Question

What is the `sin^(-1)(sqrt(3)/2)`?

Answer 1

`sin^(-1)(sqrt(3)/2) = pi/3` (radians)

Explanation:

`sin^(-1)(sqrt(3)/2) = theta`
`color(white)("XX")`means
`sin(theta)=sqrt(3)/2`
`color(white)("XXXX")`with the restriction that `theta in [-pi/2,pi/2]`

The possible triangles with this ratio of `y` to `"hypotenuse"` (i.e. `sin`) are pictured below:

...but note that only `pi/3` falls within the restricted range for `sin^(-1)`

Answer 2

`theta` = 60° or 120°

Explanation:

It helps if one recognises the lengths 3 and `sqrt2` as
being two of the sides in one of the special triangle,
namely 30°, 60° 90°.

Sketch this by drawing an equilateral triangle of sides 2 units, and drawing in a line of symmetry to form two right-angled triangles.

The hypotenuse is of length 2 units and the opposite side
is of length `sqrt3`. The angle is therefore 60°.

However, the sign is positive. Sin is positive in the first and second quadrants.

Therefore `theta` = 60° or 120°