Chapter H.2, Problem 32E

To determine

To calculate:

The point of intersection of the given curves:

  $r^2=\sin 2\theta ,r^2=\cos 2\theta$

Answer to Problem 32E

The point of intersection of the given curves $\left\{\left(\frac{\sqrt{2}}{2},\frac{\pi }{8}\right),\left(\frac{\sqrt{2}}{2},\frac{\pi }{8}\right),\left(\frac{\sqrt{2}}{2},\frac{\pi }{8}\right),\left(\frac{\sqrt{2}}{2},\frac{3\pi }{8}\right),\left(\frac{\sqrt{2}}{2},\frac{5\pi }{8}\right),\left(\frac{\sqrt{2}}{2},\frac{7\pi }{8}\right),\left(\frac{\sqrt{2}}{2},\frac{9\pi }{8}\right),\left(\frac{\sqrt{2}}{2},\frac{11\pi }{8}\right),\left(\frac{\sqrt{2}}{2},\frac{13\pi }{8}\right),\left(\frac{\sqrt{2}}{2},\frac{15\pi }{8}\right)\right\}$

Explanation of Solution

Given information:

The point of intersection of given curves

  $r^2=\sin 2\theta ,r^2=\cos 2\theta$

Calculation:

Consider these curves

  $r^2=\sin 2\theta$ And $r^2=\cos 2\theta$

Graph the two polar coordinates.

  

These appear to be nine point of interaction including the origin.

  $\begin{array}{l}{r}^2=r^3\\ \left|r\right|=\left|r\right|\\ r=\pm r\\ \sin 2\theta =\pm \cos 2\theta \\ \tan 2\theta =\pm 1\\ 2\theta =\pm \frac{\pi }{4}\\ \theta =\pm \frac{\pi }{8}\end{array}$

From the symmetry of the figure, see that each point of intersection other than the origin is equivalent from the origin and that each is offset from the major axis by the anfle of plus or minus $\theta =\pi /8$ .

The radius of each of the curves at the point is:

  $\begin{array}{l}\left|r\right|=\left|\sin 2\theta \right|\\ \left|r\right|=\left|\sin \left[2\left(\frac{\pi }{8}\right)\right]\right|\\ \left|r\right|=\left|\sin \frac{\pi }{4}\right|\\ \left|r\right|=\frac{\sqrt{2}}{2}\end{array}$

The 9 points of intersection and in polar coordinates the points:

  $\left\{\left(\frac{\sqrt{2}}{2},\frac{\pi }{8}\right),\left(\frac{\sqrt{2}}{2},\frac{\pi }{8}\right),\left(\frac{\sqrt{2}}{2},\frac{\pi }{8}\right),\left(\frac{\sqrt{2}}{2},\frac{3\pi }{8}\right),\left(\frac{\sqrt{2}}{2},\frac{5\pi }{8}\right),\left(\frac{\sqrt{2}}{2},\frac{7\pi }{8}\right),\left(\frac{\sqrt{2}}{2},\frac{9\pi }{8}\right),\left(\frac{\sqrt{2}}{2},\frac{11\pi }{8}\right),\left(\frac{\sqrt{2}}{2},\frac{13\pi }{8}\right),\left(\frac{\sqrt{2}}{2},\frac{15\pi }{8}\right)\right\}$

Now, the graph the two polar coordinates with intersection points.