<s>[INST] Find complex solutions Find all solutions of the equation $4\sin(z) + 5 = 0$. 
So $4\sin(z) + 5 = 4\sin(x+iy) + 5 = 0$. Hence $\sin(x+iy) = \frac{-5}{4}$. So $x+iy = \sin^{-1}(\frac{-5}{4})$ which gives us $x+iy =$ ...
I am not sure what is really being asked here or how to achieve it. Help?
 [/INST] $\displaystyle\sin z=-\frac54$
$\displaystyle\cos z=\pm\sqrt{1-\frac{25}{16}}=\pm\frac{3i}4$
Using Euler formula, $$e^{iz}=\cos z+i\sin z=\pm\frac{3i}4+i\left(-\frac54\right)=-\frac{(5\mp3)i}4=-2i\text{ or }-\frac i2$$
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