Converting Rectangular Equations to Polar Equations

Overview

• Rectangular Equation: Contains x or y variables (e.g., x + y = 4, x^2 + y^2 = 9).
• Polar Equation: Contains r and θ (theta) variables (e.g., r = 4, θ = π/3, r = 7sinθ).
• Goal: Convert rectangular equations to polar equations using specific formulas.

Important Formulas

• x^2 + y^2 = r^2
• tan(θ) = y / x
• x = rcos(θ)
• y = rsin(θ)

Step-by-Step Examples

1. Example: x = 8

   • Use x = rcos(θ).
   • Replace x with rcos(θ):
     • 8 = rcos(θ)
   • Isolate r:
     • r = 8 / cos(θ)
     • r can be written as r = 8cos^(-1)(θ) or 8sec(θ).

2. Example: y = 5

   • Use y = rsin(θ).
   • Replace y with rsin(θ):
     • 5 = rsin(θ)
   • Isolate r:
     • r = 5 / sin(θ)
     • r = 5csc(θ).

3. Example: 5x + 4y = 8

   • Use x = rcos(θ) and y = rsin(θ).
   • Replace x and y:
     • 5rcos(θ) + 4rsin(θ) = 8
   • Factor out r:
     • r(5cos(θ) + 4sin(θ)) = 8
   • Isolate r:
     • r = 8 / (5cos(θ) + 4sin(θ))

4. Example: x^2 + y^2 = 16

   • Use x^2 + y^2 = r^2.
   • Replace x^2 + y^2 with r^2:
     • r^2 = 16
   • Take the square root:
     • r = 4

5. Example: (x - 3)^2 + y^2 = 9

   • Expand and simplify:
     • x^2 - 6x + 9 + y^2 = 9
   • Combine like terms:
     • x^2 + y^2 - 6x = 0
   • Replace with polar equivalents:
     • r^2 - 6rcos(θ) = 0
   • Factor out r:
     • r(r - 6cos(θ)) = 0
   • Solutions:
     • r = 0 or r = 6cos(θ)

6. Example: x^2 + (y + 4)^2 = 16

   • Expand and simplify:
     • x^2 + y^2 + 8y + 16 = 16
   • Combine like terms:
     • x^2 + y^2 + 8y = 0
   • Replace with polar equivalents:
     • r^2 + 8rsin(θ) = 0
   • Factor out r:
     • r(r + 8sin(θ)) = 0
   • Solutions:
     • r = 0 or r = -8sin(θ)

7. Example: y^2 = 4x

   • Use x = rcos(θ) and y = rsin(θ).
   • Replace y^2 and x:
     • (rsin(θ))^2 = 4rcos(θ)
   • Simplify and isolate r:
     • r^2sin^2(θ) = 4rcos(θ)
   • Factor and solve:
     • r(sin^2(θ) - 4cos(θ)) = 0
   • Solutions:
     • r = 0 or r = 4cos(θ)/sin^2(θ)
     • Further simplification: r = 4cot(θ)csc(θ)

8. Example: x^2 = 8y

   • Use x = rcos(θ) and y = rsin(θ).
   • Replace x^2 and y:
     • r^2cos^2(θ) = 8rsin(θ)
   • Simplify and isolate r:
     • rcos^2(θ) = 8sin(θ)
     • r = 8tan(θ)sec(θ)

9. Example: x^2 + y^2 = 6x + 4y

   • Use x = rcos(θ) and y = rsin(θ).
   • Replace all terms:
     • r^2 = 6rcos(θ) + 4rsin(θ)
   • Divide by r:
     • r = 6cos(θ) + 4sin(θ)
   • Note: r = 0 is another possible solution.

10. Example: y = √3x

    • Use x = rcos(θ) and y = rsin(θ).
    • Replace y and x:
      • rsin(θ) = √3rcos(θ)
    • Divide by r and solve for θ:
      • tan(θ) = √3
      • θ = π/3

Key Points

• Always replace x and y with their polar equivalents.
• Simplify to express equations in terms of r and θ.
• Isolate r whenever possible.
• Multiple solutions (including r = 0) may be valid but are sometimes omitted.

Conclusion

The conversion between rectangular and polar equations relies on a solid understanding of trigonometric identities and algebraic manipulation. Practice with various examples to master the technique.