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

2 Answers
Mar 6, 2018

60 degrees or 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:sqrt3:2 ratio as seen below. Thus, we use the definition of sine to see that the 60 degree angle is opposite the sqrt3 side and the hypotenuse is the 2 side. Thus, since
sin(60^o)=sqrt3/2, sin^(-1)(sqrt3/2)=60^o
enter image source here

Mar 6, 2018

pi/3; (2pi)/3

Explanation:

Find arcsin (sqrt3/2)
sin x = sqrt3/2
Trig table gives as solution:
arc x = pi/3, or x = 60^@
Unit circle gives another arc x that has the same sin value (sqrt3/2)
x = pi - pi/3 = (2pi)/3, or x = 120^@