precalculus: rectangle to polar coordinates

Question

1. (-sqrt(3)/2, -1/2)
2. (-3,3)

Answer 1

To get the radius r, use r = sqrt(x² + y²)

For the angle, you need to figure out what quadrant the point is in. And also use then use your knowledge of the sin and cos for the common angles (30^o, 45^o, 60^o) to figure out the angle.

So for 1, (-√3/2/-1/2)

First, r = sqrt( (-√3/2) + (-1/2) ) = sqrt( 3/4 + 1/4) = 1

For the angle, from your knowledge of the common angles, you know that if the cos (the x) = √3/2 and the sin (the y) = 1/2 then the "reference' angle is 30^o in Quadrant I. But (-√3/2/-1/2) is in Quadrant III. And the angle in Quadrant III for that reference angle is 180 + 30 = 210^o

The polar coordinate is (1, 210^o)

For (-3,3), you can calculate that the r = 3√2. And that the angle is in Quadrant II, and because the x and y are the same length, the "reference" angle is 45^o and that angle in Quadrant II for that angle is 180 - 45 = 135^o.

The polar coordinate is (3√2, 135^o)