Problem 69 - Essential Calculus

Question

Sketch a graph of the polar equation. $$ r=3-2 \cos \theta $$

Step-by-Step Solution

Answer

The graph of the polar equation \( r = 3 - 2\cos\theta \) is a limaçon without a loop shifted 3 units from the origin.

Step 1: Identify the type of polar graph

This is a type of limaçon function which is a polar graph equation, where \( \textit{r} \) depends on the trigonometric function of \( \textit{θ} \). Limaçon functions can vary in shape, some having loops and some do not. To determine this, we will need to inspect the coefficients in the equation more.

Step 2: Determine if the graph has a loop or not

Whether it has a loop depends on the ratio of coefficients of \( r \) and the trigonometric function. Compare the absolute values of the independent term 3 (which is responsible for shifting the graph from the origin) and the coefficient of \( \cos \theta \), 2. Since \( |3| > |2| \), there will be no loop in the graph.

Step 3: Plot the graph using points

To plot the graph of this equation, select a series of values for \( \theta \) and compute the corresponding values of \( r \). You might start with \( \theta = 0 \), \( \frac{\pi}{2} \), \( \pi \), \( \frac{3\pi}{2} \), and \( 2\pi \). Plot these \( (r,\theta) \) points in the polar plane and then smoothly connect the dots. Remember the point (r, θ) is plotted a distance |r| from the origin and an angle θ from the polar axis (positive x-axis).

Step 4: Complete the graph

After plotting some key points and understanding the symmetry of the function, complete the graph by sketching it smoothly. Make sure your graph represents the correct type of limaçon function without a loop.