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Thread: cosine of arcsine of y?

  1. #1
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    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)
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  2. #2
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    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 $\sqrt{1-x^2}$

    $\arcsin(x)=\theta$

    $\cos(\arcsin(x))=\cos(\theta)=\dfrac{\sqrt{1-x^2}}{1} = \sqrt{1-x^2}$
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  3. #3
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    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 $\theta= arcsin(x)$, $cos(arcsin(x))= \pm\sqrt{1- sin^2(arcsin(x))}= \pm \sqrt{1- x^2}$
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