## Pre-calculus ch. 6

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landinrosalie  on May 9, 2012

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# Pre-calculus ch. 6

 All polar coordinates(r, θ+2nπ) or (-r, θ+2nπ+π)
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All polar coordinates (r, θ+2nπ) or (-r, θ+2nπ+π)
Polar coordinates vs. rectangular coordinates x=rcosθ
y=rsinθ
r²=x²+y²
x-axis symmetry replace (r,θ) with (r,-θ) and (-r, π-θ)
y-axis symmetry replace (r,θ) with (-r,-θ) and (r, π-θ)
origin symmetry replace (r,θ) with (-r,θ) and (r, π+θ)
The rose curve r=acos(nθ) or r=asin(nθ)
Analyzing the rose curve Domain: (-∞, ∞)
Range: [-|a|, |a|]
Max r-value: |a|
# of petals: n (if n is odd) and 2n (if n is even)
Symmetry:
- n even, symmetric about the x,y, and origin
- n odd, r=acos(nθ) symmetric about x-axis
- n odd, r=asin(nθ) symmetric about y-axis
The limacon curve r=a±bcosθ or r=a±bsinθ
Analyzing the limacon curve Domain: (-∞,∞)
Range: [a-b, a+b]
Max r-value: a+b
Symmetry:
- r=a±bcosθ symmetric about the x
- r=a±bsinθ symmetric about the y
Inner loop a/b < 1
Cardioid a/b = 1
Dimpled Limacon 1 < a/b < 2
Convex Limacon a/b ≥ 2
The spiral graph of Archimedes r= ±θ
Analyzing the spiral graph of Archimedes Domain: (-∞,∞)
Range: (-∞,∞)
no max r-value
The lemniscate curve r²=a²sin2θ or r²=a²cos2θ
Analyzing the lemniscate curve Domain: cos2θ ≥ 0 or sin2θ ≥ 0
(MUST SOLVE INEQUALITIES)
Range: [-a, a]
symmetry:
r²=a²sin2θ is symmetric about y,x and origin
r²=a²cos2θ is symmetric about the origin
max r-value: a
Absolute value of a complex number |z|=|a+bi|=√(a²+b²)
Trigonometric form of a complex number z=a+bi turns into z=r(cosθ+isinθ)
multiplication of complex numbers z₁·z₂=r₁r₂[cos(θ₁+θ₂)+isin(θ₁+θ₂)]
division of complex numbers z₁/z₂=r₁/r₂[cos(θ₁-θ₂)+isin(θ₁-θ₂)]
De Moivre's Therorem zⁿ=[r(cosθ+isinθ)]ⁿ turns into rⁿ(cosnθ+isinnθ)
Nth roots of complex numbers ⁿ√r(cos[(θ+2πk)/n]+isin[(θ+2πk)/n])

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