The Cartesian coordinates of a point are given.?

Question

(3, −5)
(i) Find polar coordinates (r, θ) of the point, where
r > 0 and 0 ≤ θ < 2π.

(ii) Find polar coordinates (r, θ) of the point, where
r < 0 and 0 ≤ θ < 2π.

Answer 1

(3, −5)
(i) Find polar coordinates (r, θ) of the point, where
r > 0 and 0 ≤ θ < 2π.

I always suggest that you make a little picture, and calculate the triangle's hypotenuse. In this case, you have a point in the 4th quadrant, and the hypotenuse is √34. Use arctangent function (with calculator set to radians) to find the angle (it will be negative). Then add 2π to make the angle positive.

 

For second part, using negative r, that goes the wrong way (relative to the little picture), into quadrant 2, so you have to change the angle to agree with that wrong way position. Then the negative r takes you to the 4th quadrant--two wrongs make a right.