Polar Coordinates - Complete Interactive Lesson
Part 1: Polar Coordinate System
📈 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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Part 2: Converting Coordinates
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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Part 3: Polar Graphs
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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Part 4: Rose Curves & Limacons
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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Part 5: Polar Equations
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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Part 6: Problem-Solving Workshop
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)
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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Part 7: Review & Applications
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)
Concept Check 🎯
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