Question

Use a graphing utility to graph the polar equation. Find an interval for \(\boldsymbol{\theta}\) over which the graph is traced only once. $$ r=2+\sin \theta $$

Short answer

The interval over which the graph is traced only once is from \( \theta = \frac{3\pi}{2} \) to \( \theta = \frac{5\pi}{2} \)

Step-by-step solution

Graph the polar equation

Use a graphing utility to graph the polar equation \( r = 2 + \sin \theta \). This creates a graph that resembles a circle with a small inner loop. You may notice that each point on the graph is traced twice except the points on the inner loop which are traced once, over certain values of \( \theta \).

Identify the maximum and minimum values of r

The range of values for \( r \) is determined by the \( 2 + \sin \theta \) part of the equation. As \( \sin \theta \) varies from -1 to 1, \( r \) varies from 1 to 3. The maximum value of 3 occurs when \( \sin \theta = 1 \) and the minimum value of 1 occurs when \( \sin \theta = -1 \). Therefore, the graph has a maximum radius of 3 and an inner loop of radius 1.

Find the interval for \( \theta \)

The graph begins to be traced out when \( \theta = 0 \). The graph completes a full loop and returns to the initial starting point when \( \theta = 2\pi \). However, the graph overlaps itself on this interval except the inner loop. The inner loop starts forming at \( \theta = \frac{3\pi}{2} \) and completes at \( \theta = \frac{5\pi}{2} \). Therefore, the interval over which the graph is traced only once is from \( \theta = \frac{3\pi}{2} \) to \( \theta = \frac{5\pi}{2} \).