Question

Graphing Polar Equations Sketch a graph of the polar equation

$r=\sin 2\theta$.

Short answer

The equation $r=\sin 2\theta$can be graphed as.

Step-by-step solution

Step 1. Given information.

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

Step 2. Graph of the polar equation.

We will refer to the table of Common polar graphs to see that which form our equation is, and then sketch the graph accordingly.

Given the Polar equation $r=\sin 2\theta$.
Compared to the Common polar graphs equation, we can see that our equation is of the form $r=a\sin n\theta$.

So, it's a rose graph.

Here$H=2$is even.

So, we will have $2n=4$ leaves with the leaves ends being at angles where $\left|\sin 2\theta \right|$ is maximized so

$\begin{array}{l}\left|\sin 2\theta \right|=1\\ \theta =\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}\end{array}$

Step 3. By using the Common polar graphs equation, the graph of r=sin2θ  is: