Question

Find the points of intersection of the graphs of the equations. $$ \begin{array}{l} r=1+\cos \theta \\ r=3 \cos \theta \end{array} $$

Short answer

The points of intersection of the graphs of the given equations are \((1.5, \pi/3)\) and \((1.5, 5\pi/3)\) in polar form.

Step-by-step solution

Equate the Two Equations

To find the points of intersection, both equations should be set equal to each other. Thus, write the equation as \(1 + \cos \theta = 3\cos \theta\).

Solve for Cosine Theta

The next step is to isolate \(\cos \theta\) on one side. So, subtract \( \cos \theta\) from both sides of the equation, resulting in \(1 = 2\cos \theta\). Then, divide both sides by 2 to isolate \(\cos \theta\). Therefore, \(\cos \theta = 1/2\).

Solve for Theta

Knowing that the cosine of an angle is 1/2, find the value of \(\theta\). In the unit circle, \(\cos \theta = 1/2\) at \(\theta = \pi/3\) and \(\theta = 5\pi/3\). So our solutions for \(\theta\) are \(\pi/3\) and \(5\pi/3\).

Substitute Theta Values in Original Equations

Substitute the \(\theta\) values found in both the original equations to find the corresponding \(r\) values. Substituting \(\theta = \pi/3\) in \(r=3\cos\theta\), we get \(r = 3*\cos(\pi/3)\) which simplifies to \(r = 3*(1/2) = 1.5\). Similarly, substituting \(\theta = 5\pi/3\) in \(r=3\cos\theta\), we get \(r = 3*\cos(5\pi/3)\) which simplifies to \(r = 3*(1/2) = 1.5\) too. So, the coordinate points in polar form for the intersections are \((1.5, \pi/3)\) and \((1.5, 5\pi/3)\).