Find the exact value of the trigonometric function $\sin 7\pi/ 6$

Question

I am finding it a little difficult to solve this problem. The reference angle for $\sin 7\pi/6$ is sin 30 degrees (I think) which is sine 1/2. But that is not the answer. How do I sove this problem?

Answer 1

The radian angle $\frac{7 \pi}{6}$ is in the third quadrant of the unit circle, and so using your reference angle, the sine of that angle is negative. Hence $$\sin\left( \frac{7 \pi}{6} \right)=-\sin\left( \frac{\pi}{6} \right)=-\frac{1}{2}.$$

Answer 2

The first thing I do when I get a weird angle is convert the angle into degrees to figure out where the angle actually is, so for your angle we have:

$$\frac{7\pi}{6} = 210^\circ$$

We know what $210^\circ$ is in the third quadrant and we have to figure out what sign $\sin$ is in that quadrant. The quickest way to do that is to use the CAST rule

The CAST rule tells us when $\sin, \cos, \tan$ are positive or negative. If you look at the third quadrant then you will see that only $\tan$ is positive there and therefore, $\sin$ is negative. Using that information, we can figure out the sign and then get our answer which is $-\frac{1}{2}$.