Problem 93 - Algebra and Trigonometry

Question

Graph \(r_{1}\) and \(r_{2}\) in the same polar coordinate system. What is the relationship between the two graphs? $$r_{1}=4 \cos 2 \theta, r_{2}=4 \cos 2\left(\theta-\frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

The two graphs are equivalent in shape and size, but \(r_{2}\) is a rotation of \(r_{1}\) counter-clockwise by an angle of \(\frac{\pi}{4}\) radians.

Step 1: Graph \(r_{1}=4 \cos 2 \theta\) in Polar Coordinates

To graph the polar equation \(r_{1}=4 \cos 2 \theta\), create a table of values for \(\theta\) and \(r_1\). Quite often, it is most convenient to work on an interval \([0, 2\pi]\) or \([-pi, pi]\) for \(\theta\). After creating the table, plot the points and draw the curve that connects these points.

Step 2: Graph \(r_{2}=4 \cos 2(\theta - \frac{\pi}{4})\) in Polar Coordinates

To plot \(r_{2}=4 \cos 2(\theta - \frac{\pi}{4})\), follow the same process as above. Generate a table of values for \(\theta\) and \(r_2\), then plot these points. Since \(\frac{\pi}{4}\) is subtracted from \(\theta\) in the polar equation, this will cause a shift to the graph. Each plotted point will rotate counter-clockwise by \(\frac{\pi}{4}\) radians around the pole.

Step 3: Analyze the Relationship Between the Two Graphs

Now that both \(r_{1}\) and \(r_{2}\) have been graphed, we can compare the two. The two graphs are of same shape because the absolute value of the coefficients of \(\cos\) are the same for both equations, but \(r_{2}\) is rotated \(\frac{\pi}{4}\) radians from \(r_{1}\). This is due to the \(\frac{\pi}{4}\) subtracted from \(\theta\) in the function of \(r_{2}\).