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Thread: Can someone explain this formula of inverse sine.

  1. #1
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    Can someone explain this formula of inverse sine.

    sin(t)=a
    t=(-1)^k sin^(-1) (a) + πk

    please explain I do not understand the (-1)^k and
    πk

    Since sine has a period of 2
    π shouldnt the equation for all solution be t=sin^(-1)(a) +2πk?
    where does the (-1)^k and
    πk come from. How is this derived?
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  2. #2
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    Re: Can someone explain this formula of inverse sine.

    Quote Originally Posted by mjzxc View Post
    sin(t)=a
    t=(-1)^k sin^(-1) (a) + πk
    please explain I do not understand the (-1)^k and πk
    Since sine has a period of 2π shouldnt the equation for all solution be t=sin^(-1)(a) +2πk?
    where does the (-1)^k and πk come from. How is this derived?
    Frankly I have no idea what you meant in the above posting.
    To say that $\sin(\theta)=a$ means that $-1\le a\le 1$ but that says nothing about $\large\bf{\theta}$.
    If $\frac{-\pi}{2}\le\theta\le\frac{\pi}{2}$ then $\sin(\theta)$ takes on every possible value in $[-1,1]$.
    However, because these functions are periodic then $\sin(\theta)=\sin(\theta\pm 2k\pi),~k\in\mathbb{Z}$

    Now let us discuss the basics. The function $\arcsin(t)$ maps the interval $[-1,1]\to \left[ {\frac{{ - \pi }}{2},\frac{\pi }{2}} \right]$
    Now look at this webpage.
    While the principal value is $\dfrac{-\pi}{4}$ there are infinitely many other solutions: $\dfrac{-\pi}{4}+2k\pi,~k\in\mathbb{Z}$
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  3. #3
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    Re: Can someone explain this formula of inverse sine.

    Quote Originally Posted by Melody2 View Post
    Plato knows 'everything'
    And his explanation if the only one with any value. (According to him).
    So I will excuse myself. Sorry for any inconvenience.
    @Melody2
    It may have been that I pointed out a more compact way of doing it.
    In any case PLEASE DO NOT DELETE YOUR POST. It may help the student more than I can.
    It has been at least eleven years since I have taught an undergraduate course.
    So you may well be much be better than I at helping this student.
    Last edited by topsquark; Apr 13th 2017 at 01:44 AM.
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  4. #4
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    Re: Can someone explain this formula of inverse sine.

    I hope the following graphical and algebraic discussion helps.



    Thanks from Melody2
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