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

how do i get to the answer

1 Answer
Mar 28, 2018

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}#.