Chapter 11, Problem 32RE

To determine

To determine: The equivalent polar equation which replaces the Cartesian equation.

Answer to Problem 32RE

The required polar equation is $r=2\sin \theta$ .

Explanation of Solution

Given information:

The Cartesian equation : $x^2+y^2-2y=0$

Calculation:

The given Cartesian equation is $x^2+y^2-2y=0$ .

To replace the Cartesian equation by a polar equation, use the polar-rectangular equations and formulas $x=r\cos \theta$ , $y=r\sin \theta$ , $r^2=x^2+y^2$ , $\sec \theta =\frac{1}{\cos \theta }$ , $\tan \theta =\frac{\sin \theta }{\cos \theta }$ and $\tan \theta =\frac{y}{x}$ .

Rewrite the expression as:

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

Simplify the above equation further.

  $\begin{array}{c}{r}^2-2r\sin \theta =0\\ \frac{r^2}{r}=\frac{2\cancel{r}\sin \theta }{\cancel{r}}\\ r=2\sin \theta \end{array}$

Hence, the required polar equation is $r=2\sin \theta$ .