# How do you find arcsin(-sqrt3/2) without a calculator?

Sep 19, 2015

First way: Use the Trig Table of Special Arcs
$\sin x = \left(- \frac{\sqrt{3}}{2}\right)$ --> arc x = -60 deg and arc x = -120 deg.

Second way: Use the trig unit circle.
In the triangle OMm, angle M = 60 deg ; OM = 1 unit ; Mm = 1/2.

Therefor,${\left(O m\right)}^{2} = 1 - \frac{1}{4} = \frac{3}{4}$ --> $O m = \frac{\sqrt{3}}{2}$

sin (-60) = sin (-120) = - $\frac{\sqrt{3}}{2}$

Sep 19, 2015

240

#### Explanation:

Since the value of the sine is negative, it should be in the third quadrant.

As we know Sin (π/3) = sqrt(3)/2
Hence sin^-1(-sqrt(3)/2) = (π+π/3)
Hence, Arcsin (-sqrt(3)/2) = 4π/3
In degrees, take π radians = 180 degrees.
Hence $A r c \sin \left(- \frac{\sqrt{3}}{2}\right) = 240 \mathrm{de} g r e e s$