Question

use reference angles to find the exact value of the following expression. do not use a calculator. csc(5π/3) determine the reference angle for 5π/3. the reference angle is . (type your answer in radians. use integers or fractions for any numbers in the expression. type an exact answer, using π as needed.)

Explanation:

Step1: Determine the quadrant

The angle $\theta=\frac{5\pi}{3}$ lies in the fourth - quadrant since $ \frac{3\pi}{2}<\frac{5\pi}{3}<2\pi$.

Step2: Calculate the reference angle

The formula for the reference angle $\theta'$ of an angle $\theta$ in the fourth - quadrant is $\theta' = 2\pi-\theta$. So, $\theta'=2\pi-\frac{5\pi}{3}=\frac{6\pi - 5\pi}{3}=\frac{\pi}{3}$.

Step3: Recall the cosecant function property

We know that $\csc\theta=\frac{1}{\sin\theta}$. Also, in the fourth - quadrant, $\sin\theta<0$. And $\sin\frac{5\pi}{3}=-\sin\frac{\pi}{3}$ (because of the angle - relationship in the unit - circle). Since $\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}$, then $\sin\frac{5\pi}{3}=-\frac{\sqrt{3}}{2}$.

Step4: Calculate the cosecant value

$\csc\frac{5\pi}{3}=\frac{1}{\sin\frac{5\pi}{3}}=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2\sqrt{3}}{3}$.

Answer:

The reference angle is $\frac{\pi}{3}$; $\csc\frac{5\pi}{3}=-\frac{2\sqrt{3}}{3}$