Definition: Polar coordinates locate a point by its distance r from the origin and its angle theta from the positive x-axis, written as the ordered pair (r, theta).
How do polar coordinates work?
Instead of moving right and up, you turn through an angle theta and then walk r units along that ray. The conversions are x = r cos theta, y = r sin theta, and r2 = x2 + y2.
How do you convert a point?
Worked example: the polar point (6, 2 pi/3) has x = 6 cos(120 degrees) = 6(-1/2) = -3 and y = 6 sin(120 degrees) = 6(√3/2) = 3√3, about 5.20. So the rectangular point is (-3, 5.20).
Reverse direction: the point (-3, 3) has r = square root of 18, about 4.24. The angle is in quadrant II, so theta = 135 degrees.
The same point has many polar names, such as (6, 2 pi/3 + 2 pi) or (-6, 5 pi/3).
What mistake do students make?
Trusting the inverse tangent in the wrong quadrant. Mathbot plots the point first and then picks the angle that lands there.