Bounds for phase of sum of complex numbers

Question

The magnitude of the sum of complex numbers $z_1$ and $z_2$ is bound by $| |z_1| - |z_2| | \leq |z_1+z_2| \leq |z_1| + |z_2|$. I'm looking for a similar inequality for the phase of the sum $\angle(z_1+z_2)$.

For example, given $z_1 = 10e^{j\cdot 2}$ and $z_2 = 1e^{j\cdot \angle z_2}$ where $-\pi \leq \angle z_2 \leq \pi$, $|z_1+z_2|$ varies between 9 and 11 for all $\angle z_2$. $\angle(z_1+z_2)$ varies between 2-0.100167 and 2+0.100167.

I want a formula to get the phase's bounds, especially a term for the 0.100167 figure. I cannot assume $\angle z_2$, so the formula should not include that term. Intuitively, $|z_1|>>|z_2|$ seems to have something to do with it, so I expect the formula to contain the terms $|z_1|=10$ and $|z_2|=1$, in addition to $\angle z_1 = 2$.

Answer 1

I tried visualizing the sum $z_1+z_2$ as 2 vectors on a complex plane, $z_1$ having a fixed phase and $z_2$ varying its phase (compass would trace a circle):

1. If $|z_1|<|z_2|$, then $-\pi \leq \angle(z_1+z_2) \leq \pi$ (all angles).
2. If $|z_1| \geq |z_2|$, then $\angle z_1 - \arcsin(|z_2|/|z_1|) \leq \angle(z_1+z_2) \leq \angle z_1 + \arcsin(|z_2|/|z_1|) $. (the change in phase $|\angle(z_1+z_2)-\angle z_1|$ seems maximized when $z_2$ forms a right angle with $z_1+z_2$).

It checks out in the example: $\arcsin(1/10) \approx 0.1001674211615598$