# How do you find the exact value of csc ((5pi)/6) using the half angle formula?

Jun 29, 2016

$\csc 5 \frac{\pi}{6} = 2.$

#### Explanation:

Half-angle formula for $\sin$ is : $\sin \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}} ,$ where sign is to be taken properly.

Putting, $\theta = 5 \frac{\pi}{3}$, we get,
$\sin \left\{\frac{5 \frac{\pi}{3}}{2}\right\} = \sin \left(5 \frac{\pi}{6}\right) = \pm \sqrt{\frac{1 - \cos 5 \frac{\pi}{3}}{2}}$

Since, $\sin \left(5 \frac{\pi}{6}\right) = \sin \left(\pi - \frac{\pi}{6}\right) , 5 \frac{\pi}{6}$ lies in the $I {I}^{n d}$ Quadrant, $+ v e$ sign has to be taken

But, $\cos 5 \frac{\pi}{3} = \cos \left(2 \pi - \frac{\pi}{3}\right) = \cos \left(\frac{\pi}{3}\right) = \frac{1}{2.}$
$\therefore \sin 5 \frac{\pi}{6} = \sqrt{\frac{1 - \frac{1}{2}}{2}} = \sqrt{\frac{1}{4}} = \frac{1}{2.}$

Hence, $\csc \left(5 \frac{\pi}{6}\right) = \frac{1}{\sin} \left(5 \frac{\pi}{6}\right) = 2.$

Jun 30, 2016

= 2

#### Explanation:

We can evaluate csc ((5pi)/6) without using half angle formula.
$\csc \left(\frac{5 \pi}{6}\right) = \frac{1}{\sin} \left(\frac{5 \pi}{6}\right)$.
Find $\sin \left(\frac{5 \pi}{5}\right) .$
Trig table, and unit circle -->
$\sin \left(\frac{5 \pi}{6}\right) = \sin \left(- \frac{\pi}{6} + \frac{6 \pi}{6}\right) = \sin \left(- \frac{\pi}{6} + \pi\right) =$
$= \sin \left(\frac{\pi}{6}\right) = \frac{1}{2}$
Therefor,
$\csc \left(\frac{5 \pi}{6}\right) = \frac{1}{\sin} = 2$