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 intersection points in Cartesian coordinates are: \((\sqrt{3}, 1), (-\sqrt{3}, 1), (-\sqrt{3}, -1), (\sqrt{3}, -1)\)

Step-by-step solution

Set the equations equal to each other

Since we're looking for intersection points, make the two equations equal: \(4 \sin 2 \theta = 2\). This simplifies to: \(\sin 2 \theta = \frac{1}{2}\).

Solve for theta

Solving for \(\theta\), gives us \(\theta = \frac{\pi}{12}, \frac{5\pi}{12}, \frac{13\pi}{12}, \frac{17\pi}{12}\). These are the polar angles of the intersection points when \(r>0\).

Convert to Cartesian coordinates

Cartesian coordinates can be found using the formulas \(x = r \cdot \cos \theta\) and \(y = r \cdot \sin \theta\). Using the values of \(\theta\) from step 2 and \(r=2\) we obtain the points: \((\sqrt{3}, 1), (-\sqrt{3}, 1), (-\sqrt{3}, -1), (\sqrt{3}, -1)\)