I am trying to write this fraction in polar form (4+10i)/(24i-5) . I am having trouble to get the angle of the polar conversion. I know that in order to get the angle I need to write arctan(10/4)-arctan(-24/5) which gives me 146.43 degrees. But the correct answer is 33.57 degrees when I do it by looking at the polar plane.
What am I doing wrong?
Since $-\frac{24}{5}$ is negative, $\tan^{-1} -\frac{24}{5}$ gives you the angle between $-90^\circ$ and $0$ in the fourth quadrant where the imaginary part is negative and the real part is positive. However, the number is $24i-5$, so we want the angle to be where the imaginary part is positive and the real part is negative, so add $180^\circ$ to get $\tan^{-1}\left(-\frac{24}{5}\right)+180^\circ$. This brings us into the second quadrant, which is where we want to be.
Now, subtract the first angle from the second angle to get the correct answer:
$$\tan^{-1} \frac{10}{4}-\left(\tan^{-1}\left(-\frac{24}{5}\right)+180^\circ\right) \approx -33.57^\circ$$