Question

use reference angles to find the exact value of the following expression. do not use a calculator. sin(-210°). determine the reference angle for -210°. the reference angle is □°.

Explanation:

Step1: Find the positive - equivalent angle

Add 360° to - 210°.
$-210^{\circ}+360^{\circ}=150^{\circ}$

Step2: Determine the reference angle

Since 150° is in the second - quadrant ($90^{\circ}<150^{\circ}<180^{\circ}$), the reference angle $\theta_{r}=180^{\circ}-150^{\circ}=30^{\circ}$

Step3: Find the sine value

The sine function has the property $\sin(-\alpha)=-\sin\alpha$. Also, for an angle in the second - quadrant, $\sin\theta=\sin(180^{\circ}-\theta)$. So, $\sin(-210^{\circ})=\sin(150^{\circ})=\sin(180^{\circ} - 30^{\circ})=\sin30^{\circ}=\frac{1}{2}$

Answer:

The reference angle is $30^{\circ}$ and $\sin(-210^{\circ})=\frac{1}{2}$