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Thread: inverse trig functions

  1. #1
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    inverse trig functions

    how can I simplify this: tan(arcsin(x))
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  2. #2
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    Hello, xxju1anxx!

    Simplify: .$\displaystyle \tan\bigg[\arcsin(x)\bigg]$
    An inverse trig function is an angle.

    Let $\displaystyle \theta \,=\,\arcsin(x) \quad\Rightarrow\quad\sin\theta \,=\,x$

    We have: .$\displaystyle \sin\theta \:=\:\frac{x}{1} \:=\:\frac{opp}{hyp}$

    $\displaystyle \theta$ is in a right triangle with: .$\displaystyle opp = x,\;hyp = 1$
    . . Using Pythagorus, we find that: .$\displaystyle adj = \sqrt{1-x^2}$
    Hence: .$\displaystyle \tan\theta \:=\:\frac{opp}{adj} \:=\:\frac{x}{\sqrt{1-x^2}}$

    Therefore: .$\displaystyle \tan\bigg[\arcsin(x)\bigg] \;=\;\tan(\theta) \;=\;\frac{x}{\sqrt{1-x^2}} $

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