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Chapter 6: Problem 36

What is the relationship between the point with polar coordinates (5,0.2) and the point with polar coordinates (5,-0.2)\(?\)

Short Answer

Expert verified
The relationship between the points with polar coordinates (5, 0.2) and (5, -0.2) is that they are reflections of each other about the x-axis. Both points have the same radius (5) but different angles (0.2 and -0.2), indicating that they are symmetric with respect to the x-axis. Converting the points to Cartesian coordinates also confirms their reflection, as their x-coordinates are the same and their y-coordinates are opposites.

Step by step solution

01

Plot the points in the polar coordinate system

To get a better understanding of the relationship between the points, we should first plot them on the polar coordinate system. Remember that in polar coordinates, (r, 胃) represents the point located at a distance r from the origin, making an angle 胃 with the positive x-axis. For the point (5, 0.2), r = 5 and 胃 = 0.2, plot this point with respect to the origin. For the point (5, -0.2), r = 5 and 胃 = -0.2, plot this point with respect to the origin.
02

Observe the relationship in the polar coordinate system

With the polar coordinates plotted, we notice that both points share the same radius, as they are both located at a distance of 5 units from the origin. However, their angles are different - one makes a positive angle (0.2) with the positive x-axis, while the other makes a negative angle (-0.2) with the positive x-axis. This tells us that the two points are symmetric with respect to the x-axis. In simpler words, we can say that the two points are reflections of each other about the x-axis.
03

Convert the points to Cartesian coordinates

To further understand the relationship between the two points, let's convert them to Cartesian coordinates. We can use the polar-to-Cartesian coordinate conversion formula: \( x = r \cdot \cos{\theta} \) \( y = r \cdot \sin{\theta} \) For the point (5, 0.2): x = \( 5 \cdot \cos{0.2} \) y = \( 5 \cdot \sin{0.2} \) For the point (5, -0.2): x = \( 5 \cdot \cos{(-0.2)} \) y = \( 5 \cdot \sin{(-0.2)} \)
04

Observe the relationship in Cartesian coordinates

By comparing the Cartesian coordinates of the two points, we can observe their relationship more clearly. We can see that the x-coordinates are the same for both points, which indicates that they are aligned vertically. We also notice that the y-coordinates are opposites of one another, which confirms that these points are reflections about the x-axis.

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