Problem 32 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph of this polar equation has y-axis symmetry. The plot forms a circle centered at a radial distance of 3 from the pole, extending to a maximum radial distance of 4, and mirroring such pattern across the y-axis.

Step 1: Test for symmetry

Start with the given polar equation \(r=3+\sin \theta \). Apply the tests for symmetry. For the x-axis symmetry, replace \(\theta\) with \(-\theta\), for y-axis symmetry, replace \(\theta\) with \(\pi-\theta\), and for symmetry about the origin, replace \(\theta\) with \(-\theta\) and \(r\) with \(-r\).

Step 2: Calculate the tests

Upon testing, it can be seen that the equation is not symmetrical about the x-axis or the origin, but only the y-axis. The equation \(r = 3 + \sin(\pi-\theta)\) simplifies to \(r = 3 + \sin \theta \). Hence, the graph will have symmetry about the y-axis.

Step 3: Graph the equation

For graphing the equation, draw a polar axis and then sketch the plot by substituting for various values of theta in the range 0 to \(2\pi\). You will observe a circular plot starting from the pole where \(r \) = 3 when \( \theta = 0\), increasing up to a maximum of \(r = 4\) at \(\theta = \pi/2\), and then decreasing back to \(r = 3\) at \(\theta = \pi\), following a mirror image across the y-axis for the second half due to its y-axis symmetry.