prove $\sin(3z)=A$ has a solution for every $A\in\mathbb{C}$ without Picards theorem

Question

I want to prove that $\sin(3z)=A$ has a solution for every $A\in\mathbb{C}$. I read a lot about a Picard theorem, but I've never had this before and I still need to prove this. I'm not sure how to do this. Should I write $$\sin(z)=\frac{e^{iz}-e^{-iz}}{2i}$$ or should I write it in a power series? Hope someone can help me!

Answer 1

$$\begin{array}{rcl} \sin(3z) &=& A \\ \dfrac{e^{3iz}-e^{-3iz}}{2i} &=& A \\ \dfrac{e^{6iz}-1}{2ie^{3iz}} &=& A \\ e^{6iz}-1 &=& 2Aie^{3iz} \\ \left(e^{3iz}\right)^2 - 2Aie^{3iz} - 1 &=& 0 \\ e^{3iz} &=& \dfrac{2Ai \pm \sqrt{-4A^2+4}}{2} \\ 3iz &=& \ln \left(\dfrac{2Ai \pm \sqrt{-4A^2+4}}{2}\right) \\ z &=& \dfrac1{3i}\ln \left(\dfrac{2Ai \pm \sqrt{-4A^2+4}}{2}\right) \\ \end{array}$$