AP Calculus BC

The Polar Coordinate System and Plotting in Polar Coordinates

The polar coordinate system

The polar system names a point by a radius and an angle, $(r, \theta)$, instead of $x$ and $y$. $\theta$ is measured counterclockwise from the positive x-axis, and $r$ is the distance from the origin along that angle. A negative $r$ means going the opposite direction, the same distance backward.

Plotting points

Each round below gives a polar coordinate, sometimes written with a negative $r$ or an angle outside $[0, 2\pi)$. Click the point it names, then give that point's standard coordinates, with $r > 0$ and $0 \le \theta < 2\pi$.

Converting between systems

$x = r\cos\theta \qquad y = r\sin\theta$

$r^2 = x^2 + y^2 \qquad \theta = \tan^{-1}\left(\frac{y}{x}\right)$

The $\tan^{-1}$ formula only gives an angle in quadrant I or IV. If $(x, y)$ is in quadrant II or III, adjust $\theta$ using the reference angle and the quadrant, the same way you already find $\sin\theta$ and $\cos\theta$ from a reference angle on the unit circle.

Alternate coordinates

Every point has more than one valid pair of polar coordinates:

• Adding or subtracting a full turn, $2\pi$, to $\theta$ lands back on the same point.
• Negating $r$ and adding $\pi$ to $\theta$ also lands on the same point, since it points the same distance in the opposite direction.

Recognizing when two different-looking coordinate pairs name the same point works the same way in reverse.

Review

This challenge mixes plotting, conversion, and alternate coordinates.