Question

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

Answer 1

`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`

Answer 2

`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^@`