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Math Help - Exact value of arcsin(2/3)-arcsin(-2/3)

  1. #1
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    Exact value of arcsin(2/3)-arcsin(-2/3)

    What is the Exact value of arcsin(2/3)-arcsin(-2/3)?
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  2. #2
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    Re: Exact value of arcsin(2/3)-arcsin(-2/3)

    Use WolframAlpha!

    It says 2arcsin \frac {2}{3} There is no convenient form of the exact value of the question. Are you looking for ten significant figures? Nine? Eight? Twenty? Use WolframAlpha! :P
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  3. #3
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    Re: Exact value of arcsin(2/3)-arcsin(-2/3)

    I guess it should be in terms of pi or square root.
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  4. #4
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    Re: Exact value of arcsin(2/3)-arcsin(-2/3)

    Quote Originally Posted by yeoky View Post
    I guess it should be in terms of pi or square root.
    I'm not seeing any wizardry that would allow this to be evaluated exactly. Are you sure you have the problem correct?
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  5. #5
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    Re: Exact value of arcsin(2/3)-arcsin(-2/3)

    $\displaystyle \sin^{-1}(x) = \sum_{k\ge 0} \dfrac{x^{1+2 k} \left(\tfrac{1}{2}\right)_k}{k!+2 k\cdot k!}$

    Plug in $x = \dfrac{2}{3}$

    The $\left(\tfrac{1}{2}\right)_k$ is the Pochammer symbol, meaning falling factorial.
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  6. #6
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    Re: Exact value of arcsin(2/3)-arcsin(-2/3)

    Quote Originally Posted by SlipEternal View Post
    $\displaystyle \sin^{-1}(x) = \sum_{k\ge 0} \dfrac{x^{1+2 k} \left(\tfrac{1}{2}\right)_k}{k!+2 k\cdot k!}$

    Plug in $x = \dfrac{2}{3}$

    The $\left(\tfrac{1}{2}\right)_k$ is the Pochammer symbol, meaning falling factorial.
    that's some wizardry all right!
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  7. #7
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    Re: Exact value of arcsin(2/3)-arcsin(-2/3)

    Quote Originally Posted by LimpSpider View Post
    Use WolframAlpha!
    Click Here
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  8. #8
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    Re: Exact value of arcsin(2/3)-arcsin(-2/3)

    I wonder if this is what they are after.

    $\arcsin(\frac 2 3) - \arcsin(- \frac 2 3)=2 \arcsin(\frac 2 3)$

    $y = 2 \arcsin(\frac 2 3)$

    $\sin(y) = \sin\left( 2 \arcsin(\frac 2 3)\right)=$

    $2 \sin\left(\arcsin(\frac 2 3)\right)\cos\left(\arcsin(\frac 2 3)\right)=$

    $2(\frac 2 3)(\frac {\sqrt{5}} 3)=\frac {4 \sqrt{5}} 9$

    $\sin(y)=\dfrac{4 \sqrt 5} 9$

    $y = \arcsin\left(\dfrac{4 \sqrt 5} 9\right)$

    It is the correct value.
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  9. #9
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    Re: Exact value of arcsin(2/3)-arcsin(-2/3)

    Complex Solution
    Click Here
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