# Thread: Complex Values and Sin function

1. ## 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.

2. 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.

3. Originally Posted by blorpinbloo
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.