Chapter 6.5, Problem 71E

Solution

To graph: The given three equations and describe their relation with each other.

The graph $r_2$ is obtained when the graph of $r_1$ is rotated by $\frac{\pi }{4}$ radians in counter-clockwise direction.

The graph $r_3$ is obtained when the graph of $r_1$ is rotated by $\frac{\pi }{3}$ radians in counter -clockwise direction.

Given information:

The given equation are $r_1=2\sec \theta$ , $r_2=2\sec \left(\theta -\frac{\pi }{4}\right)$ , and $r_3=2\sec \left(\theta -\frac{\pi }{3}\right)$ .

Calculation:

Plot the graph for $r_1=2\sec \theta$ .

  

Figure (1)

Plot the graph for $r_2=2\sec \left(\theta -\frac{\pi }{4}\right)$ .

  

Figure (2)

Plot the graph for $r_3=2\sec \left(\theta -\frac{\pi }{3}\right)$ .

  

Figure (3)

The graph $r_2$ is obtained when the graph of $r_1$ is rotated by $\frac{\pi }{4}$ radians in counter-clockwise direction.

The graph $r_3$ is obtained when the graph of $r_1$ is rotated by $\frac{\pi }{3}$ radians in counter -clockwise direction.