Questions and Answers
What does the Law of Sines state?
What is the standard form of the Law of Cosines?
a^2 = b^2 + c^2 - 2bc cos A
What is the formula for the area of an oblique triangle?
Area = 1/2(base)(height)
What does Heron's Area Formula calculate?
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What is the trigonometric form of a complex number?
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What does DeMoivre's Theorem state regarding powers of complex numbers?
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What is the formula for the nth root of a complex number?
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The reciprocal identities include sin x = 1/csc x.
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What is the Pythagorean identity for sin and cos?
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What does the Fibonacci Sequence define recursively?
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What is the formula for the sum of a finite arithmetic sequence?
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What is the standard form for the nth term of an arithmetic sequence?
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Which formula gives the sum of an infinite geometric sequence?
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The formula for polar coordinates includes x = r cos θ.
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Study Notes
Law of Sines
- Expresses the relationship between angles and sides in a triangle: Sin A/a = Sin B/b = Sin C/c.
- Can also be written in reciprocal form: a/Sin A = b/Sin B = c/Sin C.
Law of Cosines
- Standard Form: a² = b² + c² - 2bc cos A, b² = a² + c² - 2ac cos B, c² = a² + b² - 2ab cos C.
- Alternative Form: Cos A = (b² + c² - a²) / (2bc), Cos B = (a² + c² - b²) / (2ac), Cos C = (a² + b² - c²) / (2ab).
Area of an Oblique Triangle
- Height given by h = b sin A; area can be calculated using Area = 1/2(base)(height).
- Different forms of area: Area = 1/2(c)(bsinA), 1/2(a)(b)sinC, 1/2(a)(c)sinB.
Heron's Area Formula
- Area of a triangle determined by side lengths a, b, c: Area = √s(s-a)(s-b)(s-c) where s = (a + b + c) / 2.
Trigonometric Form of a Complex Number
- A complex number can be expressed as a = r cos θ and b = r sin θ, with r = √(a² + b²).
- Written as a + bi = r(cos θ + i sin θ) and tan θ = b/a.
Product and Quotient of Two Complex Numbers
- Product: z1 z2 = r1 r2 [cos(θ1 + θ2) + i sin(θ1 + θ2)].
- Quotient: z1 / z2 = r1 / r2 [cos(θ1 - θ2) + i sin(θ1 - θ2)].
Powers of Complex Numbers - DeMoivre's Theorem
- For a complex number z = r(cos θ + i sin θ) and integer n, z^n = r^n(cos(nθ) + i sin(nθ)).
Nth Root of a Complex Number
- The complex number z = r(cos θ + i sin θ) has n distinct nth roots: nth root of r[cos(θ + 2πk/n) + i sin(θ + 2πk/n)], with k = 0, 1, 2,..., n - 1.
Reciprocal Identities
- sin x = 1/csc x; csc x = 1/sin x.
- cos x = 1/sec x; sec x = 1/cos x.
- tan x = 1/cot x; cot x = 1/tan x.
Quotient Identities
- tan x = sin x / cos x; cot x = cos x / sin x.
Pythagorean Identities
- sin² x + cos² x = 1.
- tan² x + 1 = sec² x.
- 1 + cot² x = csc² x.
Cofunction Identities
- sin (π/2 - x) = cos x; cos (π/2 - x) = sin x.
- tan (π/2 - x) = cot x; cot (π/2 - x) = tan x.
- sec (π/2 - x) = csc x; csc (π/2 - x) = sec x.
Even/Odd Identities
- sin (-x) = -sin x; csc (-x) = -csc x.
- cos (-x) = cos x; sec (-x) = sec x.
- tan (-x) = -tan x; cot (-x) = -cot x.
Triple Angle Formulas
- sin(3x) = 3sin x - 4sin³ x; cos(3x) = 4cos³ x - 3cos x.
Sum and Difference Formulas
- For sin: sin(x + y) = sin x cos y + cos x sin y; sin(x - y) = sin x cos y - cos x sin y.
- For cos: cos(x + y) = cos x cos y - sin x sin y; cos(x - y) = cos x cos y + sin x sin y.
- For tan: tan(x + y) = (tan x + tan y) / (1 - tan x tan y); tan(x - y) = (tan x - tan y) / (1 + tan x tan y).
Multiple Angle and Formulas
- sin(2x) = 2 sin x cos x; cos(2x) has three forms: cos² x - sin² x, 1 - 2sin² x, 2cos² x - 1.
- tan(2x) = (2 tan x) / (1 - tan² x).
Power-Reducing Formulas
- sin² x = (1 - cos(2x)) / 2; cos² x = (1 + cos(2x)) / 2.
- tan² x = (1 - cos(2x)) / (1 + cos(2x)).
Half-Angle Formulas
- sin(x/2) = ±√(1 - cos x) / 2; cos(x/2) = ±√(1 + cos x) / 2.
- tan(x/2) = (1 - cos x) / sin x or sin x / (1 + cos x).
Product-to-Sum Formulas
- sin x sin y = 1/2 [cos(x - y) - cos(x + y)].
- cos x cos y = 1/2 [cos(x - y) + cos(x + y)].
- sin x cos y = 1/2 [sin(x + y) + sin(x - y)]; cos x sin y = 1/2 [sin(x + y) - sin(x - y)].
Sum-to-Product Formulas
- sin x + sin y = 2sin((x + y)/2)cos((x - y)/2); sin x - sin y = 2cos((x + y)/2)sin((x - y)/2).
- cos x + cos y = 2cos((x + y)/2)cos((x - y)/2); cos x - cos y = -2sin((x + y)/2)sin((x - y)/2).
The Fibonacci Sequence: A Recursive Sequence
- Defined recursively: a₀ = 1, a₁ = 1, aₖ = aₖ₋₂ + aₖ₋₁ for k ≥ 2.
Factorial Notation
- Factorial of a positive integer n: n! = 1 × 2 × 3 × ... × n.
- Special case: 0! = 1.
Summation Notation
- Sum of the first n terms: Σ (aᵢ) = a₁ + a₂ + a₃ + ... + aₙ, with i as the index of summation.
Series
- Sum of the first n terms of a sequence called a finite series or partial sum: Sₙ = a₁ + a₂ + ... + aₙ = Σ (aᵢ).
Infinite Series
- Sum of all terms in an infinite sequence: Σ (aᵢ) = a₁ + a₂ + a₃ + ... , represented as an infinite series.
The nth Term of an Arithmetic Sequence
- Has the form: aₙ = d n + c, where d is the common difference, c = a₁ - d.
The Sum of a Finite Arithmetic Sequence
- Given n terms, the sum is: Sₙ = n/2 (a₁ + aₙ).
Geometric Sequences and Series
- A sequence is geometric if ratios of consecutive terms are constant. The common ratio r: a₂ / a₁ = a₃ / a₂ = ... .
The nth Term of a Geometric Sequence
- Given by: aₙ = a₁ * r^(n-1), with r as the common ratio.
The Sum of a Finite Geometric Sequence
- With common ratio r ≠ 1: Sₙ = Σ (a₁ r^(i - 1)) = a₁ (1 - rⁿ) / (1 - r).
The Sum of an Infinite Geometric Sequence
- If |r| < 1, the sum S = a₁ / (1 - r); if |r| ≥ 1, the series diverges.
Increasing Annuity
- A = P(1 + r/n)^(nt).
Polar Coordinates
- Each point represented as (r, θ) in the polar plane: x = r cos θ, y = r sin θ.
- Relationship: tan θ = y/x and r² = x² + y².
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