Question

use reference angles to find the exact value of the following expression. do not use a calculator. csc $\frac{5pi}{3}$ select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. csc $\frac{5pi}{3}$ = (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the answer.) b. the answer is undefined.

Explanation:

Step1: Determine the reference angle

The angle $\frac{5\pi}{3}$ is in the fourth - quadrant. To find the reference angle $\theta_{r}$, we use the formula $\theta_{r}=2\pi-\theta$ for angles in the fourth - quadrant. So, $\theta_{r}=2\pi-\frac{5\pi}{3}=\frac{6\pi - 5\pi}{3}=\frac{\pi}{3}$.

Step2: Recall the cosecant function relation

We know that $\csc\theta=\frac{1}{\sin\theta}$. Also, in the fourth - quadrant, $\sin\theta$ is negative. And $\sin\frac{5\pi}{3}=-\sin\frac{\pi}{3}$ (because sine is negative in the fourth - quadrant and $\sin(2\pi - \alpha)=-\sin\alpha$). Since $\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}$, then $\sin\frac{5\pi}{3}=-\frac{\sqrt{3}}{2}$.

Step3: Calculate the cosecant value

Since $\csc\frac{5\pi}{3}=\frac{1}{\sin\frac{5\pi}{3}}$, substituting $\sin\frac{5\pi}{3}=-\frac{\sqrt{3}}{2}$, we get $\csc\frac{5\pi}{3}=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2\sqrt{3}}{3}$.

Answer:

A. $\csc\frac{5\pi}{3}=-\frac{2\sqrt{3}}{3}$