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Chapter 6: Problem 19

Test for symmetry and then graph each polar equation. $$r=2+\cos \theta$$

Short Answer

Expert verified
The polar equation \( r = 2 + \cos \theta \) is symmetric with respect to the polar axis. The graph of the equation forms a circle with a smaller circle 'indent' or 'loop' in the positive x-direction.

Step by step solution

01

Testing For Symmetry

First, let's test for symmetry. A polar equation is symmetric with respect to the polar axis if it remains the same when \( \theta \) is replaced with \( -\theta \). Try substituting \( -\theta \) for \( \theta \) in the given polar equation: \( r = 2 + \cos(-\theta) \). We know that \( \cos(-\theta) = \cos \theta \) so the equation simplifies to \( r = 2 + \cos \theta \) which is the original polar equation. Thus, the equation has symmetry with respect to the polar axis.
02

Graphing The Polar Equation

Now, let's graph the polar equation. Start by setting up a polar coordinate system with 胃 on the x-axis and r on the y-axis. Pick some values of 胃 from 0 to 2蟺 and calculate the corresponding values of r. For example, when 胃 = 0, r = 2 + cos(0) = 3. When 胃 = 蟺/2, r = 2 + cos(蟺/2) = 2 and so on. Plot these points on the coordinate system and then join them smoothly. The graph shape for this particular polar equation is a circle with a smaller circle 'indent' or 'loop' in the positive x-direction.

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