Question

How do you evaluate `arcsin -1/2`?

Answer 1

Consider an equilateral triangle with sides of length 2.
Each of the internal angles will be `pi/3` (i.e. `60^o`).

Now split the triangle into two mirror image right angled triangles.
The shortest side of each will have length `1`, and the smallest angle opposite it will be `pi/6` (i.e. `30^o`).

Then by definition `sin(pi/6) = 1/2` - the length of the shortest side divided by the length of the hypotenuse.

Now `sin(-theta) = -sin(theta)`, so

`sin(-pi/6) = -sin(pi/6) = -1/2`

The range of `arcsin` is `-pi/2 <= theta <= pi/2`.

`-pi/6` lies in this range so `arcsin(-1/2) = -pi/6`