Lecture Notes: Converting Rectangular Equations to Polar Equations

Introduction

• The objective is to convert rectangular equations (involving x's and y's) into polar equations (involving r's and thetas).
• The lecture aims to provide an in-depth understanding of the conversion process.

Definitions

Rectangular Equations

• Can be graphed on an XY coordinate plane using x's and y's.

Polar Equations

• Involve r's and thetas, referring to a polar coordinate system with a polar axis and a pole (origin).

Importance of Conversion

• Graphing Functions: Rectangular equations work well for functions.
• Non-functions: For shapes like circles, ellipses, cardioids, etc., polar equations are easier to graph.

Conversion Process

1. Identify x² + y²: If the equation includes x² + y², replace it with r².
2. Substitutions: For any x or y variables that cannot be converted directly:
   • Use the relationships:
     • x = r cos(θ)
     • y = r sin(θ)

Example Conversions

Example 1: 2x² + 2y² = 3

• Factor out the 2:
  • 2(x² + y²) = 3
• Substitute x² + y² with r²:
  • 2r² = 3
• Solve for r:
  • r² = 3/2
  • r = √(6)/2 (indicating a circle with radius √(6)/2).

Example 2: x² + y² = x

• Replace x with r cos(θ):
  • r² = r cos(θ)
• Rearrange:
  • r(r - cos(θ)) = 0
  • Solutions: r = 0 or r = cos(θ).

Example 3: x² = 4y

• Replace x with r cos(θ) and y with r sin(θ):
  • (r cos(θ))² = 4(r sin(θ))
• Simplify:
  • r² cos²(θ) = 4r sin(θ).

Additional Conversion Considerations

• If unable to factor, revert back to individual variables and make appropriate substitutions.
• Be cautious when dividing by r; ensure that r = 0 is considered as a solution when applicable.

Graphing Lines in Polar Coordinates

Vertical Line Example: x = 3

• Substitute:
  • r cos(θ) = 3
• Results in:
  • r = 3 sec(θ).

Horizontal Line Example: y = 7

• Substitute:
  • r sin(θ) = 7
• Results in:
  • r = 7 csc(θ).

Diagonal Line Example: y = √3x

• Rewrite as:
  • y/x = √3
• Use tangent:
  • θ = tan⁻¹(√3) or θ = π/3.

Conclusion

• Understanding when to convert and the proper methods for conversion is crucial for graphing functions and shapes effectively.
• The next lecture will cover converting from polar equations back to rectangular equations.