Pre-calculus ch. 6
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Created by:
landinrosalie on May 9, 2012
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23 terms
English | Math / Symbols |
|---|---|
 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: (-∞,∞) Symmetry about the y 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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