8.4 Polar coordinates: graphs  (Page 3/16)

To find the zeros, set $r$ equal to zero and solve for $\theta .$

Substitute any one of the $\theta$ values into the equation. We will use $0.$

The points $(0,0)$ and $(0,\pm n\pi )$ are the zeros of the equation. They all coincide, so only one point is visible on the graph. This point is also the only polar axis intercept.

To find the maximum value of the equation, look at the maximum value of the trigonometric function $\sin \theta ,$ which occurs when $\theta =\frac{\pi }{2}\pm 2k\pi$ resulting in $\sin \left(\frac{\pi }{2}\right)=1.$ Substitute $\frac{\pi }{2}$ for $\mathrm{\theta .}$

First, testing the equation for symmetry, we find that the graph is symmetric about the polar axis. Next, we find the zeros    and maximum $\left|r\right|$ for $r=4\cos \theta .$ First, set $r=0,$ and solve for $\theta$ . Thus, a zero occurs at $\theta =\frac{\pi }{2}\pm k\pi .$ A key point to plot is $\left(0,\text{}\text{}\frac{\pi }{2}\right).$

To find the maximum value of $r,$ note that the maximum value of the cosine function is 1 when $\theta =0\pm 2k\pi .$ Substitute $\theta =0$ into the equation:

The maximum value of the equation is 4. A key point to plot is $(4,0).$

As $r=4\cos \theta$ is symmetric with respect to the polar axis, we only need to calculate r -values for $\theta$ over the interval $[0,\pi ].$ Points in the upper quadrant can then be reflected to the lower quadrant. Make a table of values similar to [link] . The graph is shown in [link] .