How do you convert six units at 30 degrees Polar Coordinate into a Cartesian Coordinate?



I would start by sketching a diagram.

In my diagram O is the pole, which will be the origin in the Cartesian system, and P is the point with polar coordinates (r,theta) = (6,30o). I drew a perpendicular line from P to Q on the polar line, which will be the X-axis in the Cartesian system. I now have a right triangle POQ where the length of OQ is the X-coordinate of P and the length of PQ is the Y-coordinate of P. I can find these lengths in two ways.

First method:

I know that any triangle with angles 30, 60 and 90 degrees has side lengths in the ratio if 1 to 2 to √3. (These triangles come up so often that this is a fact I remember.) 2 is the length of the hypotenuse, 1 is the length of the shorter leg and √3 is the length of the longer leg. In the triangle above the length of the hypotenuse is 6 units so the length of the shorter leg, |PQ| is half of that which is 3 units. The length of the longer leg is √3 times the length of the shorter leg so |OQ| = √3 3. Thus P has Cartesian coordinates (3√3, 3).

Second Method:

Use trig functions.

sin(30o) = |PQ|/|OP||PQ|/61/2 and therefore |PQ| = 3.

cos(30o) = |OQ|/|OP||OQ|/6√3/2 and therefore |PQ| = 3 √3.

Thus P has cartesian coordinates (3 √3, 3)