Question

Graphing Polar Equations

Sketch a graph of the polar equation$r^2=\cos 2\theta$

Short answer

The graph of the given polar equation is-

Step-by-step solution

Step 1. Given information

A polar equation is given as$r^2=\cos 2\theta$

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$r^2=\cos 2\theta$,

Let$\theta =0$,

$\begin{array}{l}r=\sqrt{\cos 2\theta }\\ r=\sqrt{\cos 2\left(0\right)}\\ r=\sqrt{1}\\ r=\pm 1\end{array}$

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

For $r=1$

We get,$\begin{array}{l}x=1\\ y=0\end{array}$

For $r=-1$

We get, $\begin{array}{l}x=-1\\ y=0\end{array}$

Similarly,

$\begin{array}{l}\theta =0.5,\\ r=\sqrt{\cos 2\theta }\\ r=\sqrt{\cos 1}\\ r=\pm 0.73\end{array}$

Plot the points on graph-