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Chapter 6: Problem 37

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

Short Answer

Expert verified
The point with polar coordinates (5,0.2+蟺) is the reflection of the point with polar coordinates (5,0.2) through the origin in the Cartesian plane.

Step by step solution

01

Convert polar coordinates to Cartesian coordinates

Given the Polar coordinates (r, 胃), the Cartesian coordinates (x,y) can be found using the following formulas: x = r*cos(胃) y = r*sin(胃) We need to convert both the points (5,0.2) and (5,0.2+蟺) to Cartesian coordinates. Let's first convert the point (5,0.2): x1 = 5*cos(0.2) = \(5 \cos(0.2)\) y1 = 5*sin(0.2) = \(5 \sin(0.2)\) Now, let's convert the point (5,0.2+蟺): x2 = 5*cos(0.2+蟺) = \(5 \cos(0.2+\pi)\) y2 = 5*sin(0.2+蟺) = \(5 \sin(0.2+\pi)\)
02

Analyze their positions in Cartesian plane

We know that adding 蟺 to the angle 胃 in polar coordinates results in a 180-degree rotation about the origin. So, when we add 蟺 to the angle 0.2 in the polar coordinates: x2 = \(5\cos(0.2+\pi) = -5\cos(0.2)\) y2 = \(5\sin(0.2+\pi) = -5\sin(0.2)\) Comparing these coordinates with x1 and y1, we can observe: x2 = -x1 and y2 = -y1
03

Conclusion

Based on the analysis of the Cartesian coordinates, we can conclude that the point with polar coordinates (5,0.2+蟺) is the reflection of the point with polar coordinates (5,0.2) through the origin in the Cartesian plane.

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Most popular questions from this chapter

Show that \(\overline{z^{n}}=(\bar{z})^{n}\) for every complex number \(z\) and every positive integer \(n\).

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