Problem 58 - Precalculus : Building Concepts and Connections

Question

Convert each of the given polar equations to rectangular form. $$\theta=\pi$$

Step-by-Step Solution

Answer

The rectangular form of the polar equation \(\theta = \pi\) is \(x = 0\).

Step 1: Analyze the given polar equation

The equation in this exercise is \(\theta = \pi\). In polar coordinates, \(\theta = \pi\) means that every point on the line has an angle of \(\pi\) radians with respect to the positive x-axis. This line is the negative x-axis itself.

Step 2: Convert to rectangular coordinates

In rectangular coordinates, the negative x-axis is represented by the equation \(x = 0\). Hence, the rectangular form of the given polar equation \(\theta = \pi\) is \(x = 0\).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a radius and an angle. This system is particularly useful when dealing with circular or spiral shapes and provides an alternative to the more familiar rectangular (or Cartesian) coordinates. In polar coordinates, each point is described by \(r, \, \theta\), where \(r\) is the distance from the origin, and \(\theta\) is the angle from the positive x-axis. This angle is measured in radians, where one full rotation (360 degrees) is equivalent to \(2\pi\) radians.

• \(r\) - Radial distance from the origin
• \(\theta\) - Angle measured in radians from the positive x-axis

Often, polar equations relate \(r\) and \(\theta\), defining paths or other entities in terms of these values. A key understanding is how these polar coordinates can be visualized on a graph, with the radial lines radiating from the origin and crossing the angles defined by \(\theta\). With polar coordinates, the flexibility to describe complex curves and directions makes them a powerful tool for addressing specific geometric problems.

Conversion of Equations

Converting between polar and rectangular forms is a common task in mathematics. Understanding this conversion is crucial for solving problems that involve both coordinate systems. For polar coordinates \( (r, \theta)\), the equations that relate them to rectangular coordinates \( (x, y)\) are:

• \(x = r \cos(\theta)\)
• \(y = r \sin(\theta)\)

These equations allow us to switch from polar equations to rectangular by replacing \( r \cos(\theta)\) and \( r \sin(\theta)\) with \( x\) and \( y\), respectively. Conversely, to convert from rectangular to polar:

• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\)

Understanding these equations helps in solving geometry problems involving paths or shapes that naturally fit one coordinate system or the other. Being able to convert and translate these equations allows for easier computations and insights into the nature of the geometric figures involved.

Negative X-axis

The negative x-axis in both rectangular and polar coordinates holds specific geometric significance. In polar coordinates, a line that extends along the negative x-axis can be described by an angle \(\theta = \pi\). This means every point on this line sustains a consistent angle of \(\pi\) radians from the positive x-axis.In rectangular coordinates, the negative x-axis is simpler to represent as a vertical line directly along the x-axis plane where x-values are zero, implying points on this line adhere to the equation \(x < 0\) and \(y = 0\). However, for the purpose of conversion as seen in the exercise, often such descriptions correspond to \(x = 0\) in the absence of specific radial measurement, emphasizing directionality.Points along the negative x-axis reveal how both coordinate systems express the same geometric path differently. Each description serves an advantage, with polar coordinates highlighting directionality and angular measures, while rectangular coordinates simplify collinear relationships directly along axes. This understanding is pivotal in translating between the two systems accurately.