Converting Polar Coordinates

Given the point P has Cartesian coordinates   \( (x,y) \)   and polar coordinates   \( (r, \theta) \)

\begin{align} x & = r\,cos \theta \quad \quad \quad \quad \frac{y}{x} = tan\, \theta\\ \\ y & = r\,sin\theta \quad \quad \quad \quad r^2 = x^2 + y^2 \\ \end{align}

Example - (Polar to Cartesian)

Convert this polar equation of a circle into a Cartesian equation

\(r = 2a \,cos \theta \)

\begin{align} r & =2a \,cos \theta \ \\ \\ r & = 2a \left( \frac{x}{r} \right)\\ \\ r^2 & = 2ax \\ \\ 2ax & = x^2 + y^2 \\ \\ 0 & = x^2 + y^2 - 2ax \end{align}

Example - (Cartesian to Polar)

Convert this Cartesian equation of a parabola into a polar equation

\(y= 2x^2 \)

\begin{align} y & =2x^2 \\ \\ rsin \theta & = 2(rcos \theta) ^2\\ \\ rsin \theta & = 2r^2cos^2 \theta \\ \\ \frac {1}{r} & = \frac {2 cos^2 \theta}{sin \theta} \\ \\ r & = \frac{sin \theta}{2cos^2 \theta} \\ \\ r & = \frac{1}{2} \,tan \theta \, sec \theta \end{align}