Graphing Polar Equations

This tutorial explains how to graph polar equations by hand (sketching) in order to build a deep conceptual understanding. Several examples with detailed solutions are presented. Points in polar coordinates are written as \( (R, t) \), where \( R \) is the polar distance and \( t \) is the polar angle. The point-by-point method is used throughout.

Helpful references: Points in polar coordinates | Free polar graph paper

Examples on Graphing Polar Equations

Example 1

Graph the polar equation

\[ R = 4 \cos t \]

and identify the graph.

Solution to Example 1

We first construct a table of values using special angles. It is helpful to find values of \( t \) where \( R \) is maximum, minimum, or zero.
- \( R \) is maximum (\( R = 4 \)) when \( t = 0 \)
- \( R \) is minimum (\( R = -4 \)) when \( t = \pi \)
- \( R = 0 \) when \( t = \frac{\pi}{2} \)

\( t \)	\( R \)
\( 0 \)	\( 4 \)
\( \frac{\pi}{6} \)	\( 3.5 \)
\( \frac{\pi}{4} \)	\( 2.8 \)
\( \frac{\pi}{3} \)	\( 2 \)
\( \frac{\pi}{2} \)	\( 0 \)
\( \frac{2\pi}{3} \)	\( -2 \)
\( \frac{3\pi}{4} \)	\( -2.8 \)
\( \frac{5\pi}{6} \)	\( -3.5 \)
\( \pi \)	\( -4 \)

We stop at \( t = \pi \) since the values repeat beyond this point.
The points are plotted on a polar coordinate system. Special attention must be given to points with negative values of \( R \).

Polar coordinate system with plotted points

Finally, the points are joined with a smooth curve.

Graph of the polar equation R = 4 cos t

Example 2

Graph the polar equation

\[ R = 2 + 2 \sin t \]

and identify the graph.

Solution to Example 2

We construct a table of values using special angles.
- \( R \) is maximum (\( R = 4 \)) at \( t = \frac{\pi}{2} \)
- \( R = 0 \) at \( t = \frac{3\pi}{2} \)

\( t \)	\( R \)
\( 0 \)	\( 2 \)
\( \frac{\pi}{6} \)	\( 3.0 \)
\( \frac{\pi}{4} \)	\( 3.4 \)
\( \frac{\pi}{3} \)	\( 3.7 \)
\( \frac{\pi}{2} \)	\( 4 \)
\( \frac{2\pi}{3} \)	\( 3.7 \)
\( \frac{3\pi}{4} \)	\( 3.4 \)
\( \frac{5\pi}{6} \)	\( 3 \)
\( \pi \)	\( 2 \)
\( \frac{7\pi}{6} \)	\( 1 \)
\( \frac{5\pi}{4} \)	\( 0.6 \)
\( \frac{4\pi}{3} \)	\( 0.3 \)
\( \frac{3\pi}{2} \)	\( 0 \)

Graph of the polar equation R = 2 + 2 sin t

Example 3

Graph the polar equation

\[ R = 4 \cos(2t) \]

and identify the graph.

Solution to Example 3

Maximum values \( R = 4 \) occur at \( t = 0 \) and \( t = \pi \).
Minimum values \( R = -4 \) occur at \( t = \frac{\pi}{2} \) and \( t = \frac{3\pi}{2} \).

\( t \)	\( R \)
\( 0 \)	\( 4 \)
\( \frac{\pi}{6} \)	\( 2 \)
\( \frac{\pi}{4} \)	\( 0 \)
\( \frac{\pi}{3} \)	\( -2 \)
\( \frac{\pi}{2} \)	\( -4 \)
\( \frac{2\pi}{3} \)	\( -2 \)
\( \frac{3\pi}{4} \)	\( 0 \)
\( \frac{5\pi}{6} \)	\( 2 \)
\( \pi \)	\( 4 \)

Graph of the polar equation R = 4 cos(2t)

More References on Polar Coordinates

Polar Coordinates – Additional Resources