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 for the two equations are \((1.5, 60°)\) and \((1.5, 300°)\), or in radians, \((1.5, \pi/3)\) and \((1.5, 5\pi/3)\).

Step-by-step solution

Set the Equations Equal to Each Other

In order to find the points of intersection, you first set the two equations equal to each other. This gives us: \(1 + \cos \theta = 3 \cos \theta\).

Rearrange the Equation

Rearrange the equation to isolate the cosine function. The equation becomes: \(\cos \theta = 0.5\).

Solving for theta

To solve for θ, use the arccos or inverse cosine function to get: \(\theta = \arccos(0.5)\). This yields \(\theta = 60°\) and \(\theta = 300°\) (or in radians, \(\theta = \pi/3\) and \(\theta = 5\pi/3\)).

Substituting Back for r

Substitute the calculated θ values back into either of the original equations to get the corresponding r values. By substituting the θ values in the second equation \(r = 3 \cos \theta\), for \(\theta = 60°\), \(r = 3 \cos 60° = 1.5\) and for \(\theta = 300°\), \(r = 3 \cos 300° = 1.5\).