Question

use reference angles to find the exact value of the following expression. do not use a calculator. determine the reference angle for $csc\frac{5pi}{3}$. the reference angle is $\frac{pi}{3}$. (type your answer in radians. use integers or fractions for any numbers in the expression. type an exact answer, using $pi$ as needed.) 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 expression. rationalize the denominator.) b. the answer is undefined.

Explanation:

Step1: Recall the definition of cosecant

The cosecant function is defined as $\csc\theta=\frac{1}{\sin\theta}$. We need to find $\sin(\frac{5\pi}{3})$.

Step2: Determine the reference - angle

The angle $\theta = \frac{5\pi}{3}$ is in the fourth - quadrant. The reference angle $\theta_{r}$ for $\theta=\frac{5\pi}{3}$ is $2\pi-\frac{5\pi}{3}=\frac{\pi}{3}$.

Step3: Find the sine value

In the fourth - quadrant, $\sin\theta$ is negative. Since $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$, then $\sin(\frac{5\pi}{3})=-\frac{\sqrt{3}}{2}$.

Step4: Calculate the cosecant value

Since $\csc\theta=\frac{1}{\sin\theta}$, then $\csc(\frac{5\pi}{3})=\frac{1}{\sin(\frac{5\pi}{3})}=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2\sqrt{3}}{3}$.

Answer:

$-\frac{2\sqrt{3}}{3}$