I am trying to convert the following complex number into polar coordinates: $$A = -3 + i4.$$
I got $A = 5e^{-i53.13},$ by plugging in $\arctan\frac4{-3}= -53.13º.$ But the given solution uses $126.76º$ instead. I notice that the given angle is obtained via 180º - 53.13º, but I do not understand why this is the case?
This is not a check-my-work question; rather, I want to understand how $126.76º$ is equivalent to $-53.13º.$ If an angle is negative, it means that we are going clockwise with respect to the real axis; thus, to get the angle in the counterclockwise direction, shouldn't it be $360º - 53.13º ?$
Observe that $$\tan (120º)=\tan (-60º)=-\sqrt3,$$ and$$\arctan(-\sqrt3)=-60º\ne120º.$$
The point is that $\arctan$ has range $(-90º,90º),$ and that $$\large\tan\theta=s\;\iff \;\text{for some integer } k,\;\theta=\arctan(s)+180ºk.$$
Since $\arctan\left(\frac4{-3}\right)= -53.13º$ and $\,A=−3+4i\,$ lies in the second quadrant, $A$'s polar angle $\theta$ can be found using the above identity with $k=1.$
So, $\theta=126.76º,$ which matches the answer given to you.
Note that $$\large A\ne5e^{i126.76}\\\large A\ne5e^{i126.76º}\\\large A\ne5e^{i2.21\text{rad}}\\\large A=5e^{i2.21}.$$