# Thread: cosine of arcsine of y?

1. ## cosine of arcsine of y?

I am doing a differential equation, where I have an initial condition, so I find the general solution, then I need to find explicit solution, and I have everything correctly, but I would like to know the rationale or logic of this part in my solution:

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

2. ## Re: cosine of arcsine of y?

consider a right triangle with hypotenuse length 1, side opposite angle $\theta$ of length $x$

by Pythagoras the side adjacent $\theta$ is $\sqrt{1-x^2}$

$\arcsin(x)=\theta$

$\cos(\arcsin(x))=\cos(\theta)=\dfrac{\sqrt{1-x^2}}{1} = \sqrt{1-x^2}$

3. ## Re: cosine of arcsine of y?

Equivalently, $sin^2(\theta)+ cos^2(\theta)= 1$ so $cos^2(\theta)= 1- sin^2(\theta)$ and $cos(\theta)= \pm\sqrt{1- sin^2(\theta)}$.

Taking $\theta= arcsin(x)$, $cos(arcsin(x))= \pm\sqrt{1- sin^2(arcsin(x))}= \pm \sqrt{1- x^2}$