What is the effect of taking the sine of inverse cosine?

Question

How can I evaluate the sine of an inverse cosine?

for example: sin(arccos((x)^1/2))

Answer 1

This is the full solution. I apologize that I do not know how to make/insert I diagram. Draw a right triangle with one leg of length $\sqrt{x}$ and hypotenuse $1$. By the Pythagorean theorem, we can conclude that the other leg has length $1-x^2$. Let $\theta$ be the angle so that the $\sqrt{x}$ leg is adjacent to it. Thus, $\cos(\theta) =\sqrt{x}$. Thus, $\theta = \arccos(\sqrt{x})$. Now consider $\sin(\theta) = \sqrt{1^2-x^2}/1=\sqrt{1-x^2}$ And that $\sin(\theta)=\sin(\arccos(\sqrt{x}))$ Thus, we have the final answer we wanted, $\sqrt{1-x^2}$.