Question

How do you evaluate `csc^-1(-sqrt2)` without a calculator?

Answer 1

`(5pi)/4; (7pi)/4`

Explanation:

`csc x = -sqrt2` --> `sin x = 1/(csc) = - 1/sqrt2 = - sqrt2/2`
`csc^-1(-sqrt2) --> arcsin (-sqrt2/2)`
Trig table and unit circle give 2 solutions:

`x = -pi/4`, or `x = (7pi)/4` (co-terminal), and
`x = pi - (-pi/4) = (5pi)/4`