To graph in the rectangular coordinate system we construct a table of $x$ and $y$  values. To graph in the polar coordinate system we construct a table of $r$ and $\theta$ values. We enter values of $\theta$ into a polar equation and calculate $r$. However, using the properties of symmetry and finding key values of $\theta$ and $r$ means fewer calculations will be needed.

Investigating Rose Curves

Polar equations can be used to generate unique graphs. The following type of polar equation produces a petal-like shape called a rose curve. Although the graphs look complex, a simple polar equation generates the pattern. The formulas that generate the graph of a rose curve are given by:

 $\displaystyle r=a\cdot\cos \left( n\theta \right) \qquad \text{and} \qquad r=a\cdot\sin \left( n\theta \right) \qquad \text{where} \qquad a\ne 0$  

If $n$ is even, the curve has $2n$ petals. If $n$ is odd, the curve has $n$ petals.

Rose Curves

Complex graphs generated by the simple polar formulas that generate rose curves:$r=a\:\cos n\theta$ and $r=a\:\sin n\theta$ where $a≠0$.

Investigating the Archimedes' Spiral

Archimedes’ spiral is named for its discoverer, the Greek mathematician Archimedes ($c. 287 BCE - c. 212 BCE$), who is credited with numerous discoveries in the fields of geometry and mechanics.

The formula that generates the graph of the Archimedes’ spiral is given by:

$\displaystyle r=θ \qquad \text{for} \qquad \theta\geq 0$ 

As $\theta$ increases, $r$ increases at a constant rate in an ever-widening, never-ending, spiraling path.

Archimedes' Spiral

The formula that generates the graph of a spiral is $r=θ$ for $θ≥0$.