# How do you express sin(pi/12) * cos(( 5 pi)/8 )  without products of trigonometric functions?

Feb 27, 2016

$\sin \left(\frac{\pi}{12}\right) \cdot \cos \left(\frac{5 \pi}{8}\right) = - \frac{\sqrt{\left(1 - \frac{\sqrt{3}}{2}\right) \left(1 - \frac{\sqrt{2}}{2}\right)}}{2}$

#### Explanation:

$\left[1\right] \text{ } \sin \left(\frac{\pi}{12}\right) \cdot \cos \left(\frac{5 \pi}{8}\right)$

Think of $\frac{\pi}{12}$ as $\frac{\frac{\pi}{6}}{2}$.

$\left[2\right] \text{ } = \sin \left(\frac{\frac{\pi}{6}}{2}\right) \cdot \cos \left(\frac{5 \pi}{8}\right)$

Half angle identity: $\sin \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos \left(x\right)}{2}}$
**$\frac{\pi}{12}$ is in the first quadrant, and sine is positive in the first quadrant. Therefore, we know that $\sin \left(\frac{\frac{\pi}{6}}{2}\right) = + \sqrt{\frac{1 - \cos \left(\frac{\pi}{6}\right)}{2}}$

$\left[3\right] \text{ } = \sqrt{\frac{1 - \cos \left(\frac{\pi}{6}\right)}{2}} \cdot \cos \left(\frac{5 \pi}{8}\right)$

Evaluate $\cos \left(\frac{\pi}{6}\right)$

$\left[4\right] \text{ } = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} \cdot \cos \left(\frac{5 \pi}{8}\right)$

Think of $\frac{5 \pi}{8}$ as $\frac{\frac{5 \pi}{4}}{2}$.

$\left[5\right] \text{ } = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} \cdot \cos \left(\frac{\frac{5 \pi}{4}}{2}\right)$

Half angle identity: $\cos \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos \left(x\right)}{2}}$
**$\frac{5 \pi}{8}$ is in the second quadrant, and cosine is negative in the second quadrant. Therefore, we know that $\cos \left(\frac{\frac{5 \pi}{4}}{2}\right) = - \sqrt{\frac{1 + \cos \left(\frac{5 \pi}{4}\right)}{2}}$

$\left[5\right] \text{ } = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} \cdot \left(- \sqrt{\frac{1 + \cos \left(\frac{5 \pi}{4}\right)}{2}}\right)$

Evaluate $\cos \left(\frac{5 \pi}{4}\right)$

$\left[6\right] \text{ } = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} \cdot \left(- \sqrt{\frac{1 + \left(- \frac{\sqrt{2}}{2}\right)}{2}}\right)$

Simplify.

$\left[7\right] \text{ } = \textcolor{b l u e}{- \frac{\sqrt{\left(1 - \frac{\sqrt{3}}{2}\right) \left(1 - \frac{\sqrt{2}}{2}\right)}}{2}}$

I think you can leave it at that.