Problem 55 - Algebra and Trigonometry

Question

Convert the polar equation to rectangular coordinates. $$ r=1+\cos \theta $$

Step-by-Step Solution

Answer

The rectangular form of the polar equation is \( x^2 - x + y^2 = 1 \).

Step 1: Identify Polar Equation Components

The polar equation given is \( r = 1 + \cos \theta \). In polar coordinates, \( r \) represents the distance from the origin to a point, and \( \theta \) is the angle with respect to the positive x-axis.

Step 2: Use Conversion Formulas

To convert from polar to rectangular coordinates, use the formulas \( x = r \cos \theta \) and \( y = r \sin \theta \). Additionally, we know \( \cos \theta = \frac{x}{r} \) and \( \sin \theta = \frac{y}{r} \).

Step 3: Substitute in the Polar Equation

Substitute \( r \cos \theta = x \) and \( r \sin \theta = y \) into the polar equation. We have \( r = 1 + \frac{x}{r} \).

Step 4: Multiply Both Sides by \( r \)

Multiply both sides of the equation \( r = 1 + \frac{x}{r} \) by \( r \) to eliminate the fraction: \( r^2 = r + x \).

Step 5: Use Identity for \( r^2 \)

Recall that \( r^2 = x^2 + y^2 \), an identity in polar to rectangular conversions. Substitute this into the equation: \( x^2 + y^2 = r + x \).

Step 6: Substitute \( r \) Back with Equation

Since \( r = 1 + \cos \theta \) and we know \( \cos \theta = \frac{x}{r} \), substitute \( r = 1 + \frac{x}{r} \) directly again if needed. However, \( r = 1 + \frac{x}{r} \) implies another round of substitution which was already managed. Continuing with substitute realignment: \( y^2 = 1 \).

Step 7: Complete Rectangular Form

Rewriting the combined forms of identity, we isolate x: \( x^2 - x + y^2 = 1 \). Simplify further like a circle form by completing the square if needed but recognize the original function symmetry. However, no further isolation or completing square gives better insight as identity holds at core.

Key Concepts

Polar Coordinates

Polar coordinates are a way to describe a point on a plane using a distance and an angle, rather than the traditional x and y coordinates. In this system, any point is defined by:

• r: The distance from the origin to the point.
• θ (theta): The angle from the positive x-axis to the line connecting the origin to the point.

Unlike the Cartesian system, where you move horizontally and vertically to reach a point, polar coordinates work by moving along a radius at a certain angle. This can be particularly useful in contexts where circular motion or angles are involved, such as navigation and robotics.
For instance, the polar equation you are working with is represented as\[r = 1 + \cos \theta.\]This means the distance from the origin changes with the angle, making it a dynamic representation.

Rectangular Coordinates

Rectangular coordinates, more commonly known as Cartesian coordinates, use two values to pinpoint a location on a plane: the x-coordinate and the y-coordinate. Unlike polar coordinates, these coordinates tell you how far along the x-axis and y-axis a point is located.

• x: The horizontal distance from the y-axis.
• y: The vertical distance from the x-axis.

These coordinates are particularly useful because they provide an easy way to measure distances and perform arithmetic operations for equations that involve straight lines.
In the Cartesian plane, every point is represented uniquely by an (x, y) pair. For example, converting from polar to rectangular coordinates involves deriving these specific x and y values from given r and θ values.

Coordinate Conversion

Converting between polar and rectangular coordinates can be essential when dealing with problems that involve different types of geometric and trigonometric analysis. The conversion process helps translate an angle and distance into a more familiar x and y format for further calculation or graphing. Here are the essential conversion formulas:

• From polar to rectangular:
  • \( x = r \cos \theta \)
  • \( y = r \sin \theta \)
• From rectangular back to polar:
  • \( r = \sqrt{x^2 + y^2} \)
  • \( \theta = \tan^{-1}(y/x) \)

By using these formulas, one can tackle mathematical problems that require changing the form of an expression or equation, such as transforming the polar equation \( r = 1 + \cos \theta \) into its rectangular equivalent. This equation, when transformed, illustrates a particular geometric shape and helps to visualize the relationship between these two coordinate systems.