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Chapter 9: Problem 21

Use a graphing utility to approximate the points of intersection of the graphs of the polar equations. Confirm your results analytically. $$ \begin{array}{l} r=2+3 \cos \theta \\ r=\frac{\sec \theta}{2} \end{array} $$

Short Answer

Expert verified
The points of intersection of the two polar equations can be visually approximated on a graph and then confirmed precisely by analytically solving the equations for \(\theta\) and \(r\). The process involves graphing the equations, setting them equal to each other, solving for \(\theta\) and subsequently for \(r\). The exact points will vary depending on the solutions for \(\theta\) and \(r\).

Step by step solution

01

Graph the equations

Plot the given equations on a graphing tool. For instance, use an online graphing calculator that supports polar coordinates, and sketch the graphs of the two polar functions \(r=2+3 \cos \theta\) and \(r=\frac{\sec \theta}{2}\). Observe the points where these graphs intersect.
02

Analytical confirmation - Set equations equal to each other

To find the intersection points analytically, set the two equations equal to each other and solve for \(\theta\).So, we have \(2+3 \cos \theta = \frac{\sec \theta}{2}\). This equation can be rewritten as \(2+3 \cos \theta = \frac{1}{2 \cos \theta}\).
03

Analytical confirmation - Solve for \(\theta\)

Now, multiply both sides of the equality by \(2\cos\theta\) to get rid of the fraction. This results in \(4\cos\theta + 6\cos^2\theta = 1\). Rewrite this as a quadratic equation \(6\cos^2\theta + 4\cos\theta - 1 = 0\). Solve this quadratic equation for \(cos\theta\), then find \(\theta\) by using the arccosine function.
04

Find corresponding \(r\) values

Substitute each value of \(\theta\) obtained in the previous step into either of the initial polar equations to find the corresponding \(r\) values. That will give you the points of intersection in polar coordinates.

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