Answer
$(a)$
We have to find the reference angle for $\theta$, that is the acute angle formed by the terminal side of $\theta$ and the $x$-axis.
Find the quadrant where the terminal side is located and using the quadrant find its sign.
And calculate using the information.
$(b)$
$\sin \frac{5\pi}{6}=\frac{1}{2}$
Work Step by Step
$(a)$
To find the value of a trigonometric function of an angle $\theta$ we have to follow the following steps :
At first find the reference angle for $\theta$, that is the acute angle formed by the terminal side of $\theta$ and the $x-axis.$
Next find the quadrant where the terminal side is located and using the quadrant find its sign.
Using the above mentioned step we can calculate the value of the function at $\theta$.
$(b)$
The terminal side of the angle is located in the second quadrant, so the reference angle is :
$\pi - \frac{5\pi}{6}=\frac{\pi}{6}$
Also note that sine is positive in Quadrant $II$ and we can write :
$\sin \frac{5\pi}{6}=\sin \frac{\pi}{6}=\frac{1}{2}$
$\sin \frac{5\pi}{6}=\frac{1}{2}$
