Question

Graphing Polar Equations

Sketch a graph of the polar equation$r=3+6\sin \theta$

Short answer

The graph of the given polar equation is-

Step-by-step solution

Step 1. Given information

A polar equation is given as$r=3+6\sin \theta$

Step 2. Concept used

For sketching the graph, first, find different values r for different$\theta$. After that, convert them into x and y by using equations-

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

Once we have the points draw a graph by plotting them.

Step 3. Calculation

The polar equation is given as$r=3+6\sin \theta$,

Let$\theta =0$,

$\begin{array}{l}r=3+6\sin \theta \\ r=3+6\sin \left(0\right)\\ r=3\end{array}$

Substitute the value in formulas$\begin{array}{l}x=r\cos \theta \\ y=r\sin \theta \end{array}$,

We get, $\begin{array}{l}x=3\\ y=0\end{array}$

Similarly,$\begin{array}{l}\theta =\frac{3\pi }{2},\\ r=3+6\sin \theta \\ r=3+6\sin \left(\frac{3\pi }{2}\right)\\ r=3-6\\ r=-3\end{array}$

We get, $\begin{array}{l}x=0\\ y=3\end{array}$

Plot the points on graph-