Problem 58 - Algebra and Trigonometry

Question

Convert the polar equation to rectangular coordinates. $$ r=2-\cos \theta $$

Step-by-Step Solution

Answer

\(x^2 + y^2 = 2\sqrt{x^2 + y^2} - x\) is the rectangular form.

Step 1: Understand Polar to Rectangular Conversion

Recall that in polar coordinates, a point is represented as \((r, \theta)\), where \(r\) is the radius and \(\theta\) is the angle. In rectangular coordinates, it's represented as \((x, y)\). The conversions between these systems are given by: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, \(r = \sqrt{x^2 + y^2}\), and \(\cos \theta = \frac{x}{r}\).

Step 2: Replace \(r\) with \(\sqrt{x^2 + y^2}\)

In the given equation \(r = 2 - \cos \theta\), replace \(r\) with \(\sqrt{x^2 + y^2}\) to start converting into rectangular coordinates: \[ \sqrt{x^2 + y^2} = 2 - \cos \theta \]

Step 3: Express \(\cos \theta\) in Rectangular Form

Use the identity \(\cos \theta = \frac{x}{r}\) and substitute it into the equation:\[ \sqrt{x^2 + y^2} = 2 - \frac{x}{\sqrt{x^2 + y^2}} \]

Step 4: Eliminate the Square Root

Multiply both sides of the equation by \(\sqrt{x^2 + y^2}\) to eliminate the square root from the denominator on the right-hand side:\[ x^2 + y^2 = (2\sqrt{x^2 + y^2}) - x \]

Step 5: Rearrange the Equation

Rearrange the equation to bring all terms to one side:\[ x^2 + y^2 - 2\sqrt{x^2 + y^2} + x = 0 \]

Step 6: Simplify and Solve if Possible

An explicit form in rectangular coordinates can be complex for some equations. In this case, the equation already shows a relationship. However, further simplification might typically involve assuming specific conditions to solve for a more insightful solution.

Key Concepts

Understanding Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are one of the most common systems used in mathematics to describe the position of a point in a plane. This system is based on two components:

• **x-coordinate**: Represents the horizontal distance of the point from the origin.
• **y-coordinate**: Represents the vertical distance of the point from the origin.

These coordinates are expressed as a pair \( (x, y) \).
The origin, where \( x = 0 \) and \( y = 0 \), is the central point equidistant from all other points in this system.
Compared to the polar coordinate system, rectangular coordinates are conceptually simpler for visualizing straight lines and performing basic arithmetic operations. However, certain calculations, like rotations or dealing with curves, might become complex.

Exploring Polar Coordinates

Polar coordinates provide a different way of representing points in a plane, particularly useful for circular and angular problems.
Instead of using a grid pattern, like rectangular coordinates, polar coordinates use:

• **r (radius)**: The straight-line distance from the origin to the point.
• **\( \theta \) (angle)**: The direction of the radius line, measured in radians (or degrees) from the positive x-axis.

Points are identified by the ordered pair \( (r, \theta) \).
In this system, it is significant that different pairs of values can represent the same point due to periodicity and symmetry in angles.
This characteristic is especially helpful when dealing with problems involving rotation or circular motion.
For example, polar coordinates are ideally suited for describing the paths of particles moving around a central point.

Using Trigonometric Identities for Conversion

Trigonometric identities are essential tools in converting between polar and rectangular coordinates. Trigonometry connects angles with linear distances, which is crucial in these transformations.
The main identities used include:

• \( x = r \cos \theta \): Relates the x-coordinate to the radius and angle.
• \( y = r \sin \theta \): Connects the y-coordinate similarly.
• \( r = \sqrt{x^2 + y^2} \): Finds the radius when \(x\) and \(y\) are known.
• \( \cos \theta = \frac{x}{r} \): Provides a way to express \(\cos \theta\) using rectangular coordinates.

These identities allow for the interchange of polar and rectangular forms, enabling us to leverage the strengths of both methods for various problem-solving scenarios.
Understanding how each identity functions helps grasp the interactions between distances and angles, facilitating efficient coordinate transformation.

Navigating Coordinate Transformation

Coordinate transformation is the process of converting points expressed in one coordinate system to another system. It's a vital concept when switching between polar and rectangular coordinates, especially in calculus and geometry.
Here’s how it’s typically approached:

• Start with the given equation in one system (polar or rectangular).
• Apply trigonometric identities and geometric relationships to express variables in terms of each other.
• Ensure all terms are consistently transformed to the target coordinate system.

For instance, to transform the polar equation \( r = 2 - \cos \theta \) into rectangular coordinates, we used:- Substitute \( r = \sqrt{x^2 + y^2} \) and \( \cos \theta = \frac{x}{r} \).- Manipulate the equation to the form \( x^2 + y^2 - 2\sqrt{x^2 + y^2} + x = 0 \), a step toward expressing it entirely in \( x \) and \( y \).
Such transformations allow the flexibility to use the most appropriate system for solving mathematical problems, understanding patterns, or simulating real-world scenarios.