# What is the principal argument of $-5-5i$?

When i calculated it by $\tan^{-1}\dfrac{y}{x}$, I got $\dfrac{\pi}{4}$, then i added $\pi$ to make it in the right quadrant, so my final answer is $\dfrac{5\pi}{4}$

However, the correct answer is $-\dfrac{3\pi}{4}$...why is that?

• The "principal argument" is often taken to be normalized to lie in $(-\pi,\pi]$. Then instead of adding $\pi$, you should have subtracted $\pi$. – Daniel Fischer Jan 12 '14 at 15:32
• Now thats a good answer...thanks! :) – Mohdak Jan 12 '14 at 15:37

Using the definition of atan2, $$\arg(-5-5i)=\arctan\left(\frac{-5}{-5}\right)-\pi=\arctan1-\pi=\frac\pi4-\pi$$
• Your answer is absolutely correct except the notation that you need to use $\mathrm{Arg}$. – NasuSama Jan 12 '14 at 15:44