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Chapter 6: Problem 62

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$\theta=\frac{\pi}{3}$$

Short Answer

Expert verified
The rectangular form of the given polar equation \(\theta = \frac{\pi}{3}\) is \(x = \frac{1}{2}\). The graph of this rectangular equation will be a vertical line passing through (0.5, 0) on x-axis and extends infinitely in both positive and negative y-directions.

Step by step solution

01

Understand the given equation

The given equation is in the form of a polar coordinate where \(\theta\) is a constant, \(\theta = \frac{\pi}{3}\), which denotes a line in the polar coordinate system.
02

Convert the polar equation to rectangular equation

A polar equation of the form \(\theta = C\), where C is constant, represents a line that forms an angle C with the positive x-axis. Therefore, we have a vertical line x = r cos(\(\frac{\pi}{3}\)) = r * \( \frac{1}{2}\). But for \(\theta = \frac{\pi}{3}\), the value of r does not matter as it is a line of constant angle. Therefore, the rectangular form of the given polar equation \(\theta = \frac{\pi}{3}\) is \(x = \frac{1}{2}\).
03

Graph the rectangular equation

The rectangular equation \(x = \frac{1}{2}\) represents a vertical line passing through the point (0.5, 0) on the x-axis. This line extends infinitely in both positive and negative y-directions on the graph.

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