Question

Rectangular Equations to Polar Equations

Convert the equation to polar form.$y=x^2$

Short answer

The polar equation is$r=\tan \theta \sec \theta$

Step-by-step solution

Step 1. Given information

An equation is given in a rectangular coordinate system as$y=x^2$

Step 2. Concept used

Substitute the values in the formulas, calculate the results, and simplify the problem. For a given response, consider a certain range, radius, and angle. Otherwise, the number of equations is unlimited.

Step 3. Calculation

Formulae used,

$\begin{array}{l}x=r\cos \theta \\ y=r\sin \theta \end{array}$

Substitute the values in the formulae,

$\begin{array}{l}\Rightarrow y=x^2\\ \Rightarrow r\sin \theta \text{Invalid <m:msup> element}=\\ \Rightarrow r\sin \theta =r^2{\cos }^2\theta \\ \Rightarrow r\cos \theta =\tan \theta \\ \Rightarrow r=\tan \theta \sec \theta \end{array}$