Chapter 10

Suppose that a radio station has two broadcasting towers located along a north-south line and that the towers are separated by a distance of \(\frac{1}{2} \lambda,\) where \(\lambda\) is the wavelength of the station's broadcasting signal. Then the intensity \(I\) of the signal in the direction \(\theta\) can be expressed by the given equation, where \(I_{0}\) is the maximum intensity of the signal. (a) Plot \(I\) using polar coordinates with \(I_{0}=5\) for \(\boldsymbol{\theta} \in \mathbf{[ 0 , 2 \pi ]}\) (b) Determine the directions in which the radio signal has maximum and minimum intensity. $$I=\frac{1}{2} I_{0}[1+\cos (\pi \sin 2 \theta)]$$

Graph the polar equation for the indicated values of \(\theta,\) and use the graph to determine symmetries. $$r=2 \sin ^{2} \theta \tan ^{2} \theta ; \quad-\pi / 3 \leq \theta \leq \pi / 3$$

Graph the polar equation for the indicated values of \(\theta,\) and use the graph to determine symmetries. $$r=\frac{4}{1+\sin ^{2} \theta} ; \quad 0 \leq \theta \leq 2 \pi$$

Graph the polar equations on the same coordinate plane, and estimate the points of intersection of the graphs. $$r=8 \cos 3 \theta, \quad r=4-2.5 \cos \theta$$

Graph the polar equations on the same coordinate plane, and estimate the points of intersection of the graphs. $$r=2 \sin ^{2} \theta, \quad r=\frac{3}{4}\left(\theta+\cos ^{2} \theta\right)$$