Pre-Calculus Formulas Flashcards

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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Test your knowledge of essential pre-calculus formulas with these flashcards. Each card covers key concepts such as the Law of Sines, Law of Cosines, and more. Perfect for students looking to reinforce their understanding before exams.

Pre-Calculus Formulas Flashcards

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Questions and Answers

What does the Law of Sines state?

What is the standard form of the Law of Cosines?

What is the formula for the area of an oblique triangle?

What does Heron's Area Formula calculate?

What is the trigonometric form of a complex number?

What does DeMoivre's Theorem state regarding powers of complex numbers?

What is the formula for the nth root of a complex number?

The reciprocal identities include sin x = 1/csc x.

What is the Pythagorean identity for sin and cos?

What does the Fibonacci Sequence define recursively?

What is the formula for the sum of a finite arithmetic sequence?

What is the standard form for the nth term of an arithmetic sequence?

Which formula gives the sum of an infinite geometric sequence?

The formula for polar coordinates includes x = r cos θ.