### 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 from the origin and the angle measured from the positive x-axis to the point.
The distance from the origin, or $r$, is given by the Pythagorean theorem: $r = \sqrt{x^2 + y^2}.$
Plugging in the given coordinates, I get $r = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = 2.$
The angle, or $\theta$, is given by the inverse tangent function: $\theta = \tan^{-1}(y/x).$
Plugging in the given coordinates, I get $\theta = \tan^{-1}(-\sqrt{3}/1) = -\pi/3.$
However, this angle is in the fourth quadrant, and I want an angle in the range $[0,2\pi).$
To find the equivalent angle in the standard position, I need to add $2\pi$ to the negative angle: $\theta = -\pi/3 + 2\pi = 5\pi/3.$
Therefore, the polar coordinates of the point are $(2,5\pi/3).$