Real part of a quotient Is there some fast way to know the real part of a quotient?
$$\Re\left(\frac{z_1}{z_2}\right)$$
$z_i\in \mathbb{C}$

At user130512's request in a comment, Euler's form. (I really suspect the calculation is going to get bad).
Let $z_1 = r_1e^{i\theta_1}$ and $z_2 = r_2e^{i\theta_2}$.
Now $\frac{z_1}{z_2} = \frac{r_1}{r_2}e^{i(\theta_1 - \theta_2)}$
Assuming $z_1 = a + ib, z_2 = c+id$
$$\Re\left(\frac{z_1}{z_2}\right) = \frac{r_1}{r_2}\cos\left(\arctan\left(\frac{b}{a}\right) - \arctan\left(\frac{d}{c}\right)\right)$$
Disclaimer This is for pure amusement. No practical point in doing this. 
