Question

​Find the principal value of the following:

`\text(cosec)^-1(2/sqrt3)`

Answer 1

Let `\text(cosec)^-1(2/sqrt3)=y`

Then,

`\text(cosec)  y=2/sqrt3`

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

Thus,

`\text(cosec)  y=2/sqrt3=text(cosec)(pi/3)`

`=>y=pi/3in[-pi/2,pi/2],y!=0`

Hence, the principal value of `\text(cosec)^-1(2/sqrt3)   is   pi/3`