Question

use reference angles to find the exact value. rationalize the denominator of your answer, if needed. select \undefined\ if applicable. csc(-210°) =

Explanation:

Step1: Use the property of cosecant function

We know that $\csc(x)=\frac{1}{\sin(x)}$ and $\csc(-\theta)=-\csc(\theta)$. So, $\csc(- 210^{\circ})=-\csc(210^{\circ})$.

Step2: Find the reference - angle

The angle $210^{\circ}$ is in the third - quadrant. The reference angle $\theta_{r}=210^{\circ}-180^{\circ} = 30^{\circ}$. And in the third - quadrant, $\sin(210^{\circ})=-\sin(30^{\circ})$. Since $\sin(30^{\circ})=\frac{1}{2}$, then $\sin(210^{\circ})=-\frac{1}{2}$.

Step3: Calculate the value of $\csc(210^{\circ})$

Since $\csc(x)=\frac{1}{\sin(x)}$, when $x = 210^{\circ}$, $\csc(210^{\circ})=\frac{1}{\sin(210^{\circ})}=\frac{1}{-\frac{1}{2}}=-2$.

Step4: Calculate the value of $\csc(-210^{\circ})$

Since $\csc(-210^{\circ})=-\csc(210^{\circ})$, then $\csc(-210^{\circ})=-(-2) = 2$.

Answer:

$2$