \left[\sin \left(\frac{\pi}{3} + \frac{11 \pi}{8}\right) - \sin \left(\frac{\pi}{3} - \frac{11 \pi}{8}\right)\right]$

$= \frac{1}{2} \left[\sin \left(\frac{41 \pi}{24}\right) - \sin \left(\frac{- 25 \pi}{24}\right)\right]$

$= \frac{1}{2} \left[\sin \left(\frac{41 \pi}{24}\right) + \sin \left(\frac{25 \pi}{24}\right)\right]$

Apr 16, 2016

$- \left(\frac{1}{2}\right) \sin \left(\frac{3 \pi}{8}\right)$

#### Explanation:

$P = \cos \left(\frac{\pi}{3}\right) . \sin \left(\frac{11 \pi}{8}\right)$
Trig table -->$\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}$.
$\sin \left(\frac{11 \pi}{8}\right) = \sin \left(\frac{3 \pi}{8} + \pi\right) = - \sin \left(\frac{3 \pi}{8}\right)$
The product can be expressed as
$P = - \left(\frac{1}{2}\right) \sin \left(\frac{3 \pi}{8}\right)$

We can evaluate P by applying the identity: $\cos 2 a = 1 - 2 {\sin}^{2} a$
$\cos \left(\frac{6 \pi}{8}\right) = \cos \left(\frac{3 \pi}{4}\right) = - \frac{\sqrt{2}}{2} = 1 - 2 {\sin}^{2} \left(\frac{3 \pi}{8}\right)$
$2 {\sin}^{2} \left(\frac{3 \pi}{8}\right) = 1 + \frac{\sqrt{2}}{2} = \frac{2 + \sqrt{2}}{2}$
${\sin}^{2} \left(\frac{3 \pi}{4}\right) = \frac{2 + \sqrt{2}}{4}$
$\sin \left(\frac{3 \pi}{8}\right) = \frac{\sqrt{2 + \sqrt{2}}}{2}$ --> $\sin \left(\frac{3 \pi}{8}\right)$ is positive.
Finally,
$P = - \left(\frac{1}{2}\right) . \sin \left(\frac{3 \pi}{8}\right) = - \left(\frac{1}{4}\right) \sqrt{2 + \sqrt{2}}$