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Thread: Inverse Trig Identity

  1. #1
    Junior Member Greymalkin's Avatar
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    Inverse Trig Identity

    The actual problem is: prove that $\displaystyle sin^-1x=tan^-1{x\over {\sqrt{1-x^2}}$

    My problem is: how does $\displaystyle sin(tan^-1{x\over \sqrt{1-x^2}})$ equal x??
    I understand how $\displaystyle sin(sin^-1x)=x$ but not the rational inverse, why is it so???
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  2. #2
    MHF Contributor
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    Re: Inverse Trig Identity

    The idea is that if $\displaystyle x = \sin\alpha$, then $\displaystyle \sin^{-1}x=\alpha$. We have $\displaystyle \frac{\sin\alpha}{\sqrt{1-\sin^2\alpha}}=\tan\alpha$, so $\displaystyle \tan^{-1}\frac{x}{\sqrt{1-x^2}}$ is indeed $\displaystyle \alpha$.

    One has to take care of various subtleties such as $\displaystyle \sqrt{x^2}=|x|$ rather than $\displaystyle x$ and $\displaystyle \sin^{-1}(\sin\alpha)=\alpha$ only for $\displaystyle \alpha\in[-\pi/2,\pi/2]$.
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  3. #3
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    Re: Inverse Trig Identity

    Hello, Greymalkin!

    $\displaystyle \text{Prove: }\:\sin^{\text{-}1}\!x\:=\:\tan^{\text{-}1}\!\left(\frac{x}{\sqrt{1-x^2}}\right)$

    $\displaystyle \text{Let }\,\theta \:=\:\sin^{\text{-}1}\!x \;\;[1]$

    $\displaystyle \text{Then: }\,\sin\theta \:=\:x \:=\:\frac{x}{1} \:=\:\frac{opp}{hyp}$

    $\displaystyle \text{We see that }\theta\text{ is in a right triangle with: }\:\begin{Bmatrix}opp \:=\:x \\ hyp \:=\:1 \end{Bmatrix}$
    $\displaystyle \text{From Pythagorus: }\:adj \:=\:\sqrt{1-x^2}$

    $\displaystyle \text{Then: }\:\tan\theta \:=\:\frac{opp}{adj} \:=\:\frac{x}{\sqrt{1-x^2}}$

    $\displaystyle \text{Hence: }\:\theta \:=\:\tan^{\text{-}1}\!\left(\frac{x}{\sqrt{1-x^2}}\right) \;\;[2]$


    $\displaystyle \text{Equating [1] and [2]: }\:\sin^{\text{-}1}\!x \;=\;\tan^{\text{-}1}\!\left(\frac{x}{\sqrt{1-x^2}}\right) $
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