Converting Polar Equations to Rectangular Equations

Fundamental Formulas

• Considering a right triangle:
  • Sides: x, y and hypotenuse r
  • Angle: θ
• Essential equations for conversion:
  • x^2 + y^2 = r^2
  • x = r cos(θ)
  • y = r sin(θ)
  • tan(θ) = y / x

Difference Between Polar and Rectangular Equations

• Polar Equations: Variables r or θ (e.g., r = 5 sin(θ), r = 7)
• Rectangular Equations: Variables x and y (e.g., x^2 + y^2 = 4, x = 3, x^2 = 4y)

Conversion Examples

Example 1: r = 7

1. Square both sides: r^2 = 7^2 = 49
2. Use the identity x^2 + y^2 = r^2
3. Result: x^2 + y^2 = 49

Example 2: r = 5

1. r^2 = 25
2. x^2 + y^2 = 25
3. Result: x^2 + y^2 = 25

Example 3: θ = π/4

1. Take the tangent: tan(θ) = tan(π/4)
2. tan(θ) = y/x, tan(π/4) = 1
3. Multiply by x: y = x

Example 4: θ = 0°

1. Take the tangent: tan(θ) = tan(0°)
2. tan(0°) = 0, so y/x = 0
3. y = 0

Example 5: θ = π/2

1. Take the tangent: tan(θ) = tan(π/2)
2. tan(π/2) is undefined, so x = 0
3. Result: x = 0

Example 6: r sin(θ) = 5, r cos(θ) = 4

• r sin(θ) = y, so y = 5
• r cos(θ) = x, so x = 4

Example 7: r = 3 csc(θ)

1. csc(θ) = 1/sin(θ)
2. Multiply: r sin(θ) = 3
3. y = 3

Example 8: r = 4 sec(θ)

1. sec(θ) = 1/cos(θ)
2. Multiply: r cos(θ) = 4
3. x = 4

Example 9: r = 3 sin(θ)

1. Multiply by r: r^2 = 3r sin(θ)
2. Use identities: x^2 + y^2 = r^2, y = r sin(θ)
3. Result: x^2 + y^2 = 3y

Example 10: r = 4 cos(θ)

1. Multiply by r: r^2 = 4r cos(θ)
2. Use identities: x^2 + y^2 = r^2, x = r cos(θ)
3. Result: x^2 + y^2 = 4x

Example 11: r = 3 cos(θ) + 5 sin(θ)

1. Multiply by r: r^2 = 3r cos(θ) + 5r sin(θ)
2. Use identities: x^2 + y^2 = r^2, x = r cos(θ), y = r sin(θ)
3. Result: x^2 + y^2 = 3x + 5y

Example 12: r = 5 / (2 cos(θ) + 3 sin(θ))

1. Multiply by denominator: r(2 cos(θ) + 3 sin(θ)) = 5
2. Distribute r: 2r cos(θ) + 3r sin(θ) = 5
3. Use identities: x = r cos(θ), y = r sin(θ)
4. Result: 2x + 3y = 5

Example 13: r^2 sin(2θ) = 8

1. Use double angle formula: sin(2θ) = 2 sin(θ) cos(θ)
2. Simplify: r^2 2 sin(θ) cos(θ) = 8
3. Divide by 2: r^2 sin(θ) cos(θ) = 4
4. Use identities and simplify: xy = 4 -> y = 4/x

Example 14: r = 8 / cos(θ)

1. Multiply by cos(θ): r cos(θ) = 8
2. Result: x = 8

Example 15: r = 5 cos(θ) / sin^2(θ)

1. Multiply both sides by sin(θ): r sin(θ) = 5 cos(θ)/sin(θ)
2. Simplify: y = 5 cot(θ)
3. Use identity: cot(θ) = x/y
4. Result: y^2 = 5x

Example 16: r = sin(θ) cos^2(θ)

1. Multiply by r^3: r^4 = r^3 sin(θ) cos^2(θ)
2. Use identities and simplify
3. Result: x^2 + y^2 = x sqrt(y)