Chapter 10.3, Problem 29E

To determine

To replace: the polar equation $r=8\sin \theta$ by an equivalent Cartesian(rectangular) equation and identify or describe the graph without using a grapher.

Answer to Problem 29E

The cartesian equation for the polar equation $r=8\sin \theta$ is $x^2+{\left(y-4\right)}^2=4^2$ and it represents the equation of a circle with center $\left(0,4\right)$ and radius $4$ units.

Explanation of Solution

Given information:

The polar equation is $r=8\sin \theta$ .

Formula used:

  $x=r\cos \theta$ and $y=r\sin \theta$ .

The formula to convert polar equation to cartesian equation: $r^2=x^2+y^2$ .

Calculation:

The given equation is $r=8\sin \theta$ .

Substitute $\frac{y}{r}$ for $\sin \theta$ and $\frac{x}{r}$ for $\cos \theta$ in $r=2\cos \theta +2\sin \theta$ .

  $\begin{array}{l}r=8\frac{y}{r}\\ r^2=8y\end{array}$

Substitute $8y$ for $r^2$ in $r^2=x^2+y^2$ .

  $\begin{array}{l}{x}^2+y^2=8y\\ x^2+y^2-8y=0\end{array}$

Convert $x^2-2x$ into a perfect square by adding and subtracting $1$ . Similarly, convert $y^2-2y$ into a perfect square by adding and subtracting $1$ .

  $x^2+y^2-8y+16-16=0$

  $x^2+{\left(y-4\right)}^2-16=0$

  $x^2+{\left(y-4\right)}^2=16$

  $x^2+{\left(y-4\right)}^2=4^2$

Therefore, the cartesian equation for the polar equation $r=8\sin \theta$ is $x^2+{\left(y-4\right)}^2=4^2$ .

Hence, the polar equation represents the equation of a circle with center $\left(0,4\right)$ and radius $4$ units.