Find the principal values of each of the following: cosec^-1(2cos 2π/3)

Question

Find the principal values of each of the following:

cosec^-1(2cos \(\frac{2\pi}3\))

Answer 1

\(\therefore\) The question converts to cosec^-1(-1)

Now,

The range of principal value of cosec^-1 is 

\([\frac{-\pi}2,\frac{\pi}2]\) - {0} and cosec\((\frac{-\pi}2)=-1\)

Therefore, the principal value of cosec^-1(2cos\(\frac{2\pi}3\)) is \(\frac{-\pi}2.\)