Problem 18 - Algebra and Trigonometry Real Mathematics, Real People

Question

Test for symmetry with respect to the line \(\theta=\pi / 2,\) the polar axis, and the pole. $$r=\frac{2}{1+\sin \theta}$$

Step-by-Step Solution

Answer

The given polar equation \(r=\frac{2}{1+\sin \theta}\) is symmetric with respect to the polar axis, but not with respect to the line \(\theta=\pi / 2\) or the pole.

Step 1: Symmetry with respect to the line \(\theta=\pi / 2\)

To check for symmetry with respect to the line \(\theta=\pi / 2\), one must replace \(\theta\) by \(\pi -\theta\) in the original equation. If the resulting equation is equivalent to the original one, then the given equation is symmetric with respect to the line \(\theta=\pi / 2\).

Step 2: Symmetry with respect to the polar axis

To check for symmetry with respect to the polar axis, \(\theta\) should be replaced with \(-\theta\) in the original equation. If the resulting equation is the same as the initial equation, it means that the original polar function has symmetry with respect to the polar axis.

Step 3: Symmetry with respect to the pole

In order to see if the equation is symmetric with respect to the pole, \(r\) must be replaced with \(-r\) in the original equation. If the equation obtained is equivalent to the original one, then the polar equation is symmetric relative to the pole. However, in this case, since it's impossible to substitute \(r\) with \(-r\) without changing the equation, it can be concluded that it doesn't have symmetry with respect to the pole.