Question

How do you simplify `Sin(Cos^-1 x)`?

Answer 1

`sin(cos^(-1)(x)) = sqrt(1-x^2)`

Explanation:

Let's draw a right triangle with an angle of `a = cos^(-1)(x)`.

As we know `cos(a) = x = x/1` we can label the adjacent leg as `x` and the hypotenuse as `1`. The Pythagorean theorem then allows us to solve for the second leg as `sqrt(1-x^2)`.

With this, we can now find `sin(cos^(-1)(x))` as the quotient of the opposite leg and the hypotenuse.

`sin(cos^(-1)(x)) = sin(a) = sqrt(1-x^2)/1 = sqrt(1-x^2)`