Problem 91 - Precalculus with Limits

Question

In Exercises 85-108, convert the polar equation to rectangular form. \(\theta=11\pi/6\)

Step-by-Step Solution

Answer

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

Step 1: Recognizing Geometric Meaning of Constant \(\theta\)

A constant \(\theta\) in polar coordinates represents a ray originating from the origin making an angle of \(\theta\) with positive x-axis. When \(\theta\) is \(11\pi/6\) or \(\pi/6\) (since \(2\pi=12\pi/6\), they represent the same direction), it is in direction of the positive x axis and supposedly x is going to be a nonzero constant.

Step 2: Conversion to Rectangular form

Because this ray doesn't cross the y-axis, y can take on any value. Hence, the ray represents the entire x-axis. Thus the rectangular equation of this ray is \(x = 0\).

Key Concepts

Polar Coordinates

Polar coordinates are a method to represent a point in a two-dimensional plane using a distance and an angle. Unlike the rectangular or Cartesian system that uses an x-y grid, polar coordinates use:

• the radial distance from the origin, denoted usually as \(r\), and
• the angular coordinate (angle), often symbolized by \(\theta\).

For example, the point (\(r, \theta\)) specifies a location by describing how far it is from the origin and the direction or angle from the positive x-axis. By using angles, polar coordinates are particularly useful in contexts where symmetry, rotations, or circular patterns are involved.

When processing polar equations like \(\theta = 11\pi/6\), we basically consider this:\(\theta\) defines an angle from the positive x-axis, mapping out a line making this angle. It's essential to note, angles can be expressed in different equivalent measures, such as radians or degrees.

Rectangular Coordinates

Rectangular coordinates describe a point's location using two distances along the axes from the origin: x and y. In coordinate geometry, these are often called Cartesian coordinates. The main features include:

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

For instance, a point (x, y) could be plotted to show exactly where it lies in rectangular space. This straightforward description, relying purely on horizontal and vertical measurements, suits many applications that deal with linear paths and movements.

The conversion from polar to rectangular coordinates is achieved using specific formulas: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). For \(\theta = 11\pi/6\), this indicates that any point described can be understood as lying along the axis parallel to the x-axis, extending infinitely in both directions.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves studying geometry using a coordinate system. It connects geometric constructs, like lines and shapes, to algebraic equations. This field bridges the understanding between algebra and geometry, allowing for precise calculations and graphical representation of shapes and figures.

Key elements of coordinate geometry include:

• Equations: translating geometric figures into algebraic expressions.
• Visualization: graphically representing shapes and paths on coordinate planes.
• Transformation: converting between different forms, such as from polar to rectangular coordinates.

Consider our exercise, which involves conversion. By relating a polar angle, \(\theta=11\pi/6\), to its rectangular form, we see it describes a line, highlighting how coordinate geometry makes geometric interpretations accessible through algebra. Understanding these conversions enhances our ability to analyze patterns and systems in a mathematical framework.