Problem 93 - Algebra and Trigonometry Real Mathematics, Real People

Question

In the rectangular coordinate system, each point \((x, y)\) has a unique representation. Explain why this is not true for a point \((r, \theta)\) in the polar coordinate system.

Step-by-Step Solution

Answer

In the rectangular coordinate system, every point has a unique representation based on x and y coordinates. However, in the polar coordinate system, a point does not have a unique representation since the same point can be represented using different (r, θ) pairs due to the periodic nature of trigonometric functions. Moreover, the point can also be represented with negative distance and an additional π to the angle, creating yet another representation for the same point.

Step 1: Understanding the Rectangular Coordinate System

In the rectangular coordinate system, a point is uniquely defined by its x and y coordinates. A change in either x or y results in a different point. Hence, every point in this system has a unique representation with a pair of coordinates (x, y).

Step 2: Understanding the Polar Coordinate System

In the polar coordinate system, points are defined by a distance from the origin (r) and an angle (θ). The r value defines how far the point is from the origin, while theta measures the angle from the positive x-axis to the line connecting the origin to the point.

Step 3: Demonstrating Non-Uniqueness in the Polar Coordinate System

A point in the polar coordinate system does not have a unique representation as the same point can be represented using different (r, θ) pairs. For example, a point could be represented by the coordinates (r, θ) and again by the coordinates (r, θ+2π) due to the periodic nature of trigonometric functions. Similarly, a point can also be represented by (-r, θ+π), another set of coordinates. This shows that a single point can have multiple representations in the polar coordinate system and that is why it's not unique in this system.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a way of representing points in a plane using two perpendicular axes. Typically labeled as the x-axis and y-axis, these axes allow every point to have a specific pair of numbers. This pair, \(x, y\), denotes the point's location in a two-dimensional space.

• The x-coordinate indicates the horizontal position.
• The y-coordinate indicates the vertical position.

The beauty of this system lies in its uniqueness. Every distinct point in the rectangular coordinate system is represented by a unique set of coordinates. Change either the x or y value, and you have a completely different point. This makes it straightforward and unambiguous in indicating positions on a plane.

Unique Representation

In the Cartesian or rectangular coordinate system, a key advantage is the unique representation of each point. No two distinct points can share the same pair of coordinates \(x, y\).

• This uniqueness simplifies calculations such as finding distance, slope, or identifying geometric shapes like lines and curves.
• Ensures precise communication of location without ambiguity.

However, in the polar coordinate system, this principle of unique representation does not hold. While the Cartesian system maintains consistency, the polar system allows multiple (r, \( \theta \)) pairs to denote the same point due to angular periodicity and flexibility in direction.

Trigonometric Functions

Trigonometric functions play a vital role in connecting different coordinate systems. In polar coordinates, these functions define the angle \( \theta \), which is the primary difference from rectangular coordinates.

• Functions such as sine and cosine help convert polar coordinates into rectangular ones: \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\).
• Periodicity of trigonometric functions implies that angles normal to \(360^\circ\) or \(2\pi\) radians effectively represent the same direction.

This cyclic nature of angles in trigonometric functions leads to multiple valid expressions for the same polar point. For instance, a point initially described by \((r, \theta)\) could equivalently be \( (r, \theta + 2\pi) \) or \((-r, \theta + \pi)\).

Coordinate Systems

Coordinate systems like rectangular and polar systems are essential for understanding spatial relationships and geometric properties.

• Rectangular Coordinate System: Characterized by two perpendicular axes, x and y, that define locations using ordered pairs. This system is intuitive for linear calculations and graphical plots.
• Polar Coordinate System: Uses a radius and an angle to define positions, capturing rotational symmetries and radial distances more naturally.

Each system has its strengths and specific applications. The rectangular system is preferred for precise, linear definitions, while the polar system excels in scenarios involving rotations, circles, and oscillatory motions.
By understanding both systems, one can adeptly switch between them, leveraging trigonometric conversions, to solve a variety of mathematical problems.