Problem 33 - Algebra and Trigonometry

Question

Test for symmetry and then graph each polar equation. $$r \cos \theta=-3$$

Step-by-Step Solution

Answer

The polar equation \(r \cos \theta=-3\) is symmetric with respect to the pole, but not the polar axis. The graph of this equation represents a vertical line passing through -3 on the x-axis.

Step 1: Testing for symmetry

For \(r \cos \theta=-3\), go ahead and replace \(-r\) for \(r\) and \(-\theta\) for \(\theta\). The result is \(-r \cdot \cos(-\theta) = -3\). Due to the even symmetry of the cosine function, we get \(-r \cos \theta=-3\). This polar equation changes, so it's not symmetric about the polar axis.

Step 2: Testing for pole symmetry

Replace \(\theta\) with \(-\theta\) in the original equation: \(r \cos(-\theta) = -3\), which simplifies to \(r \cos \theta = -3\). The equation doesn't change, so the graph is symmetric with respect to the pole.

Step 3: Graphing the function

To graph \(r \cos \theta=-3\), recall that its rectangular form is \(x=-3\). This represents a vertical line cutting the x-axis at -3. In polar coordinates, it's an entire line going through pole (-3,0) and extending to the negative and positive direction of the y-axis. Graph the line on the coordinate system.