Question

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

Answer 1

Please see the explanation.

Explanation:

Let's rationalize the denominator by multiplying the argument by 1 in the form of `sqrt(2)/sqrt(2)`:

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

The denominator becomes 2 and the two numerators are multiplied:

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

`sqrt(2)/2` is a well know value for the sine and the cosine. It is the point where they are equal to each other for the same value of `theta`; that value is `theta = pi/4` but, because of the negative sign, the angle must be in either the 3rd or the 4th quadrant.

To make it the 3rd quadrant add `pi` to `pi/4`

`theta = pi + pi/4`

`theta = (5pi)/4`

To make it the 4th quadrant subtract `pi/4` from `2pi`:

`theta = 2pi - pi/4`

`theta = (7pi)/4`