Solve complex equation $\cos z = j\pi$

Question

How do you solve

$$\cos z = j\pi$$

I am not sure what to do with the right side. Can someone help me please?

Thank you

Answer 1

Note

$$\cos z=\sin(2n\pi+\frac\pi2 -z)=-i \sinh(i2n\pi+i \frac\pi2-iz)=i \pi$$

Then,

$$i2n\pi+i \frac\pi2-iz=-\sinh^{-1}\pi$$

which yields the solutions

$$z=2n\pi+ \frac\pi2-i \sinh^{-1}\pi$$