### Instruction: Convert the point $(1,-\sqrt{3})$ in rectangular coordinates to polar coordinates.  Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2 \pi.$.### Response: To convert from rectangular to polar coordinates, I need to use the formulas $r = \sqrt{x^2 + y^2}$ and $\theta = \tan^{-1}(y/x),$ where I have to be careful about the quadrant of the point.
Plugging in the given coordinates, I get $r = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{4} = 2.$
For $\theta,$ I have $\theta = \tan^{-1}(-\sqrt{3}/1) = \tan^{-1}(-\sqrt{3}),$ which is the same as $\theta = -\pi/3,$ but that is not in the required interval.
To find the equivalent angle in the interval $[0,2\pi),$ I have to add $2\pi$ to $-\pi/3,$ which gives me $\theta = 5\pi/3.$
So the polar coordinates are $(2,5\pi/3).$