Question

Graphing Polar Equations

Use a graphing device to graph the polar equation. Choose the domain $\theta$ to make sure you produce the entire graph

$r=1+2\sin \left(\frac{\theta }{2}\right)$

Short answer

The graph of the given polar equation is-

Step-by-step solution

Step 1. Given information

A polar equation is given as $r=1+2\sin \left(\frac{\theta }{2}\right)$

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-

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

To convert rectangular to polar coordinates use formulas-

$\begin{array}{l}r=\sqrt{x^2+y^2}\\ \tan \theta =\frac{y}{x}\end{array}$

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

Step 3. Calculation

The polar equation is given as$r=1+2\sin \left(\frac{\theta }{2}\right)$,

Let$\theta =0$,

$\begin{array}{l}r=1+2\sin \frac{\theta }{2}\\ r=1+2\sin \left(0\right)\\ r=1\end{array}$

Similarly,

Let$\theta =1$,

$\begin{array}{l}r=1+2\sin \frac{\theta }{2}\\ r=1+2\sin \frac{1}{2}\\ r=1.94\end{array}$

Let$\theta =2$,

$\begin{array}{l}r=1+2\sin \frac{\theta }{2}\\ r=1+2\sin \left(1\right)\\ r=2.68\end{array}$

Plot the points on graph-