Argument of (5,-5) in polar co-ordinate

Question

I am trying to convert $\ [5,-5] $ into polar form.I determined the argument of this co-ordinate. it is $\ \theta = \tan^{-1} \frac{-5}{5}= \frac{-\pi}{4}$. As the co-ordinate is fourth-quadrant.So,its range should be $\ \frac{3\pi}{2} \lt \theta \lt 2\pi $ So,I modify it like this,

argument z $ \ = 2\pi-\frac{\pi}{4}=\frac{7\pi}{4} $.But the answer is $\ \frac{-\pi}{4} $

I don't know what i have done wrong.

Answer 1

It is just a question of which representation is desired. $\frac {-\pi}4$ and $\frac {7\pi}4$ represent the same direction. Presumably whoever made the answer key expects arguments to be in the range $(-\pi,\pi]$ or $[-\pi,\pi)$and you are expecting arguments to be in the range $[0,2\pi)$. They both work.