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Math Help - Complex Values and Sin function

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    Complex Values and Sin function

    Here is the question: "Prove that sin takes on every complex value."

    I know sin(z) = [e^(iz) - e^(-iz)]/2i

    for complex z.

    But I'm having a hard time converting that right hand side into something which reasonably shows that sin is able to take on any complex value.
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    Super Member Aryth's Avatar
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    This proof may include the proof that the exponential can take on all complex values except 0, which would then mean that you would have to show that sin can take the complex value of 0 to complete the proof.
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  3. #3
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    Quote Originally Posted by blorpinbloo View Post
    Here is the question: "Prove that sin takes on every complex value."

    I know sin(z) = [e^(iz) - e^(-iz)]/2i

    for complex z.

    But I'm having a hard time converting that right hand side into something which reasonably shows that sin is able to take on any complex value.
    Show that, for any complex number, y, you can solve sin(z)= \frac{e^{iz}- e^{-iz}}{2i}= y.

    For example, you can multiply through by 2ie^{iz} and get e^(2iz)- 1= 2ie^{iz}y. Now let v= e^{iz} and you have a quadratic equation to solve for v.
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