How do you evaluate sin^-1(sqrt3/2) without a calculator?

Mar 6, 2018

$60$ degrees or $\frac{\pi}{3}$ radians

Explanation:

There is no easy way to do this but through memorization. Recall that in a right triangle with angle measures 30-60-90 degrees, the side lengths are at a $1 : \sqrt{3} : 2$ ratio as seen below. Thus, we use the definition of sine to see that the 60 degree angle is opposite the $\sqrt{3}$ side and the hypotenuse is the $2$ side. Thus, since
$\sin \left({60}^{o}\right) = \frac{\sqrt{3}}{2}$, ${\sin}^{- 1} \left(\frac{\sqrt{3}}{2}\right) = {60}^{o}$

Mar 6, 2018

pi/3; (2pi)/3

Explanation:

Find $\arcsin \left(\frac{\sqrt{3}}{2}\right)$
$\sin x = \frac{\sqrt{3}}{2}$
Trig table gives as solution:
arc $x = \frac{\pi}{3}$, or $x = {60}^{\circ}$
Unit circle gives another arc x that has the same sin value (sqrt3/2)
$x = \pi - \frac{\pi}{3} = \frac{2 \pi}{3}$, or $x = {120}^{\circ}$