real and imaginary part in $\sin z$ where z is complex

Question

I wanted to know, how can I determine the real and imaginary part in

$\sin z$ where $z \in \Bbb{C}$?

Well, this is a part of a series of questions comprising the same in

$\log z$ and $\tan^{-1} z$

I was able to solve this but no idea on how to solve for $\sin z$.

Any help appreciated.

Thanks.

score 14 · Accepted Answer · answered Jul 22 '13 at 16:27

14

If $z=x+iy, \sin(x+iy)=\sin x\cos (iy)+\cos x\sin(iy)=\sin x\cosh y+i\cos x\sinh y$ using the relationship between Hyperbolic & Trigonometric ratios

answered Jul 22 '13 at 16:27

lab bhattacharjee

score 7 · Answer 2 · answered Jul 22 '13 at 16:10

7

Hint

$$\sin z= \frac{1}{2i}\left(e^{iz}-e^{-iz}\right)\\ e^{ix}=\cos x+i\sin x$$

answered Jul 22 '13 at 16:10

score 4 · Answer 3 · answered Jul 22 '13 at 16:10

4

HINT: Use the Euler formula for $\sin(z)$. Be sure to take care and define $z=x+iy$ where $x,y\in\Bbb R$.

answered Jul 22 '13 at 16:10

Clayton

score 4 · Answer 4 · answered Jul 22 '13 at 16:12

4

We know that $$\sin z= \frac{e^{iz}-e^{-iz}}{2i}$$

From this definition we can get the real and imaginary parts.

answered Jul 22 '13 at 16:12

guestgoijeoiejr

4 Answers4