Inverting complex cosine

Question

I have been working out problem 3a in chapter 1 section 3 in Basic Complex Analysis by Marsden. He asks to solve $$ \cos z=\frac{3}{4}+\frac{i}{4} $$

After putting cosine in its exponential form and attempting to solve for $z$ via the quadratic formula I obtained: $$e^{iz}=\frac{3+i \pm\sqrt{6i-8}}{4} $$ My question is, how do I know which root to use, and is there an easy way to find the roots for this particular problem without taking a stab in the dark?

Answer 1

Use

$$\arccos(z)=-i\log(z\pm{i}\sqrt{1-z^2})$$

Answer 2

The square root of a complex number $c=a+bi$ is $$ \sqrt{(|c|+a)/2}+sign^+(b)\, i\sqrt{(|c|-a)/2} $$ where $sign^+(0)=+1$. The other square root is, as usual, obtained by multiplying with $-1$.

$$ \sqrt{6i-8}=\sqrt{(10-8)/2}+i\sqrt{(10+8)/2}=1+3i $$