<s>[INST] Solve $\cos(z)=\frac{3}{4}+\frac{i}{4}$ I tried solving this using the definition of $cos(z)=\frac{e^{iz}+e^{-iz}}{2}$ and equating it to $\frac{3}{4}+\frac{i}{4}$ and converting it to a complex quadratic equation through a substitution $t=e^{iz}$ and finding roots via the complex quadratic formula but it didn't seem to work. I would prefer solutions via elementary methods.
Here is my attempt: 
By definition we have $\frac{e^{iz}+e^{-iz}}{2}=\frac{3}{4}+\frac{i}{4} \implies e^{iz}+e^{-iz}=\frac{3}{2}+\frac{i}{2}$. Let $t=e^{iz}$ so then we have $t+\frac{1}{t}=\frac{3}{2}+\frac{i}{2}$ and if we multiply both sides by $t$ we have $t^2+1=(\frac{3}{2}+\frac{i}{2})t$ and hence $t^2+(\frac{3}{2}+\frac{i}{2})t+1=0$ By the quadratic formula for complex numbers we have, $a=1, b=\frac{3}{2}+\frac{i}{2}, c=1 \implies z=\frac{-(\frac{3}{2}+\frac{i}{2}) \pm \sqrt{(\frac{3}{2}+\frac{i}{2})^2-4(1)(1)}}{2(1)}$. Simplyifing we have $z=\frac{-\frac{3}{2}-(\frac{1}{2})i \pm \sqrt{-2+(\frac{3}{2})i}}{2}$ We wish to express $-2+(\frac{3}{3})i$ in polar form so we have $|-2+(\frac{3}{2})i|=\frac{5}{2}$. Now equating the real and imaginary parts we have $\frac{5}{2}\cos(\theta)=-2 \implies \cos(\theta)=-\frac{4}{5}$ and $\frac{5}{2}\sin(\theta)=\frac{3}{2} \implies \sin(\theta)=\frac{3}{5}$. From this we have $\tan(\theta)=-\frac{3}{4} \implies \theta=\arctan(-\frac{3}{4}) \approx -.6435$ rad. So we have $w=-2+(\frac{3}{2})i=\frac{5}{4}(\cos(\theta)+i\sin(\theta))=\frac{5}{4}e^{i\theta}$. By Proposition 1.3.12 we have $\sqrt{w}=\sqrt{\frac{5}{4}}e^{\frac{i\theta}{2}}=\frac{\sqrt{5}}{2}e^{\frac{i\theta}{2}}$. Similarily for $-\frac{3}{2}-(\frac{1}{2})=\frac{\sqrt{10}}{2}e^{i\varphi}$ Where $\varphi=\arctan(\frac{1}{3})$.  So finally we have $z=\frac{-(\frac{3}{2}+\frac{i}{2}) \pm \sqrt{(\frac{3}{2}+\frac{i}{2})^2-4(1)(1)}}{2(1)}=\frac{\sqrt{10}e^{i\varphi} \pm \sqrt{5}e^{\frac{i\theta}{2}}}{4}$ as solutions to $\cos(z)=\frac{3}{4}+\frac{i}{4}$.
 [/INST] I agree with your approach for the most part, but I think you've messed up on calculating $b$ in the quadratic formula:
$$e^{iz} + e^{-iz} -\frac{3}{2} - \frac{i}{2} = 0$$
Using $t = e^{iz}$:
$$t + \frac{1}{t} -\frac{3}{2} - \frac{i}{2} = 0$$
Multiplying by $t$:
$$t^2\color{red}{-}\left(\frac{3}{2} + \frac{i}{2}\right)t + 1  = 0$$
This is where your solution starts to go wrong.  Those minus signs are just lying in wait for the innocent mathematician! :P
Thus:
$$\begin{align}
x &= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\
&= \frac{\left(\frac{3}{2} + \frac{i}{2}\right) \pm \sqrt{\left(\frac{3}{2} + \frac{i}{2}\right)^2 - 4}}{2}\\
&= \frac{\left(\frac{3}{2} + \frac{i}{2}\right) \pm \sqrt{\left(\frac{9}{4} - \frac{1}{4}+\frac{3}{2}i\right) - 4}}{2}\\
&= \frac{\left(\frac{3}{2} + \frac{i}{2}\right) \pm \frac{1}{2}\sqrt{-8+6i}}{2}\\
&= \frac{3+i \pm (3i+1)}{4}\\
&= \cdots
\end{align}$$
Going from line three to line four:
$$\begin{align}
\frac{\left(\frac{3}{2} + \frac{i}{2}\right) \pm \sqrt{\left(\frac{9}{4} - \frac{1}{4}+\frac{3}{2}i\right) - 4}}{2} &= \frac{\left(\frac{3}{2} + \frac{i}{2}\right) \pm \sqrt{\left(\frac{8}{4}+\frac{3}{2}i\right) - 4}}{2}\\
&= \frac{\left(\frac{3}{2} + \frac{i}{2}\right) \pm \sqrt{\left(2+\frac{3}{2}i\right) - 4}}{2}\\
&= \frac{\left(\frac{3}{2} + \frac{i}{2}\right) \pm \sqrt{-2+\frac{3}{2}i}}{2}\\
&= \frac{\left(\frac{3}{2} + \frac{i}{2}\right) \pm \sqrt{-\frac{8}{4}+\frac{6}{4}i}}{2}\\
&= \frac{\left(\frac{3}{2} + \frac{i}{2}\right) \pm \frac{1}{2}\sqrt{-8+6i}}{2}\\
\end{align}$$
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