Calculus expressions

Question

Find the exact value of the given expression.

 

sin(cos^-1 x)

Answer 1

Let θ = cos^-1x.  Then, cosθ = x = x/1

 

Draw a right triangle with acute angle θ, label the side adjacent to θ as x and label the hypotenuse 1.  By the Pythagorean Theorem, the side opposite to θ is √(1-x²).  

 

So, sin(cos^-1x) = sinθ = opposite/hypotenuse = √(1-x²).