Question

​Find the principal value of the following:

`cosec^-1(2cos  (2pi)/3)`

Answer 1

Let `cosec^-1(2cos  (2pi)/3)=y`

Then,

`cosec  y=2cos  (2pi)/3`

We know that the range of the principal value branch is `[-pi/2,pi/2]-{0}.`

Thus,

`cosec  y =2cos  (2pi)/3=2xx(-1)/2=-1=cosec(-pi/2).`

`=>y=-pi/2in[-pi/2,pi/2],y!=0`

Hence, the principal value of `cosec^-1(2cos  (2pi)/3)   is   -pi/2.`