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Thread: Inverse Trigonometric Fnnction and when they cancel what their inverse does

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    Inverse Trigonometric Fnnction and when they cancel what their inverse does

    When do the inverse trigonometric functions cancel each other? I mean that while sin(arcsin(2/3))=2/3, you can't just do that here: arcsin(sin(5π/13)). when do they cancel each other and when you need to be careful? Thanks.

    In my homework I have examples that show that sin(arcsin(2/3))=2/3. So it is clear that the functions cancel each other. But in another example: arcsin(sin(5π/13π). Here, it shows a little work before they cancel what sin does: arcsin(sin(5π/13)=arcsin(sin(π−5π/13))=2π/7.




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    Re: Inverse Trigonometric Fnnction and when they cancel what their inverse does

    Quote Originally Posted by IloveIl View Post
    When do the inverse trigonometric functions cancel each other? I mean that while sin(arcsin(2/3))=2/3, you can't just do that here: arcsin(sin(5π/13)). when do they cancel each other and when you need to be careful? Thanks.

    In my homework I have examples that show that sin(arcsin(2/3))=2/3. So it is clear that the functions cancel each other. But in another example: arcsin(sin(5π/13π). Here, it shows a little work before they cancel what sin does: arcsin(sin(5π/13)=arcsin(sin(π−5π/13))=2π/7.



    Have a look at this table.

    Try to explain to yourself. Lookup the domain & range of the $\arcsin$ function.
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    Re: Inverse Trigonometric Fnnction and when they cancel what their inverse does

    but 5π/13 is outside the domain of arcsine.
    true, however the expression is $\arcsin\left[\sin\left(\dfrac{5\pi}{13}\right)\right]$, not $\arcsin\left(\dfrac{5\pi}{13}\right)$

    $0 < \sin\left(\dfrac{5\pi}{13}\right) < 1$ is in the domain of the arcsine function ...
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    Re: Inverse Trigonometric Fnnction and when they cancel what their inverse does

    Accidental double post.
    Last edited by HallsofIvy; Oct 2nd 2017 at 04:52 AM.
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    Re: Inverse Trigonometric Fnnction and when they cancel what their inverse does

    First, the trig functions, because they are not "one to one" do not have true "inverses". In order to have inverses, we have to restrict the domains. For example, we can take x from -\frac{\pi}{2} to \frac{\pi}{2} without values repeating. So arcsin(y) always lies between -\frac{\pi}{2} and \frac{\pi}{2}.

    But your example, arcsin(sin(\frac{5\pi}{13})), does not illustrate that. \frac{5\pi}{13} is less than \frac{\pi}{2} so arcsin(sin(\frac{5\pi}{13}))= \frac{5\pi}{13}! Using a calculator, \frac{5\pi}{13}= 1.2083, sin(1.2083)= 0.9350, and arcsin(0.9350)= 1.2083.
    Last edited by HallsofIvy; Oct 2nd 2017 at 04:53 AM.
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