# For what values of $z$ is $\sin z$ real/imaginary

For what values of $z$ is $\sin z$, $\cos z$ and $\tan z$ real/imaginary.

If someone points me in the right direction for the first one I think I will be able to solve the rest on my own.

• How do you define $\sin \sqrt {-1}$? – Apurv Oct 22 '15 at 7:39
• @Apurv: $$\sin\sqrt{-1}=\dfrac{e^{i\sqrt{-1}}-e^{-i\sqrt{-1}}}{2i}$$google.lk/… – Bumblebee Oct 22 '15 at 7:50
• @Nilan, thanks. – Apurv Oct 22 '15 at 7:55

$$\sin z=\sin(x)\cosh(y)+i\cos(x)\sinh(y)$$ real part$=0$ then $\sin(x)\cosh(y)=0$ iff $sinx=0\Rightarrow x=k\pi$.
Imaginary part $=0 \Rightarrow \cos(x)\sinh(y)=0$ thus $\cos x=0$ or $\sinh y=0$ which for first one $x=k\pi+\frac{\pi}{2}$ anf for the second one $y=0$
• $$\sin z=\sin(x)\cosh(y)+i\cos(x)\sinh(y)$$ This I have never seen before, can you tell me how to derive it? You don't have to do it yourself if you don't want to, just tell me how? You write $\sin z$ in the exponential form and then add and subtract something? – user269620 Oct 22 '15 at 8:12