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

1 Answer
May 9, 2016

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

Explanation:

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

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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)