Question

use reference angles to find the exact value of the following expression, csc(-\frac{\pi}{6}). select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. csc(-\frac{\pi}{6}) = (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. rationalize the denominator.) b. the answer is undefined.

Explanation:

Step1: Recall the definition of cosecant

$\csc(x)=\frac{1}{\sin(x)}$, so $\csc(-\frac{\pi}{6})=\frac{1}{\sin(-\frac{\pi}{6})}$.

Step2: Use the property of sine function

The sine - function is an odd function, i.e., $\sin(-x)=-\sin(x)$. So, $\sin(-\frac{\pi}{6})=-\sin(\frac{\pi}{6})$.

Step3: Find the value of $\sin(\frac{\pi}{6})$

We know that $\sin(\frac{\pi}{6})=\frac{1}{2}$. Then $\sin(-\frac{\pi}{6})=-\frac{1}{2}$.

Step4: Calculate $\csc(-\frac{\pi}{6})$

Since $\csc(-\frac{\pi}{6})=\frac{1}{\sin(-\frac{\pi}{6})}$, substituting $\sin(-\frac{\pi}{6}) = -\frac{1}{2}$, we get $\csc(-\frac{\pi}{6})=\frac{1}{-\frac{1}{2}}=-2$.

Answer:

A. $\csc(-\frac{\pi}{6})=-2$