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Thread: cosine of arcsine of x?

  1. #1
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    cosine of arcsine of x?

    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)

    TYVM
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  2. #2
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    Re: cosine of arcsine of x?

    cosine of arcsine of x?-clipboard01.jpg
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  3. #3
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    Re: cosine of arcsine of x?

    Do you know that $sin^2(x)+ cos^2(x)= 1$?

    So $cos(x)= \sqrt{1- sin^2(x)}$.

    And $cos(arcsin(x))= \sqrt{1- sin^2(arcsin(x))}= \pm\sqrt{1- x^2}$.
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    Re: cosine of arcsine of x?

    Quote Originally Posted by HallsofIvy View Post
    Do you know that $sin^2(x)+ cos^2(x)= 1$?

    So $cos(x)= \sqrt{1- sin^2(x)}$.

    And $cos(arcsin(x))= \sqrt{1- sin^2(arcsin(x))}= \pm\sqrt{1- x^2}$.
    Yes I know that identity, lol.

    This makes perfect sense now, I totally see it. the sine and arcsine cancel leaving sqrt(1-x^2).
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