Question

Find the points of intersection of the graphs of the equations. $$ \begin{array}{l} r=4 \sin 2 \theta \\ r=2 \end{array} $$

Short answer

The points of intersection of the graphs of the equations \(r = 4 \sin(2\theta)\) and \(r = 2\) are \((2, \frac{\pi}{12})\) and \((2, \frac{5\pi}{12})\).

Step-by-step solution

Set the Two Polar Equations Equal

To find the intersections of the graphs of the two equations, we set them equal to each other, resulting in: \[4 \sin(2\theta) = 2\] or \[\sin(2\theta) = \frac{1}{2}\]

Solve for Theta

Solving the equation \(\sin(2\theta) = \frac{1}{2}\), we get: \[2\theta = \frac{\pi}{6}, \frac{5\pi}{6}\] then we divide these results by 2, getting: \[\theta = \frac{\pi}{12}, \frac{5\pi}{12}\]

Determine the Intersection Points

Next, we substitute the values of \(\theta\) into one of our original equations (it doesn't matter which one since they're equal at the points of intersection). Substituting into \(r = 2\) yields the points of intersection: \[(2, \frac{\pi}{12}), (2, \frac{5\pi}{12})\]