Question

How do you evaluate `arccos(sqrt3/2)` without a calculator?

Answer 1

`arccos(sqrt(3)/2)=30^@ (=pi/6 " radians")`

Explanation:

If we split an equilateral triangle (in which all interior angles `=60^@ or pi/3 " radians"`) into two triangles as indicated:

we will get an angle `=(60^@)/2=30^@` whose `cos` is `sqrt(3)/2`

(Note that there is also the angle `330^@` which gives this `sqrt(3)/2` ratio, but by the standard definition of `cos^(-1)` as a function we are restricted to the range `[0,pi]`