Question

How do you evaluate `tan(csc^-1(-2))` without a calculator?

Answer 1

`tan(csc^-1(sqrt(7)/2))=1/sqrt3`

Explanation:

Let `theta=csc^-1(-2)`

or

`csctheta=-2`

or

`1/sintheta=-2`

or

`sintheta=-1/2`

or

`theta=sin^-1(-1/2)`

Hence

`p/h=1/2` (Since `theta` lies in third quadrant; hence `-` sign not taken in consideration)

`costheta=b/h`

or

`costheta=sqrt(h^2-p^2)/h`

or

`costheta=sqrt((2^2-1^2))/2`

or

`costheta=sqrt(4-1)/2`

or

`costheta=sqrt3/2`

Therefore
`tantheta=sintheta/costheta`

or

`tan theta=(1/2)/((sqrt3)/2`

or

`tan theta=1/sqrt3`

or

`tan(csc^-1(sqrt(7)/2))=1/sqrt3`