# What is sin^−1(-√3 / 2)?

Feb 25, 2018

sin^-1(sqrt3/2) = 240° or 300°

#### Explanation:

A right-angled triangle with angles of 60°and 30° has sides in the ratio $1 : \sqrt{3} : 2$

The core angle of $\frac{\sqrt{3}}{2}$ is 60°

However, the negative sign indicates that we are working with an angle in the $3 r d \mathmr{and} 4 t h$ quadrants.

In the $3 r d$ quadrant: theta = 180° + 60° = 240°

In the $4 t h$ quadrant: theta = 360°-60° =300°

sin^-1(sqrt3/2) = 240° or 300°

Check using a calculator.

sin 240° = -0.866
sin 300° = -0.866

$- \frac{\sqrt{3}}{2} = - 0.866$