Problem 93 Graph $r_{1}$ and $r_{2}$ in... [FREE SOLUTION]

Chapter 7: Problem 93

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

Short Answer

Expert verified

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 by step solution

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.

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.

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 $r_{2}$.

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91影视

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

Short Answer

Step by step solution

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

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

Analyze the Relationship Between the Two Graphs

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