How to correctly get phase of a complex number

Question

If $z = -99.75 + 3i $ is a complex number.

How can I calculate the Argument of $z$?

I tried $$\arctan\left(\frac{3}{99.75}\right) = 1.72°$$ but if I try to compute it on wolfram the result is 178.712° (that should be correct) how do I get here?

Answer 1

$z=-99.75+3\iota$ is the point $P(-99.75,3)$ in the $x-y$ plane, which is in the $2^{nd}$ quadrant, so the angle $AP$ ($A$ is the origin) is making with the positive $x-$axis (which is defined as $\arg(z)$), is $$90^\circ+\angle CAP=90^\circ+\arctan\left(\dfrac{CP}{CA}\right)=90^\circ+\arctan\left(\dfrac{99.75}3\right)\approx 178.28^\circ$$

$\arg(z)$ is usually defined modulo $180^\circ$, i.e. depending on the location of $z$, the angle the vector $\vec{Az}$ (formed by joining the origin to the point $z$) makes with the positive $x-$axis, measured in the counter-clockwise direction (positive angle) or clockwise direction (negative angle), whichever gives the least absolute value.