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Chapter 6: Problem 69

Use a graphing utility to graph the polar equation. $$r=\frac{3}{\cos \theta}$$

Short Answer

Expert verified
The graph of the polar equation \( r = \frac{3}{\cos \theta} \) using the graphing utility will result in a vertical line at \( x = 3 \). This is due to the equivalent Cartesian form of the polar equation, which specifies a constant value for \( x \).

Step by step solution

01

- Understanding the polar equation

Polar equations come in many different forms representing different types of curves. The polar equation here is of the type \( r = \frac{a}{\cos \theta} \) or \( r = \frac{a}{\sin \theta} \), which represent a straight line. The equation \( r = \frac{3}{\cos \theta} \) can also be written as \( r \cos \theta = 3 \), correlating to its equivalent Cartesian equation \( x = 3 \). This represents a vertical line in the Cartesian system.
02

- Graphing the polar equation using a graphing utility

Now, use the graphing utility to plot the polar equation \( r = \frac{3}{\cos \theta} \). Remember that the polar coordinates \( (r, \theta) \) correspond to distance and angle from the origin. After plotting the equation, you will observe a vertical line at \( x = 3 \) on the polar coordinate system.
03

- Interpreting the result

Having plotted the polar equation, you can see that the graph indeed represents a vertical line at \( x = 3 \) in a Cartesian coordinate system, just as the modified polar equation \( r \cos \theta = 3 \) suggests. This verifies the equivalent Cartesian equation of the polar equation provided.

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