Problem 50 - Algebra and Trigonometry

Question

Convert the polar equation to rectangular coordinates. $$ r=6 \cos \theta $$

Step-by-Step Solution

Answer

The rectangular form is \\(x^2 - 6x + y^2 = 0\\) or \\( (x-3)^2 + y^2 = 9\\).

Step 1: Understand Polar and Rectangular Coordinates

In polar coordinates, a point is represented by \(r, \theta\), where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis. To convert to rectangular coordinates \(x, y\), use the conversions \(x = r \cos \theta\) and \(y = r \sin \theta\).

Step 2: Substitute Polar Equation into Rectangular Formula

Our given equation is \(r = 6 \cos \theta\). Multiply both sides by \(r\): \[ r^2 = 6r \cos \theta \]

Step 3: Use Rectangular Conversion Formula

Recall that \(r^2 = x^2 + y^2\) and \(r \cos \theta = x\). Substitute these into the equation: \[ x^2 + y^2 = 6x \]

Step 4: Rearrange to Rectangular Equation Form

Rearrange the equation to place all terms on one side: \[ x^2 - 6x + y^2 = 0 \]

Step 5: Complete the Square (Optional)

If needed, complete the square for the \(x\) terms to further simplify or analyze the equation: Add and subtract \(9\) (which is \(\left( \frac{6}{2} \right)^2\)) to the equation: \[(x-3)^2 + y^2 = 9\]This represents a circle with center (3, 0) and radius 3.

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way of representing points in a plane through a combination of distance and direction. In this system, every point is described by two values: \( r \) and \( \theta \).

• \( r \) represents the distance from the origin (like the radius of a circle).
• \( \theta \) denotes the angle measured from the positive x-axis.

Consider it like directing a point's location not by strict north-south-east-west lines, but rather by saying "Move 'r' units in a given direction specified by \( \theta \)." This approach is especially useful in scenarios where symmetry about a point (the origin) is involved, like in circular movements.

Rectangular Coordinates

Rectangular coordinates are the more traditional way of plotting points on a graph, utilizing horizontal and vertical distances. These are denoted as \((x, y)\).

• \( x \): The horizontal distance from the y-axis (positive towards the right).
• \( y \): The vertical distance from the x-axis (positive upwards).

This system is like navigating streets on a grid where each intersection is marked by coordinates \( (x, y) \). This is particularly effective for straightforward vertical and horizontal navigations.

Coordinate Transformation

Switching between polar and rectangular coordinates is a valuable technique, especially in mathematical fields where different perspectives or angles help clarify relationships.

• To transform from polar to rectangular, use:
  • \( x = r \cos \theta \)
  • \( y = r \sin \theta \)
• Conversely, for transforming from rectangular to polar, use:
  • \( r = \sqrt{x^2 + y^2} \)
  • \( \theta = \tan^{-1}(y/x) \)

These transformations allow for the reinterpretation of a location's position in space, facilitating a better understanding of spatial relationships and movements.

Equation Conversion

When converting an equation from polar to rectangular coordinates, like the original exercise \( r = 6 \cos \theta \), you substitute polar terms with their rectangular equivalents.

• Inicio: Multiply both sides by \( r \) yielding \( r^2 = 6r \cos \theta \).
• Step: Replace terms using known conversions. For example, use \( r^2 = x^2 + y^2 \) and \( r \cos \theta = x \).
• Resultados: Simplify into a typical rectangular equation, such as \( x^2 + y^2 = 6x \).
• Further: Optional simplification steps, like completing the square, reveal useful information (e.g., circles, ellipses).

Converting equations helps in understanding geometric shapes and areas better by peeling off their polar expressions into more familiar and analyzable rectangular forms.