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

Dec 27, 2016

#### Explanation:

Let's rationalize the denominator by multiplying the argument by 1 in the form of $\frac{\sqrt{2}}{\sqrt{2}}$:

$\theta = {\sin}^{-} 1 \left(- \frac{1}{\sqrt{2}} \frac{\sqrt{2}}{\sqrt{2}}\right)$

The denominator becomes 2 and the two numerators are multiplied:

$\theta = {\sin}^{-} 1 \left(- \frac{\sqrt{2}}{2}\right)$

$\frac{\sqrt{2}}{2}$ 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 $\theta = \frac{\pi}{4}$ 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 $\frac{\pi}{4}$

$\theta = \pi + \frac{\pi}{4}$

$\theta = \frac{5 \pi}{4}$

To make it the 4th quadrant subtract $\frac{\pi}{4}$ from $2 \pi$:

$\theta = 2 \pi - \frac{\pi}{4}$

$\theta = \frac{7 \pi}{4}$