Question

How do you find the exact value of `arcsin(1/6)`?

Answer 1

There's no way. The best we can do is the vacuous

`arcsin(1/6)`

Explanation:

Among angles that are rational multiples of `pi` or `360^circ` there aren't too many that have rational trig functions. In the first quadrant, besides `0^circ` and `90^circ,` there's `sin 30^circ`, `tan 45^circ` and `cos 60^circ`.

So we're certain the inverse sine of `1/6` is not rational. When we talk about an exact value we're willing to accept an irrational expression, integers composed via addition, subtraction, multiplication, division and root taking. That would generally mean it's the root of a low degree polynomial, which isn't the case here.

We can of course write down an infinite sum and call that the exact value. That doesn't seem right to me.