Polar Coordinates & Conversion Cheat Sheet

Key Facts

• A polar point $(r,\theta)$ means move a directed distance $r$ from the pole at angle $\theta$ measured from the positive $x$-axis.
• To convert from polar to rectangular coordinates, use $x=r\cos\theta$ and $y=r\sin\theta$.
• To convert from rectangular to polar coordinates, use $r=\sqrt{x^2+y^2}$ and $\tan\theta=\frac{y}{x}$, then adjust $\theta$ to the correct quadrant.
• The identity $r^2=x^2+y^2$ is often the fastest way to convert equations between rectangular and polar form.
• Equivalent polar coordinates include $(r,\theta+2\pi k)$ and $(-r,\theta+(2k+1)\pi)$ for any integer $k$.
• A circle centered at the pole has polar equation $r=a$, where $a>0$ is the radius.
• A ray from the pole has polar equation $\theta=\alpha$, where $\alpha$ is the angle of the ray.
• When using degrees, a full rotation is $360^{\circ}$, and when using radians, a full rotation is $2\pi$.

Vocabulary

Polar coordinate

A coordinate written as $(r,\theta)$ that gives a point by its directed distance from the pole and its angle from the polar axis.

Pole

The origin in the polar coordinate system, corresponding to the rectangular point $(0,0)$.

Polar axis

The reference ray for measuring angles in polar coordinates, usually the positive $x$-axis.

Radius $r$

The directed distance from the pole to a point, which may be positive, zero, or negative.

Argument $\theta$

The angle that gives the direction of a polar point measured from the polar axis.

Equivalent polar points

Different polar coordinate pairs that represent the same point, such as $(r,\theta)$ and $(r,\theta+2\pi)$.