Chapter 8.2, Problem 46E

To determine

To sketch the graph of the given polar equation and choose the domain of $\theta$ so that it produces the entire graph.

  $r=\sqrt{1-0.8{\sin }^2\theta }$

Answer to Problem 46E

The graph of polar equation is

  

The Domain of the complete graph is $0\le \theta \le 2\pi$

Explanation of Solution

Given: Polar equation: $r=\sqrt{1-0.8{\sin }^2\theta }$

Formula Used:

A polar equation is any equation that describes a relation between r and θ , where r represents the distance from the pole (origin) to a point on a curve, and θ represents the counter-clockwise angle made by a point on a curve, the pole, and the positive x-axis.

Also, $x=r\cos \theta$ and $y=r\sin \theta$

Domain: Choose $\theta$ from

  $\theta =\left[0\le \theta \le 2\pi \right]$

Where $r=\sqrt{1-0.8{\sin }^2\theta }$

Calculation:

Given Polar equation is $r=\sqrt{1-0.8{\sin }^2\theta }$

Drawing the polar-form curve for $r=\sqrt{1-0.8{\sin }^2\theta }$ on a graph as per the above definition of polar equation, we have:

  $\theta =\left[0\le \theta \le 2\pi \right]$

  $r=\sqrt{1-0.8{\sin }^2\theta }$

When

  $\begin{array}{l}\theta =0,r=1\\ \theta =\frac{\pi }{3},r=0.6\\ \theta =\frac{\pi }{2},r=0.4\\ \theta =\pi ,r=1\\ \theta =3\pi /2,r=0.4\\ \theta =2\pi ,r=1\end{array}$

Conclusion:

Hence, the graph of polar equation is