Question

How do you evaluate `csc^-1(sqrt2)`?

Answer 1

`pi/4`

Explanation:

Let `theta=csc^-1(sqrt2)`.

Since `csc(x)` and `csc^-1(x)` are inverse functions, this means that `csc(theta)=sqrt2`.

Another way of reaching that fact is to take the cosecant of both sides: `csc(theta)=csc(csc^-1(sqrt2))`, and since `csc(csc^-1(x))=x`, this becomes `csc(theta)=sqrt2`.

Taking the reciprocal of both sides, the equation becomes `sin(theta)=1/sqrt2=sqrt2/2`.

So, we want to find `theta`, or the angle where the value of sine is `sqrt2/2`.

This is a well known value of sine. It occurs when `theta=pi/4`, which means that `csc^-1(sqrt2)=pi/4`.