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

1 Answer
sente
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)#

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