Question

How do i find csc(7pi/3)? the answer is (2√3)/3

how do i get to the answer

Answer 1

First off, express the cosecant in terms of sine:

`\csc(\frac{7\pi}{3})=\frac{1}{\sin(\frac{7\pi}{3})}`

However, the sine is a function of period `2\pi`, so:

`\sin(\frac{7\pi}{3})=\sin(2\pi+\frac{\pi}{3})=\sin(\frac{\pi}{3})`

You know that `\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}`. Plugging into our cosecant expression:

`\csc(\frac{7\pi}{3})=\frac{1}{\frac{\sqrt{3}}{2})=\frac{2}{\sqrt{3}}`

Amplifying the fraction by the root of `3`, we get:

`\csc(\frac{7\pi}{3})=\frac{2\sqrt{3}}{3}`.