Lesson 2: Converting Between Polar and Rectangular

Estimated time: 30-35 minutes

Conversion Formulas

Polar to Rectangular: x = r cos θ, y = r sin θ

Rectangular to Polar: r² = x² + y², tan θ = y/x

Convert (4, π/3) to rectangular.

x = 4 cos(π/3) = 4(1/2) = 2. y = 4 sin(π/3) = 4(√3/2) = 2√3.

(2, 2√3)

Convert (−3, 3) to polar.

r = √(9+9) = 3√2. tanθ = 3/(−3) = −1. Since the point is in QII: θ = 3π/4.

(3√2, 3π/4)

Convert x² + y² = 25 to polar.

r² = 25, so r = 5 (a circle of radius 5).

Convert y = x to polar.

r sinθ = r cosθ ⇒ sinθ/cosθ = 1 ⇒ tanθ = 1 ⇒ θ = π/4

Convert r = 4 cosθ to rectangular.

Multiply both sides by r: r² = 4r cosθ ⇒ x² + y² = 4x

Complete the square: (x−2)² + y² = 4. Circle centered at (2,0) with radius 2.

Convert r = 2/(1 + sinθ) to rectangular.

r + r sinθ = 2 ⇒ r + y = 2 ⇒ √(x²+y²) = 2 − y

Square: x²+y² = 4 − 4y + y² ⇒ x² = −4y + 4, a parabola.

1. Convert (6, π/2) to rectangular.

x=6cos(π/2)=0, y=6sin(π/2)=6. (0, 6).

2. Convert (1, −1) to polar.

r=√2. θ=arctan(−1/1)=−π/4 (QIV), or 7π/4. (√2, 7π/4).

3. Convert r = 3 to rectangular.

r²=9 ⇒ x²+y²=9.

x = r cosθ and y = r sinθ convert polar to rectangular.
r² = x² + y² and tanθ = y/x convert rectangular to polar.
When converting equations, multiply by r or substitute x, y, r relationships strategically.
r = a cosθ and r = a sinθ are circles; θ = constant is a line through the origin.