Chapter 9, Problem 13RE

To determine

To write:the polar equation of the rectangular coordinates $\left(x,y\right)$ of the given variables $r\text{ and }\theta$ . Also, identify the equation and graph it.

Answer to Problem 13RE

The equation of the circle is $x^2+{\left(y-1\right)}^2=1$ .

Explanation of Solution

Given:

The givenequation is $r=2\sin \theta$ .

Calculation:

As it is known that $x=r\cos \theta y=r\sin \theta$ .

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

And the given equation is $r=2\sin \theta$ .

Now, r multiply both sides,

  $\begin{array}{l}{r}^2=2r\sin \theta \\ x^2+y^2=2y\{y=r\sin \theta \}\\ x^2+y^2-2y=0\\ x^2+y^2-2y+1=0+1\left\{\text{add 1 on both the sides}\right\}\\ \therefore x^2+\left(y^2-2y+1\right)=1\\ x^2+{\left(y-1\right)}^2=1\{\text{as }\left(y^2-2y+1\right)={\left(y-1\right)}^2\}\end{array}$

Therefore, it is found that the above equation is the equation of circle with center $C\left(0,1\right)$ and radius 1. Now, draw the graph of the circle.

  

Hence, polar equation of the rectangular coordinatesis $x^2+{\left(y-1\right)}^2=1$ .