Chapter 9.3, Problem 31E

To determine

To find: the polar form of the rectangular equation.

Answer to Problem 31E

The polar form is $r^2=4\sin \theta$ .

Explanation of Solution

Given information: The rectangular equation is $x^2+{\left(y-2\right)}^2=4$ .

Calculation:

The equation is $x^2+{\left(y-2\right)}^2=4$ .

Then the equation in polar form:

  $\begin{array}{l}x=r\cos \theta \\ y=r\sin \theta \\ x^2+y^2=r^2\\ {\left(r\cos \theta \right)}^2+y^2-4y+4=4\\ r^2{\cos }^2\theta +r^2{\sin }^2\theta -4r\sin \theta =0\\ r^2\left({\cos }^2\theta +{\sin }^2\theta \right)-4r\sin \theta =0\\ r^2-4r\sin \theta =0\\ r^2=4r\sin \theta \\ r^2=4\sin \theta \end{array}$