Question

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

Short answer

The points of intersection of the graphs of the given equations are \((2\sqrt{2}, \frac{\pi}{4})\) in polar coordinates and \((2, 2)\) in Cartesian coordinates.

Step-by-step solution

Convert to Cartesian Coordinates

Let's convert our polar coordinates to Cartesian coordinates. The transformation relations are \(x = r \cos \theta\) and \(y = r \sin \theta\). For the first equation, \(\theta = \frac{\pi}{4}\), these give: \(x = r \cos \frac{\pi}{4}\), \(y = r \sin \frac{\pi}{4}\). Similarly, for the second equation, \(r = 2\), Substituting \(r = 2\) into the transformation equations gives: \(x = 2 \cos \theta\), \(y = 2 \sin \theta\).

Find Intersection Points

Now to find the intersection points of the two graphs, set the x and y components of each equation equal to each other and solve for r: \[\begin{array}{c}1) r \cos \frac{\pi}{4} = 2 \cos \theta \2) r \sin \frac{\pi}{4} = 2 \sin \theta\end{array}\] From equation 1), we have \(r = 2 / cos \frac{\pi}{4} = \sqrt {2} * 2 = 2 \sqrt {2}\). Substituting r into equation 2) confirms this solution as the sin and cos of \(\pi/4\) are the same.

Convert Back to Polar Coordinates

So the polar coordinates of our intersection points are \((r, \theta) = (2\sqrt{2}, \frac{\pi}{4})\). In Cartesian coordinates, these are \((x,y) = (r\cos \theta, r\sin \theta) = (2, 2)\).