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the cartesian coordinates of a point are given (1,-3) find the polar coordinates `(r, theta)` where `r gt0` , `0<=theta<2pi` 

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Since the point (1,-3) is in the fourth quadrant, we can find the angle `theta` using the right angled triangle with sides 1, 3 and hypotenuse `sqrt{1+9}=sqrt10` .  The angle is going to then by (using the CAST rule)

`theta = 2pi - tan^{-1}(3/1)approx 5.03`

The value of r is the hypotenuse of the right angled-triangle, so `r=sqrt10` .

This means that the polar coordinates are `(r, theta)=(sqrt10, 5.03)` .

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