Polar equations (further question)

Question

Consider the polar equations of the form $r=a\cos(b\theta)$ and $r=a\sin(b\theta)$. What is the nature of the graphs of these two polar equations and then summarize some generalizations with respect to the nature of the curves generated and the two parameters $a$ and $b$.

Answer 1

All these questions can be best answered by plotting the curves in polar coordinates. First rectangular and next polar plots so we gain insight about polar curve behaviour, how the tip of vector goes around. The polar curve

$$ r= a \sin( b \,\theta) $$

is a rose.

For $b=1$ it is a circle tangent to x-axis at origin. If $b$ is odd integer, it has the same number of petals or lobes. If even integer it has double that many petals. $a$ is maximum radius from origin.

For non integral $b$ the radius goes around origin indefinitely.

For $cos$ there is a rotation offset of the entire rose.

An addition in the argument gives rise to cardioids.