🧭 Part 1 — The Polar Coordinate System
What are Polar Coordinates?
Instead of (x, y), use (r, θ) — distance from origin and angle
🗺️ Rectangular (x, y)
Go right/left x units, then up/down y units.
Like a road grid — follow the axes.
Like a road grid — follow the axes.
✈️ Polar (r, θ)
Rotate to angle θ, travel distance r from origin.
Like flying — direct path from origin.
Like flying — direct path from origin.
Polar grid — rings show r=1,2,3,4 · radial lines every 30° · Example 1 points plotted
Reading polar point (r, θ)
r = distance from origin (which ring). θ = direction to face (angle from polar axis).
Negative r: travel in the OPPOSITE direction of θ. So (−r, θ) = (r, θ+π).
Negative θ: rotate clockwise instead of counterclockwise.
📌 Example 1 — Plot: A(2, π/3), B(1, 3π/2), C(3, 4π/3), D(2, −π/6)
A(2, π/3)
r=2, θ=π/3=60°
→ Q1, ring 2
→ Q1, ring 2
B(1, 3π/2)
r=1, θ=270°
→ straight down
→ straight down
C(3, 4π/3)
r=3, θ=240°
→ Q3, ring 3
→ Q3, ring 3
D(2, −π/6)
r=2, θ=−30°
→ Q4 (clockwise)
→ Q4 (clockwise)
♾️ Part 2 — Equivalent Representations
The Same Point, Many Ways
Add ±2π to θ (same r) · add ±π to θ AND negate r
Three families of equivalent representations for (r, θ)
1) Add 2πk to θ, keep r: (r, θ+2πk) — full circle rotation, same point.
2) Add π to θ, negate r: (−r, θ+π) — flip direction, travel backwards same distance.
3) Subtract π from θ, negate r: (−r, θ−π) — same idea, other direction.
📌 Example 2 — All four represent the same point. Verify A, B, C, D all land at the position of (2, π/4).
A — Base
(2, π/4)
r=2, θ=45°
B — −2π
(2, −7π/4)
−7π/4+2π=π/4 ✓
C — +2π
(2, 9π/4)
9π/4−2π=π/4 ✓
D — neg r,+π
(−2, 5π/4)
5π/4−π=π/4, r→−r ✓
📌 Example 3 — Point (3, 5π/6). Write TWO other equivalent polar representations.
1
Subtract 2π (same r): θ = 5π/6 − 2π = −7π/6 → (3, −7π/6)2
Subtract π, negate r: θ = 5π/6 − π = −π/6, r = −3 → (−3, −π/6)3
Add π, negate r: θ = 5π/6 + π = 11π/6, r = −3 → (−3, 11π/6)Any two of: (3, −7π/6) · (−3, −π/6) · (−3, 11π/6)
🔃 Part 3 — Converting Between Systems
Conversion Formulas
x=rcosθ, y=rsinθ for polar→rect · r²=x²+y², tanθ=y/x for rect→polar
📐 Polar → Rectangular
x = r cos θ
y = r sin θ
From cosθ=x/r and sinθ=y/r → rearrange for x and y.
📐 Rectangular → Polar
r = √(x² + y²)
tan θ = y/x → sketch to find correct quadrant
⚠️ arctan(y/x) only gives Q1/Q4. For Q2/Q3: add π to the result. Always sketch first!
📌 Example 4 — Polar → Rectangular: a) (2, π) b) (4, 5π/3) c) (−3, 7π/6)
a) (2, π)
x = 2cos(π) = 2(−1) = −2
y = 2sin(π) = 2(0) = 0
(−2, 0)y = 2sin(π) = 2(0) = 0
b) (4, 5π/3)
x = 4cos(5π/3) = 4(1/2) = 2
y = 4sin(5π/3) = 4(−√3/2) = −2√3
(2, −2√3)y = 4sin(5π/3) = 4(−√3/2) = −2√3
c) (−3, 7π/6)
x = −3cos(7π/6) = −3(−√3/2) = 3√3/2
y = −3sin(7π/6) = −3(−1/2) = 3/2
(3√3/2, 3/2)y = −3sin(7π/6) = −3(−1/2) = 3/2
📌 Example 5 — Rectangular → Polar: a) (−2, 2) b) (1, −√3) c) (0, −5)
💡
Always sketch the point first to confirm which quadrant θ is in!a) (−2, 2) — Q2
r = √(4+4) = 2√2
tan θ = 2/(−2) = −1
Q2 → θ = 3π/4
(2√2, 3π/4)tan θ = 2/(−2) = −1
Q2 → θ = 3π/4
b) (1, −√3) — Q4
r = √(1+3) = 2
tan θ = −√3/1 = −√3
Q4 → θ = 5π/3
(2, 5π/3)tan θ = −√3/1 = −√3
Q4 → θ = 5π/3
c) (0, −5) — neg y-axis
r = √(0+25) = 5
tan θ = −5/0 = undef
→ θ = 3π/2
(5, 3π/2)tan θ = −5/0 = undef
→ θ = 3π/2
Quadrant trap for Rectangular → Polar
For a) (−2, 2) in Q2: tan θ = −1, but arctan(−1) = −π/4 (a Q4 answer). Since the point is in Q2, add π: θ = −π/4 + π = 3π/4 ✓. Always let the sketch guide your final angle choice.
🔢 Part 4 — Complex Numbers in Polar Form
Complex Numbers and the Complex Plane
a+bi ↔ point (a,b) ↔ polar form r cosθ + i(r sinθ)
Rectangular Form
a + bi
Point (a, b) in complex plane
Polar Form
r cosθ + i(r sinθ)
Using x=rcosθ, y=rsinθ
Finding r and θ
r = √(a²+b²)
tanθ = b/a
tanθ = b/a
Same as rect→polar!
📌 Example 6 — Complex number at (−2, −2). Express in polar form (r, θ).
1
r = √((−2)²+(−2)²) = √8 = 2√2.2
tan θ = (−2)/(−2) = 1, both negative → Q3. θ = π + π/4 = 5π/4.3
Polar form: (√8, 5π/4). Complex: √8·cos(5π/4) + i·√8·sin(5π/4) = −2−2i ✓Polar: (√8, 5π/4) = (2√2, 5π/4)
Complex polar form: √8 cos(5π/4) + i(√8 sin(5π/4))
📌 Example 7 — Polar form: 4cos(7π/6) + i·4sin(7π/6). Express as rectangular (x, y).
1
r=4, θ=7π/6. Apply x=rcosθ, y=rsinθ.2
x = 4cos(7π/6) = 4·(−√3/2) = −2√3.3
y = 4sin(7π/6) = 4·(−1/2) = −2.Rectangular: (−2√3, −2) · Complex: −2√3 − 2i
Quick Reference
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🔑 Key Formulas
Polar (r,θ): r=distance, θ=angle from polar axis
Polar→Rect: x=rcosθ · y=rsinθ
Rect→Polar: r=√(x²+y²) · tanθ=y/x (+sketch!)
(r,θ)=(r,θ+2πk)=(−r,θ+π)=(−r,θ−π)
Complex: a+bi ↔ r cosθ + i(r sinθ)
⚠️ Common Mistakes
❌ (r, θ+π) = same as (r, θ)
Adding π to θ requires NEGATING r: (r,θ) = (−r,θ+π). Without negating r, you get the OPPOSITE point.
❌ θ = arctan(y/x) always
arctan only gives Q1/Q4. For Q2/Q3, add π. Always sketch the point first to verify quadrant.
❌ Negative r means the point is invalid
Negative r is perfectly valid — it means travel opposite to θ. (−2, π/4) lands in Q3, not Q1.
❌ Complex polar form uses a and b directly as r and θ
Must compute r=√(a²+b²) and find θ from the sketch + arctan. a and b are x,y coordinates, not r,θ.