Problem 19 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The given polar equation \(r = 2 + \cos\theta\) is not symmetric about the x-axis, y-axis, or the origin. The graph is drawn by computing and plotting (r,θ) pairs given by the equation for several angles, thus portraying a cardioid shape.

Step 1: Test for Symmetry

Testing for symmetry involves the following: \nFor symmetry about the x-axis, replace \(\theta\) with \(-\theta\) and see if the equation remains the same. \nFor symmetry about the y-axis, replace \(\theta\) with \(\pi - \theta\) and see if the equation remains unchanged. \nFor symmetry about the origin, replace \(r\) with \(-r\) and check if the equation remains true. In this case, for the given equation \(r = 2 + \cos\theta\), when \(\theta\) is replaced with \(-\theta\) and \(\pi - \theta\), the equation does not remain the same. When \(r\) is replaced with \(-r\), the equation also does not remain true. Therefore, the graph is not symmetric about the x-axis, y-axis, or the origin.

Step 2: Graphing the Equation

To graph the equation, it is useful to create a table of values for several different angles and then plot these points on the polar grid. The angles, in radians, commonly used are \(0, \pi/4, \pi/2, 3\pi/4, \pi, 5\pi/4, 3\pi/2, 7\pi/4\). Substituting these angles in the given equation, the corresponding \(r\) values would be calculated. This would provide the (r,θ) pairs to be plotted to graph the equation.