Problem 72 - Essential Calculus

Question

Sketch a graph of the polar equation. $$ r=\frac{6}{2 \sin \theta-3 \cos \theta} $$

Step-by-Step Solution

Answer

The graph of the equation \(r=\frac{6}{2 \sin \theta-3 \cos \theta}\) can be sketched by choosing a range of θ values, calculating the corresponding r values, plotting these points on a polar grid, and connecting them to visualize the graph. Look out for places where the function is undefined.

Step 1: Choose values for theta

Start by choosing a range of values for θ. A complete range is usually from 0 to 2π (0 to 360 degrees), as this covers a complete circle. You can pick more specific values if needed, but a basic set to start with is every 30 or 45 degrees.

Step 2: Calculate the values of r

For each chosen value of θ, use the polar equation to find the corresponding r values. Substitute θ into the equation \(r=\frac{6}{2 \sin \theta-3 \cos \theta}\) and calculate the values.

Step 3: Plot the values

Plot the values you calculated on a polar grid. Remember that in polar coordinates, the angle θ is the counterclockwise angle from the x-axis, and r is the distance from the origin. Connect the points to visualize the graph of the equation.

Step 4: Check for Undefined Values

Take note where the denominator \(2 \sin \theta - 3 \cos \theta\) equals zero, as these values for θ will make the function undefined. You should mark these angles on your graph and determine the behavior of the graph near these points.