Question

evaluate csc(5π/3) using a calculator. give an exact value, not an approximate answer.
csc(5π/3)=□
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall the definition of cosecant

$\csc(x)=\frac{1}{\sin(x)}$, so $\csc(\frac{5\pi}{3})=\frac{1}{\sin(\frac{5\pi}{3})}$.

Step2: Find the value of $\sin(\frac{5\pi}{3})$

The angle $\frac{5\pi}{3}$ is in the fourth - quadrant. The reference angle for $\frac{5\pi}{3}$ is $2\pi-\frac{5\pi}{3}=\frac{\pi}{3}$. In the fourth - quadrant, $\sin$ is negative. And $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$, so $\sin(\frac{5\pi}{3})=-\frac{\sqrt{3}}{2}$.

Step3: Calculate $\csc(\frac{5\pi}{3})$

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:

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