Chapter 9, Problem 24RE

To determine

To sketch:the graph of the given polar equation.

Answer to Problem 24RE

  $r=1-2sin\left(-\theta \right)=1-2\sin \theta$

Explanation of Solution

Given:

The given polarequation is $r=1-2sin\theta$ .

Calculation:

As it is known that $x=r\cos \theta y=r\sin \theta$ .

The given equation is $r=1-2sin\theta$

First check for symmetry:

Polar Axis: replace $\theta \to -\theta$

  $r=1-2sin\left(-\theta \right)=1+2\sin \theta$

The test fails; therefore, the graph is not symmetric with respect to to polar axis.

The line $\theta =\frac{\pi }{2}$ : replace $\theta \to \pi -\theta$

  $r=1-2sin\left(-\theta \right)=1-2\sin \theta$

The test is satisfied; therefore, the graph is symmetric with respectthe line $\theta =\frac{\pi }{2}$ .

Now, draw the graph of the given polar equation. The graph is: