Question

Use a graphing utility to graph the polar equation and find all points of horizontal tangency. $$ r=2 \cos (3 \theta-2) $$

Short answer

The specific coordinates for the points of tangency will depend on the particular variation of the curve plotted by the graphing utility. The method to find these points requires graphing the equation and then identifying local maximum and minimum points based on the plotted graph.

Step-by-step solution

Recognize the form of the equation

The given equation is in the form of a polar curve equation. For a polar curve \(r = f(\theta)\), any point on the curve can be represented by a distance \(r\) from the origin and an angle \(\theta\) from the positive x-axis.

Graph the polar equation

Enter \(r = 2 \cos(3\theta - 2)\) into the graphing utility and plot the graph. Depending upon your utility, you may need to adjust the range of \(\theta\) to see the full picture of the curve.

Identify points of horizontal tangency

Points of horizontal tangency occur when the slope of the tangent line to the curve is zero. These points on a polar graph correspond to local maximum and minimum values of \(r\) as \(\theta\) increases. Depending upon the graph, there may be several points of horizontal tangency.

Find coordinates for the tangency points

Once the points of tangency are identified, their polar coordinates can be found by reading the \(\theta\) value off of the graph at these points, and plugging this value in to the equation to find corresponding \(r\) value.