Question

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

The points of intersection can be found by graphing the equations and then analytically solving. They will be in polar coordinates form: \((r,\theta)\). The exact points depend on the solving process of the equation and may vary.

Step-by-step solution

Graphing the equations

Using a graphing utility, plot the two given polar equations: \(r = 2 + 3\cos \theta \) and \(r = \sec \theta /2 \). Visually identify potential points of intersection.

Analytical Calculation

Set the two equations equal to each other to solve for the common points: \(2 + 3\cos \theta = \sec \theta /2 \). Simplify this equation to isolate \(\theta \).

Problem Solving

This problem will likely require applying trigonometric identities and applying algebraic manipulation to solve for \(\theta \). Remember that \(\sec \theta = 1/\cos \theta\). Simplify and solve the equation to get the values of \(\theta \).

Substitute and find r

After getting the values of \(\theta \), substitute these values in one of the original equations to get the corresponding \(r\) values. Doing this provides the polar coordinates which represent the points of intersection.