How do I write the expression cos(sin ^-1 u) as a function of u?

Question

How do I write the expression cos(sin ^-1 u) as a function of u?

Answer 1

If you look inside the parenthesis, sin^-1u = sin^-1(u/1) is some angle.  Let's call it θ.

 

 

We can rewrite this alternatively as sin(θ) = u/1.

 

 

From this and the definition of sine, the opposite side is u and the hypotenuse is 1.

 

 

From the Pythagorean theorem, the adjacent side is √(1 - u²).

 

 

So, cos(sin^-1u) = cos(θ) = √(1 - u²)/1 = √(1 - u²).