Question

The tests for symmetry given on page 574 are sufficient, but

not necessary. Explain what this means.

Short answer

There might be many test which will fail but still some graph will still exists which will be symmetric with respect to polar axis.

Step-by-step solution

Step 1. Given

A polar axis.

Step 2. Explanation

Symmetry test with respect to polar axis (x-axis)

$$\begin{gathered} replace\theta with-\theta . \\ thegraphissymmetricw.r.tpolaraxis. \end{gathered}$$

$$\begin{gathered} Symetryw.r.ttotheline\theta =\frac{\mathrm{\pi }}{2}(y-axis) \\ replace\theta by\mathrm{\pi }-\mathrm{\theta }\mathrm{if}\mathrm{the}\mathrm{resultant}\mathrm{is}\mathrm{equivalent}\mathrm{then}\mathrm{the}\mathrm{graph}\mathrm{is}\mathrm{symmetric}. \end{gathered}$$

$$\begin{gathered} Symmetricw.r.tthepole \\ replacerby-rand\theta by-\theta . \\ iftheequivalentequationresultsthenthegraphissymmetricw.r.tpole. \\ thoughmanyofequationwillfailsuchtestbutstillsomegraphwillbesymmetricw.r.tpole. \end{gathered}$$