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3.14 · Polar Coordinates — What are they?
What do (r, θ) mean in polar coordinates?
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3.14 · Convert polar → rectangular
Convert (r, θ) to rectangular (x, y).
x = ? y = ?
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3.14 · Convert rectangular → polar
Convert (x, y) to polar (r, θ).
r = ? tanθ = ?
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3.14 · Negative r — what does it mean?
In polar, what does r < 0 mean geometrically?
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3.14 · Plot: A(2, π/3)
Plot the polar point A(2, π/3). Which quadrant?
π/3 = 60°
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3.14 · Plot: B(1, 3π/2)
Plot polar point B(1, 3π/2). Which axis?
3π/2 = 270°
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3.14 · Plot: C(3, 4π/3)
Plot polar point C(3, 4π/3). Which quadrant?
4π/3 = 240°
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3.14 · Plot: D(−2, π/4)
Plot polar point D(−2, π/4). How does negative r shift the point?
Opposite direction
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✓ Convert polar → rectangular
x = r cosθ · y = r sinθ
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✓ Polar Coordinates — What are they?
r = distance from the origin (can be negative), θ = angle measured counterclockwise from the positive x-axis. The point is located at angle θ, distance r from the pole.
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✓ Negative r — what does it mean?
r < 0 means the point is in the OPPOSITE direction of θ. Go distance |r| in the direction θ + π (180° away).
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✓ Convert rectangular → polar
r = √(x²+y²) · tanθ = y/x (pick quadrant from context)
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✓ Plot: B(1, 3π/2)
3π/2 = 270° → points straight DOWN the negative y-axis. Coordinates: (0, −1)
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✓ Plot: A(2, π/3)
r = 2 (positive), θ = 60° → Q1. Coordinates: (2cos60°, 2sin60°) = (1, √3)
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✓ Plot: D(−2, π/4)
Negative r: go opposite direction from π/4. Equivalent to (2, π/4+π) = (2, 5π/4) → Q3
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✓ Plot: C(3, 4π/3)
4π/3 = 240° → Q3. Coordinates: (3cos240°, 3sin240°) = (−3/2, −3√3/2)
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3.14 · Multiple representations
Name two other equivalent ways to write the polar point (2, π/3).
Add or subtract 2π, or negate r and add π
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3.14 · Convert (−1, 3π/2) to positive r
Convert (−1, 3π/2) to an equivalent form with positive r.
Negate r, add π to θ
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3.14 · r = a (circle)
What shape does r = a trace? Describe it.
a is a constant
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3.14 · r = 2cosθ — shape?
What shape does r = 2cosθ trace? State its center and radius.
Multiply both sides by r
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3.14 · r = 2sinθ — shape?
What shape does r = 2sinθ trace? State its center and radius.
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3.14 · r = a + bsinθ (limaçon)
What shape is r = a + bsinθ? When does it have an inner loop?
When |b| > |a|
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3.14 · r = a + acosθ (cardioid)
What shape is r = a + acosθ? What makes it a cardioid vs limaçon?
When |a| = |b|
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3.14 · r = acos(nθ) — rose curve
r = 4cos(3θ). How many petals? What is the max r?
n odd → n petals; n even → 2n petals
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✓ Convert (−1, 3π/2) to positive r
(−1, 3π/2) → r = +1, θ = 3π/2 + π = 5π/2 = π/2. So: (1, π/2) ✓
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✓ Multiple representations
(2, π/3+2π) = (2, 7π/3) and (−2, π/3+π) = (−2, 4π/3). Any coterminal angle or r negation works.
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✓ r = 2cosθ — shape?
Circle. Multiply by r: r² = 2r·cosθ → x²+y² = 2x → (x−1)²+y² = 1. Center (1, 0), radius 1.
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✓ r = a (circle)
r = a traces a CIRCLE centered at the origin with radius |a|. Every point is exactly distance a from the pole.
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✓ r = a + bsinθ (limaçon)
Limaçon. Has an inner loop when |b| > |a|. Dimpled when |a| < |b| < 2|a|. Convex when |a| ≥ 2|b|.
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✓ r = 2sinθ — shape?
Circle. r² = 2r·sinθ → x²+y² = 2y → x²+(y−1)² = 1. Center (0, 1), radius 1.
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✓ r = acos(nθ) — rose curve
r = 4cos(3θ): n = 3 (ODD) → 3 petals. Max r = |a| = 4. Petals are 4 units long.
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✓ r = a + acosθ (cardioid)
Cardioid — heart-shaped curve. When |a| = |b| exactly. No inner loop. Passes through origin once.
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3.14 · r = asin(nθ) — rose curve
r = 3sin(2θ). How many petals?
n = 2 (even)
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3.14 · Trace r = 2cos(θ) petal
For r = 4cos(3θ), on [0, 2π], describe how one petal is traced.
cos(3θ) = 0 when 3θ = π/2
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3.14 · Polar graph — when is r = 0?
For r = 1 + 2sinθ, find the values of θ where r = 0 on [0, 2π].
Set 1 + 2sinθ = 0
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3.14 · Negative r region
For r = 1 + 2sinθ, on what interval is r negative?
Where sinθ < −1/2
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3.14 · Origin — when does curve pass through?
Under what condition does a polar curve pass through the origin?
r = 0 for some θ
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3.14 · Polar table of values
For r = 2 + cosθ, make a table for θ = 0, π/2, π, 3π/2, 2π. State max and min r.
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3.14 · Connect polar to rectangular
For r = 4cos(θ − π), what is the equivalent rectangular equation?
Use r² = x²+y² and x = rcosθ
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3.14 · Key formula: r² = x² + y²
State the four key conversion formulas between polar and rectangular.
r², x, y, tanθ
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✓ Trace r = 2cos(θ) petal
One petal traced on [0, π/3]: starts r=4 (θ=0), shrinks to 0 (θ=π/6), goes negative (inner), back to 0 (θ=π/3). Full petal from −π/6 to π/6.
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✓ r = asin(nθ) — rose curve
r = 3sin(2θ): n = 2 (EVEN) → 2n = 4 petals. Max r = 3.
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✓ Negative r region
sinθ < −1/2 → θ ∈ (7π/6, 11π/6). r is negative in this interval — the curve is traced in the opposite direction.
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✓ Polar graph — when is r = 0?
1 + 2sinθ = 0 → sinθ = −1/2 → θ = 7π/6 and θ = 11π/6. The curve passes through the origin at these angles.
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✓ Polar table of values
θ=0: r=3 (max) · θ=π/2: r=2 · θ=π: r=1 (min) · θ=3π/2: r=2 · θ=2π: r=3. Max=3, Min=1.
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✓ Origin — when does curve pass through?
A polar curve passes through the origin when r = 0 for some value of θ in its domain.
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✓ Key formula: r² = x² + y²
r² = x²+y² · x = rcosθ · y = rsinθ · tanθ = y/x
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✓ Connect polar to rectangular
r = 4cos(θ−π). Expand: r = 4(cosθcosπ+sinθsinπ) = −4cosθ. Then r² = −4rcosθ → x²+y²+4x=0 → (x+2)²+y²=4. Circle center (−2,0) r=2.