Equation of Cissoid of Diocles/Polar Form

From ProofWiki
Jump to navigation Jump to search

Theorem

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

$r = 2 a \sin \theta \tan \theta$


Proof

CissoidOfDiocles.png


By construction:

\(\displaystyle OS\) \(=\) \(\displaystyle 2 a \sec \theta\) Definition of Secant Function
\(\displaystyle OR\) \(=\) \(\displaystyle 2 a \cos \theta\) Definition of Cosine
\(\displaystyle OP\) \(=\) \(\displaystyle RS\) Definition of Cissoid of Diocles
\(\displaystyle \leadsto \ \ \) \(\displaystyle OP\) \(=\) \(\displaystyle OS - OR\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle r\) \(=\) \(\displaystyle 2 a \paren {\sec \theta - \cos \theta}\)
\(\displaystyle \) \(=\) \(\displaystyle 2 a \sin \theta \tan \theta\) Secant Minus Cosine

$\blacksquare$


Also see


Sources