Solve complex equation $4\cos(z)+5=0$

Question

I'm trying to solve this complex value equation

$$4\cos(z)+5=0$$

Should I use the following?

$$\sin(z) = \frac{e^{\boldsymbol{\mathrm i}z} - e^{-\boldsymbol{\mathrm i}z}}{2\boldsymbol{\mathrm i}}$$ $$ \cos(z) = \frac{e^{\boldsymbol{\mathrm i}z} + e^{-\boldsymbol{\mathrm i}z}}{2}$$

Answer 1

Note

$$-\frac54 = \cos z=-\cos(\pi-z+2\pi n)= -\cosh(-i(\pi-z+2\pi n))$$

which leads to $-i(\pi-z+2\pi n) =\pm \cosh^{-1}\frac54$ and the solutions

$$z = (1+2n)\pi \pm i\cosh^{-1}\frac54$$