# Equation of Cissoid of Diocles/Parametric Form

## Theorem

The cissoid of Diocles can be defined by the parametric equation:

$\begin {cases} x = 2 a \sin^2 \theta \\ y = \dfrac {2 a \sin^3 \theta} {\cos \theta} \end {cases}$

## Proof

 $\displaystyle r$ $=$ $\displaystyle 2 a \sin \theta \tan \theta$ Equation of Cissoid of Diocles: Polar Form $\displaystyle \leadsto \ \$ $\displaystyle r \cos \theta$ $=$ $\displaystyle 2 a \cos \sin \theta \paren {\dfrac {\sin \theta} {\cos \theta} }$ $\displaystyle r \sin \theta$ $=$ $\displaystyle 2 a \sin \theta \sin \theta \paren {\dfrac {\sin \theta} {\cos \theta} }$ $\displaystyle \leadsto \ \$ $\displaystyle x$ $=$ $\displaystyle 2 a \sin^2 \theta$ Conversion between Cartesian and Polar Coordinates in Plane $\displaystyle y$ $=$ $\displaystyle \dfrac {2 a \sin^3 \theta} {\cos \theta}$

$\blacksquare$