# Thread: solve basic trigonometric equation in complex plane

1. ## solve basic trigonometric equation in complex plane

$\displaystyle \sin z = \sin c \hfill \\$
$\displaystyle z = x + iy \hfill \\$
$\displaystyle c = a + ib \hfill \\$
$\displaystyle a,b,x,y \in \mathbb{R} \hfill \\$
$\displaystyle {\text{seperating the real and imaginary part I have the following system to solve}} \hfill \\$
$\displaystyle \left\{ \begin{gathered} \sin x\cosh y = \sin a\cosh b \hfill \\ \cos x\sinh y = \cos a\sinh b \hfill \\ \end{gathered} \right. \hfill \\$

But I don't see how I can solve this. Any idea?

2. You need to know this identity.
$\displaystyle \sin (x + yi) = \sin (x)\cosh (y) + i\left[ {\cos (x)\sinh (y)} \right]$

3. I do know this formula; this is how I reached the system I need to solve.

This is where the problem begins, at least for me.

civodul

4. Originally Posted by civodul
I do know this formula; this is how I reached the system I need to solve.
Well in that case, I would square both; add them together; use identities to eliminate one variable.

5. If $\displaystyle \sin(z)=\sin(c)$ then why can't we just write:

$\displaystyle z=-i\log\bigg(iw+\sqrt{1-w^2}\bigg),\quad w=\sin(c)$

6. Why do you not stay with your original question?
Of course there are many ways to solve this question!
Most mathematicians would have gone for $\displaystyle \sin (z) = \frac{{e^{iz} - e^{ - iz} }}{{2i}}$.

7. Hi Plato, Shawsend,