Chapter 9, Problem 7RE

(a)

To determine

To find:The equationin rectangular coordinates for given polar coordinate.

Answer to Problem 7RE

The equation in rectangular coordinates is $x^2+\left(y^2-1\right)=1$ .

Explanation of Solution

Given information:

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

Calculation:

Multiply both side by $r=2\sin \theta$ to a corresponding equation in rectangular coordinates.

  $r^2=2r\sin \theta$

Substitute $x^2+y^2$ for $r^2$ and $y$ for $r\sin \theta$ .

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

Add one on both side of the above equation.

  $\begin{array}{c}{x}^2+\left(y^2-2y+1\right)=1\\ x^2+\left(y^2-1\right)=1\end{array}$

The equation $x^2+\left(y^2-1\right)=1$ is of the form ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$ with center $\left(0,1\right)$ and radius $1$ in rectangular coordinate system.

Therefore, the equation in rectangular coordinates is $x^2+\left(y^2-1\right)=1$ .

(b)

To determine

To graph:For the equation.

Explanation of Solution

Given information:

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

Graph:

The equation $r=2\sin \theta$ is a circle with center at $\left(0,1\right)$ and radius $1$ in polar coordinate system.

The graph for the equation is shown in figure (1).

  

Figure (1)

Interpretation:Graph for the equation $r=2\sin \theta$ is shown in figure (1).