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40 Cards in this Set
- Front
- Back
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Lower sin²x
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sin²x = (1 - cos2x ) / 2
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What is the domain and range of sin-1(x)?
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D:[-1,1]
Range:[-π/2,π/2] |
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Lower cos²x
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cos²x = (1 + cos2x) / 2
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To change from POLAR to RECTANGULAR coordinates, use these formulas.
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x=rcosθ
y=rsinθ |
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Lower tan²x
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tan²x = (1 - cos2x) / (1 + cos2x)
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What is the domain and range of cos-1(x)?
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D:[-1,1]
Range:[0,π] |
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What does sin(u/2) equal?
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+ or - √( (1 - cosu) / 2 )
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A complex number z=a+bi has the polar form (or trigonometric form)...
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z=r(cosθ+sinθ*i)
where r=|z|=√(a²+b²) and tanθ=b/a The number r is the modulus of z and θ is an argument of z. |
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What does cos(u/2) equal?
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+ or - √( (1+cosu) / 2 )
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What is the domain and range of tan-1(x)?
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D: All Reals
Range:(-π/2,π/2) |
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What does tan(u/2) equal?
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(1 - cosu) / sinu
or sinu / (1 + cosu) <-- used more... |
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To change from RECTANGULAR to POLAR coordinates, use these formulas.
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r²=x²+y²
tanθ=y/x (x≠0) |
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Simplify sin2x
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2sinxcosx
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What is the domain and range of cot-1(x)?
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D: All Reals
Range: (0,π) |
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Simplify cos2x
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cos²x - sin²x
or 1-2sin²x or 2cos²x-1 |
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If two complex numbers z₁ and z₂ have the polar forms...
z₁ = r₁(cosθ₁+sinθ₁*i) and z₂ = r₂(cosθ₂+sinθ₂*i) then what are their forms when multiplied together or divided? |
z₁z₂=r₁r₂[cos(θ₁+θ₂)+i*sin(θ₁+θ₂)]
z₁/z₂=r₁/r₂*[cos(θ₁-θ₂)+i*sin(θ₁-θ₂)] |
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Simplify tan2x
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2tanx / (1 - tan²x)
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What is the domain and range of csc-1(x)?
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D: (-∞,-1] [1,∞)
Range: (0,π/2] (π, 3π/2] |
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What does Asinx + Bcosx equal if A and B are real numbers?
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ksin(x+w)
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What is DeMoivre's theorem?
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z^n=(r^n)*(cos nθ + isin nθ)
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What does k equal?
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√(A²+B²)
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What is the domain and range of sec-1(x)?
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D: (-∞,-1] [1,∞)
Range: [0,π/2) [π,3π/2) |
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What two requirements does w satisfy?
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cosw = A / √(A²+B²)
and sinw = B / √(A²+B²) |
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How do you find the nth root of complex numbers?
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for k = 0, 1, 2... n-1. For n distinct nth roots
(r^(1/n))*[cos((θ+2kπ)/n)+isin((θ+2kπ)/n] |
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What are the addition and subtraction formulas for sine?
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Sin(s+t)=sin(s)cos(t)+cos(s)sin(t)
Sin(s-t)=sin(s)cos(t)-cos(s)sin(t) |
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What are the addition and subtraction formulas for cosine?
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Cos(s+t)=cos(s)cos(t)-sin(s)sin(t)
Cos(s-t)=cos(s)cos(t)+sin(s)sin(t) |
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What are the addition and subtraction formulas for tangent?
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Tan(s+t)=tan(s)+tan(t) / 1-tan(s)tan(t)
Tan(s-t)=tan(s)-tan(t) / 1+ tan(s)tan(t) |
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What are the Pythagorean identities?
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cos²θ+sin²θ=1
tan²θ+1=sec²θ cot²θ+1=csc²θ |
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What is the Law of Sines?
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sin(A) / a = sin(B) / b = sin(C) / c
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What is the Law of Cosines?
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a² = b² + c² - 2bc cos A
b² = a² + c² - 2ac cos B c² = a² + b² - 2ab cos C |
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What is Heron's formula?
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A = √(s[s-a][s-b][s-c]) where s = 1/2(a+b+c) a.k.a half the perimeter (semiperimeter)
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What is the definition of dot product?
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If u = <a1,b1> and v = <a2,b2> are vectors, then their dot product, denoted by u*v, is defined by
u*v = a1a2 + b1b2 |
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What is the Dot Product Theorem / Angle between Two Vectors?
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cosθ = u*v / |u||v|
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What are orthogonal vectors?
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Two nonzero vectors u and v are perpendicular if and only if u*v=0
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What is the form of an arithmetic sequence?
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a+(n-1)d
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What are the formulas for the partial sum of an arithmetic sequence?
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(n/2)[2a+(n-1)d]
and n( (a+a sub n)/2 ) |
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What is the form of a geometric sequence?
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ar^(n-1)
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What is the formula for the partial sum of a geometric sequence?
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a*( (1-r^n) / (1-r) )
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What is the formula for the sum of an infinite geometric series?
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a / (1-r)
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What does (n r) stand for? (Binomial expansion...) When written it looks like n above r with parnetheses on the sides.
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(n r) = n! / ( r!(n-r)! )
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