Polar Coordinates

📈 Polar Coordinates

Part 1 of 7 — Polar Coordinate System

1. Polar coordinates

(r, θ) where r is distance from origin and θ is angle from positive x-axis

2. Multiple representations

(r, θ) = (r, θ + 2πn) = (-r, θ + π)

3. Pole is the origin (r = 0)

Pole is the origin (r = 0)

4. Polar axis is the positive x-axis direction

Polar axis is the positive x-axis direction

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Key Concepts Summary

• Polar coordinates: (r, θ) where r is distance from origin and θ is angle from positive x-axis
• Multiple representations: (r, θ) = (r, θ + 2πn) = (-r, θ + π)
• Pole is the origin (r = 0)
• Polar axis is the positive x-axis direction

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Converting Between Coordinate Systems

Part 2 of 7 — Converting Between Coordinate Systems

1. Polar to rectangular

x = r cos θ, y = r sin θ

2. Rectangular to polar

r = √(x² + y²), θ = arctan(y/x) (adjust quadrant)

3. r² = x² + y²

r² = x² + y²

4. Convert equations by substituting x, y, r, θ relationships

Convert equations by substituting x, y, r, θ relationships

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Key Concepts Summary

• Polar to rectangular: x = r cos θ, y = r sin θ
• Rectangular to polar: r = √(x² + y²), θ = arctan(y/x) (adjust quadrant)
• r² = x² + y²
• Convert equations by substituting x, y, r, θ relationships

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Polar Equations & Graphs

Part 3 of 7 — Polar Equations & Graphs

1. Circles

r = a (centered at origin), r = a cos θ or r = a sin θ (through origin)

2. Lines

θ = c (through the pole), r = a/cos θ (vertical), r = a/sin θ (horizontal)

3. Symmetry tests

replace θ with -θ (x-axis), θ with π-θ (y-axis), r with -r (origin)

4. Plot by evaluating r at several values of θ

Plot by evaluating r at several values of θ

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Key Concepts Summary

• Circles: r = a (centered at origin), r = a cos θ or r = a sin θ (through origin)
• Lines: θ = c (through the pole), r = a/cos θ (vertical), r = a/sin θ (horizontal)
• Symmetry tests: replace θ with -θ (x-axis), θ with π-θ (y-axis), r with -r (origin)
• Plot by evaluating r at several values of θ

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Classic Polar Curves

Part 4 of 7 — Classic Polar Curves

1. Rose curves

r = a cos(nθ) or r = a sin(nθ); n petals if odd, 2n petals if even

2. Limaçons

r = a + b cos θ; inner loop when |a/b| < 1, cardioid when |a/b| = 1

3. Lemniscate

r² = a² cos(2θ) or r² = a² sin(2θ)

4. Spiral of Archimedes

r = aθ, distance from pole increases with angle

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Key Concepts Summary

• Rose curves: r = a cos(nθ) or r = a sin(nθ); n petals if odd, 2n petals if even
• Limaçons: r = a + b cos θ; inner loop when |a/b| < 1, cardioid when |a/b| = 1
• Lemniscate: r² = a² cos(2θ) or r² = a² sin(2θ)
• Spiral of Archimedes: r = aθ, distance from pole increases with angle

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Complex Numbers in Polar Form

Part 5 of 7 — Complex Numbers in Polar Form

1. Complex number

z = a + bi plotted as (a, b) in complex plane

2. Polar (trigonometric) form

z = r(cos θ + i sin θ) where r = |z|

3. De Moivre's Theorem

zⁿ = rⁿ(cos(nθ) + i sin(nθ))

4. nth roots

n equally spaced roots on a circle of radius r^(1/n)

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Key Concepts Summary

• Complex number: z = a + bi plotted as (a, b) in complex plane
• Polar (trigonometric) form: z = r(cos θ + i sin θ) where r = |z|
• De Moivre's Theorem: zⁿ = rⁿ(cos(nθ) + i sin(nθ))
• nth roots: n equally spaced roots on a circle of radius r^(1/n)

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Problem-Solving Workshop

Part 6 of 7 — Problem-Solving Workshop

1. Complex number

z = a + bi plotted as (a, b) in complex plane

2. Polar (trigonometric) form

z = r(cos θ + i sin θ) where r = |z|

3. De Moivre's Theorem

zⁿ = rⁿ(cos(nθ) + i sin(nθ))

4. nth roots

n equally spaced roots on a circle of radius r^(1/n)

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Key Concepts Summary

• Complex number: z = a + bi plotted as (a, b) in complex plane
• Polar (trigonometric) form: z = r(cos θ + i sin θ) where r = |z|
• De Moivre's Theorem: zⁿ = rⁿ(cos(nθ) + i sin(nθ))
• nth roots: n equally spaced roots on a circle of radius r^(1/n)

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Review & Applications

Part 7 of 7 — Review & Applications

1. Complex number

z = a + bi plotted as (a, b) in complex plane

2. Polar (trigonometric) form

z = r(cos θ + i sin θ) where r = |z|

3. De Moivre's Theorem

zⁿ = rⁿ(cos(nθ) + i sin(nθ))

4. nth roots

n equally spaced roots on a circle of radius r^(1/n)

Check Your Understanding 🎯

Key Concepts Summary

• Complex number: z = a + bi plotted as (a, b) in complex plane
• Polar (trigonometric) form: z = r(cos θ + i sin θ) where r = |z|
• De Moivre's Theorem: zⁿ = rⁿ(cos(nθ) + i sin(nθ))
• nth roots: n equally spaced roots on a circle of radius r^(1/n)

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