### 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 find the distance and the angle of the point from the origin.
The distance, or the radius, is given by the Pythagorean theorem: $r = \sqrt{x^2 + y^2}.$
Plugging in the coordinates of the point, I get $r = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = 2.$
The angle, or the polar angle, is given by the inverse tangent of the ratio of the y-coordinate to the x-coordinate: $\theta = \tan^{-1}(y/x).$
Plugging in the coordinates of the point, I get $\theta = \tan^{-1}(-\sqrt{3}/1) = -\pi/3.$
However, this angle is negative and not in the range $[0,2\pi).$
To adjust it, I need to add $2\pi$ to it: $\theta = -\pi/3 + 2\pi = 5\pi/3.$