Identifying False Statements In Polar Coordinates Equations And Points

Navigating the realm of polar coordinates can sometimes feel like traversing a labyrinth of angles and radii. The beauty of this coordinate system lies in its ability to represent points and lines in a way that differs significantly from the familiar Cartesian plane. However, this difference also introduces nuances that can lead to confusion if not carefully examined. In this article, we will delve into a statement concerning polar coordinates and unravel the truth behind it. Our task is to identify the false statement from a set of options, focusing on the representation of lines and points in the polar plane. This exploration will not only sharpen our understanding of polar coordinates but also enhance our ability to discern subtle mathematical truths.

Deciphering Polar Equations and Points

Before we dissect the given statements, let's establish a firm grasp of the fundamentals of polar coordinates. In this system, a point in the plane is defined by its distance r from the origin (or pole) and the angle θ it forms with the positive x-axis (or polar axis). The coordinates (r, θ) uniquely identify a point, but with a twist. Unlike Cartesian coordinates, the representation in polar coordinates is not unique. For instance, adding multiples of 2π to θ doesn't alter the point's position. Furthermore, a negative r value indicates that the point lies on the opposite side of the pole along the same line defined by θ. This non-uniqueness is a key aspect to keep in mind as we evaluate the statements.

Now, consider the equation θ = π/6. This equation represents a line that passes through the origin, making an angle of π/6 radians (or 30 degrees) with the positive x-axis. The equation θ = 7π/6 also represents a line passing through the origin, but at an angle of 7π/6 radians (or 210 degrees). At first glance, these might seem like distinct lines. However, recall that polar coordinates allow for negative r values. A point with a positive r on the line θ = 7π/6 can also be represented with a negative r on the line θ = π/6. Specifically, any point (r, 7π/6) can be written as (-r, π/6). This subtle relationship is crucial for correctly interpreting polar equations.

When dealing with polar points, similar considerations apply. The point (-1, π/8) implies a distance of 1 from the origin, but in the direction opposite to the angle π/8. To find an equivalent representation, we can either add π to the angle or negate the radius. Adding π to π/8 gives us 9π/8, so (-1, π/8) is the same as (1, 9π/8). Alternatively, we can subtract π from the angle, resulting in -7π/8. This means (-1, π/8) is also equivalent to (1, -7π/8). Understanding these transformations is essential for accurately determining whether two polar points represent the same location.

Analyzing the Statements

Let's revisit the statement that needs verification: One of the following statements is false:

a. The polar equations θ = π/6 and θ = 7π/6 represent the same line. b. The polar points (-1, π/8) and (1, -7π/8) represent the same point.

Now, we dissect each statement using our understanding of polar coordinates.

Statement a claims that the polar equations θ = π/6 and θ = 7π/6 represent the same line. As discussed earlier, the equation θ = π/6 describes a line passing through the origin at an angle of π/6 with respect to the positive x-axis. Similarly, θ = 7π/6 represents a line through the origin at an angle of 7π/6. These angles are supplementary, meaning they are 180 degrees (or π radians) apart. This implies that any point on the line θ = 7π/6 can be represented with a negative radius on the line θ = π/6, and vice versa. Therefore, statement a is indeed true. Both equations define the same line in the polar plane.

Turning our attention to statement b, we are asked to determine if the polar points (-1, π/8) and (1, -7π/8) represent the same location. The point (-1, π/8) is located at a distance of 1 from the origin, but in the opposite direction of the angle π/8. To find an equivalent representation with a positive radius, we add π to the angle: π/8 + π = 9π/8. Thus, (-1, π/8) is equivalent to (1, 9π/8). Now, let's consider the point (1, -7π/8). To determine if this is the same as (1, 9π/8), we can add or subtract multiples of 2π from the angle. Adding 2π to -7π/8 yields -7π/8 + 16π/8 = 9π/8. Therefore, (1, -7π/8) is indeed the same as (1, 9π/8), and consequently, the same as (-1, π/8). Statement b is also true.

Conclusion: Unmasking the Truth in Polar Coordinates

In conclusion, after a thorough analysis of the statements concerning polar equations and points, we have determined that both statements are true. The polar equations θ = π/6 and θ = 7π/6 represent the same line due to the nature of negative radii in polar coordinates. Similarly, the polar points (-1, π/8) and (1, -7π/8) represent the same point, as can be verified by adding π to the angle and considering the periodicity of angles in polar coordinates. This exercise highlights the importance of understanding the nuances of polar coordinates, particularly the non-uniqueness of representations and the role of negative radii. By carefully considering these aspects, we can confidently navigate the polar plane and discern mathematical truths from potential fallacies.

This exploration underscores the beauty and complexity inherent in mathematical systems. Polar coordinates, while offering a different perspective from the Cartesian plane, demand a keen understanding of their unique properties. As we continue our mathematical journey, embracing these challenges and delving into the intricacies of different coordinate systems will undoubtedly enrich our understanding and appreciation of the mathematical world.