Polar coordinates and polar graphs

In the polar coordinate system, a point is located by (r, theta): its distance r from the origin (the pole) and the angle theta measured from the positive x-axis. To convert to rectangular coordinates, use x = r cos(theta) and y = r sin(theta), so (4, pi/3) becomes (2, 2 times the square root of 3). To convert back, use r^2 = x^2 + y^2 and tan(theta) = y/x, choosing the angle in the correct quadrant: (-1, 1) becomes (square root of 2, 3pi/4). Unlike rectangular coordinates, a point has many polar names, since adding 2pi to theta, or using -r with theta + pi, lands in the same place. Polar equations describe curves that are awkward in x and y: r = 3 is a circle of radius 3 centered at the pole, theta = pi/4 is a line through the pole, r = 1 + cos(theta) is a heart-shaped cardioid, and r = 3 sin(2 theta) is a rose with 4 petals. Polar graphs can be drawn by plotting points for many angles or with technology.

This explanation is AI-generated and can occasionally be wrong. 🚩 Report