Polar Equations

Rectangular equations, or Cartesian equations, containt \(x\) and \(y\) variables while polar equations contain \(r\) and \(\theta\) variables. We usually like to solve polar equations for \(r\) just like we tend to solve rectangular equations for \(y\), at least whenever reasonably possible.

\[r = \frac{5}{2\cos{\theta} + \sin{\theta}}\]

The equation above is an example of a polar equation. It is equivalent to the rectangular equation \(y = -2x + 5\).

In addition to convert coordinates between rectangular and polar coordinates, we can use the following formulas to convert rectangular equations into polar form and polar equations into rectangular form.

\[\begin{align*} x &= r\cos{\theta} \\[4pt] y &= r\sin{\theta} \end{align*}\]

\[\begin{align*}r^2 &= x^2 + y^2 \\ \tan{\theta} &= \frac{y}{x} \end{align*}\]

While not necessarily always the case, we will typically use the first two or three formulas listed above whenever converting equations. Let's take a quick look at the polar equation given above to see how we can convert it to an equivalent rectangular equation.

Example #4: Convert the polar equation below to an equivalent rectangular equation.

\[r = \frac{5}{2\cos{\theta} + \sin{\theta}}\] Solution

First, muliply the denominator across to the left side of the equation. Then distribute the \(r\) through the parenthesis. Convert the \(r\cos{\theta}\) expression to \(x\) and then \(r\sin{\theta}\) expression to \(y\). Solve for \(y\).

\[\begin{align*} r &= \frac{5}{2\cos{\theta} + \sin{\theta}} \\ r(2\cos{\theta} + \sin{\theta}) &= 5 \\ 2r\cos{\theta} + r\sin{\theta} &= 5 \\ 2x + y &= 5 \\ y &= -2x + 5 \end{align*}\]

Polar Coordinates

Polar Equations

Rectangular to Polar

Polar to Rectangular