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( Last Updated July 21th )

Pre-Calculus

Dimensions = Height x Length

Transpose: ( E^T )

[A, C] ====> [A, B]

[B, D] ====> [C, D]

[A, B] ====> [A]

=
> [B]

[A, D, G] =====> [A, B, C]

[B, E, H ]=====> [D, E, F]

[C, F, I ] =====>[ G, H, I]

[A]

[B ]====> [A, B, C]

[C]

[A ,C, E] ====> [A, B]

[B, D, F] ====> [C, D]

=
> [E, F]

[A, B ]=======> [A, C, E]

[C, D] =======>[ B, D, F]

[E, F]

Multiplying Matrices: ( X^Y)

[A, C] x [E] = [A * E + C * F] = [G]

[B, D] x [F] = [B * E + D * F] = [H]

Defined and Undefined: ( S * S )

[5, 6] = Yes

[3, 1]

[2] = No

[5]

Determinant: ( 2 * 2 )

[A, B] = ad - bc = [E]

[C, D]

Inverse: ( E^-1 ) ( 2 * 2 )

[A, B ]====> [D, -B]

[C, D ]====>[ -C, A]

=

[A, B] = ad - bc =[ F]

[C, D]

=

1F * D, -B ====> d/f, -b/f

>-C, A
> -c/f, a/f

Determinant: ( 3 * 3 )

[A, B, C]

[D, E, F] = a[ e, f ] - b[ d, f ] + c[ d, e ]

[G, H, I] = [ h, i ] [ g, i ] [ g , h ]

=

a(ei - fh) - b(di - fg) + c(dh - eg) => Simplify => One Number Variable

Imaginary units powers:

i^0 = 1

i^1 = -i

i^2 = -1

i^3 = -i

i^4 = 1

i^5 = i

i^6 = -1

i^7 = -i

i^8 = 1

If x is equal to a multiple of four then the answer is 1.

If x is one greater than a multiple of four then the answer is i.

If x is two greater than a multiple of four than the answer is -1.

If x is three greater than a multiple of four than the answer is -i.

Heron's Formula:

S = (a+b+c) / 2

A = √s(s-a)(s-b)(s-c)

√X = Y

Solving Triangles:

AB : CD = AD : DE

=

AB / BC = AD / DE = Value of x

Angle Bisector Theorem:

AB : BD = AC : CD

Degrees to Radians:

360 / 2 = 180

d * pi/180 = pi/180 radians = simplify

Unit Circle:

If positive number than the answer is the y value or middle value.

If positive pi number the answer is the x value.

1, 2

3, 4

1. If positive is y

2. If positive is x

3. If positive is x

4. If positive is y

Law of Sines Sin A/a = Sin B/b = Sin C/c

or it can be written in its reciprocal form,

a/Sin A = b/Sin B = c/Sin C



Law of Cosines Standard Form

a^2 = b^2 + c^2 - 2bccosA

b^2 = a^2 + c^2 - 2accosB

c^2 = a^2 + b^2 - 2abcosC

Alternative Form

CosA = (b^2 + c^2 - a^2)/(2bc)

CosB = (a^2 + c^2 - b^2)/(2ac)

CosC = (a^2 + b^2 - c^2)/(2ab)

Area of an Oblique Triangle To see when A is obtuse, substitute the reference angle 180 - A for A. Now the height of the triangle is given by

h = bsinA

Consequently, the area of each triangle is given by

Area = 1/2(base)(height)

Area = 1/2(c)(bsinA)

Area = 1/2(b)(c)sinA, 1/2(a)(b)sinC, 1/2(a)(c)sinB

Heron's Area Formula Given any triangle with side lengths a, b, and c, the area of the triangle is given by

Area = √s(s-a)(s-b)(s-c)

where s = (a + b + c)/(2)

Trigonometric Form of a Complex Number a = rcosθ and b = rsinθ

where r = √a^2 + b^2

Consequently, you have

a + bi = r(cosθ + i*sinθ)

tan θ = b/a

Product and Quotient of Two Complex Numbers z1z2 = r1r2[cos(θ1 + θ2) + i*sin(θ1 +θ2)] Product

z1/z2 = r1/r2[cos(θ1 - θ2) + i*sin(θ1 - θ2)] Quotient

Powers of Complex Numbers - DeMoivre's Theorem If z = r(cosθ + i*sinθ) is a complex number and n is a positive integer, then

z^n = [r(cosθ + i*sinθ)]^n

z^n = r^n(cosnθ + i*sinnθ)

Nth Root of a Complex Number For a positive integer n, the complex number z = r(cosθ + i*sinθ) has exactly n distinct nth roots given by

nth root of r[cos(θ + 2πk)/n + i*sin(θ + 2πk)/n]

where 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 sin^2 x+cos^2 x=1

tan^2 x+1= sec^2 x

1+cot^2 x=csc^2 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 sin3x = 3sinx - 4sin^3 x

cos 3x = 4cos^3 x - 3cosx

Sum and Difference Formulas sin (x + y) = sin x cos y + cos x sin y

sin (x- y) = sin x cos y - cos x sin y

cos (x + y) = cos x cos y - sin x sin y

cos (x - y) = cos x cos y + sin x sin y

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 sinx cos x

cos 2x = cos^2 x - sin^2 x

cos 2x = 1 - 2sin^2 x

cos 2x = 2cos^2 x -1

tan 2x = (2 tanx)/(1 - tan x)

* sin 4x = 2 sin 2x cos 2x

* sin 6x = 2 sin 3x cos 3x

Power-Reducing Formulas sin^2 x = (1 - cos 2x)/(2) cos^2 x = (1 + cos 2x)

tan^2 x = (1 - cos 2x)/(1 + cos 2x)

Half-Angle Formulas sin x/2 = + or - √(1 - cos x)/2

cos x/2 = + or - √1 + cos x)/2

tan x/2 = (1 - cos x)/(sin x) or (sin x)/(1 + cos x)

Product-to-Sum Formulas sinx sin y = 1/2 [cos(x - y) - cos(x + y)]

cosx cos y = 1/2 [cos(x - y) + cos(x + y)]

sinx 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 = -2 sin(x + y)/(2) sin(x - y)/(2)

The Fibonacci Sequence: A Recursive Sequence The Fibonacci Sequence is defined recursively as follows:

a to the zero = 1, a1 = 1, a to the k = a to the k - 2 + a to the k -1, where k is greater than or equal to 2

Factorial Notation If n is a positive integer, n factorial is defined as

n! = 1 2 3 4 * (n - 1) n

As a special case, zero factorial is defined as 0! = 1

2n! = 2(n!) = 2(1 2 3 *** n)

whereas (2n)! = 1 2 3 *** 2n

Summation Notation The sum of the first n terms of a sequence is represented by

n

Σ (ai) = a1 + a2 + a3 + a4 + a5 + an

i=1

where i is called the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation.

Series Consider the infinite sequence a1, a2, a3, ... ai, ...

The sum of the first n terms of the sequence is called a finite series or the partial sum of the sequence and is denoted by

n

a1 + a2 + a3 + ... + an = Σ (ai)

i = 1

Infinite Series The sum of all the terms of the infinite sequence is called an infinite series and is denoted by

∞

a1 + a2 + a3 + ... + ai + ... = Σ (ai)

i = 1

The nth Term of an Arithmetic Sequence The nth term of an arithmetic sequence has the form

an = dn + c

where d is the common difference between consecutive terms of the sequence and c = a1 - d

The Sum of a Finite Arithmetic Sequence The sum of a finite arithmetic sequence with n terms is given by

Sn = n/2 (a1 + an)

Geometric Sequences and Series A sequence is geometric if the ratios of consecutive terms are the same. So, the sequence a1, a2, a3, a4, ..., an, .... is geometric if there is a number r such that a2/a1 = a3/a2 = a4/a3 = ... r, r not equal to zero

The number r is the common ratio of the sequence.

a2 = a1 * r

The nth Term of a Geometric Sequence The nth term of a geometric sequence has the form

an = a1 * r^(n-1)

where r is the common ratio of consecutive terms of the sequence. So, every geometric sequence can be written in the following form.

a1, a2, a3, a4, a5, ... an, ...

a1, a1r, a1r^2, a1r^3, a1r^4, a1r^(n-1)

The Sum of a Finite Geometric Sequence The sum of the finite geometric sequence with common ratio r not equal to 1 is given by

n

Sn = Σ (a1r^(i - 1)) = a1 (1 - r^n/ 1 - r)

i = 1

The Sum of an Infinite Geometric Sequence If absolute value of r is less than 1, then the infinite geometric series a1 + a1r + a1r^2, + a1r^3 + ... +

has the sum

∞

S = Σ (a1r^i = (a1)/(1-r)

i = 0

Note that if the absolute value of r is greater than or equal to one, the series does not have a sum.

Increasing Annuity A = P(1 + r/n) ^ (nt)

Polar Coordinates Each point in the polar plane is assigned the point (r, θ).

Related to the rectangular coordinates,

x = r cos θ

y = r sin θ

and tan θ = y/x

r^2 = x^2 + y^2 What is your favorite subject? Business and Financial Math English History Art Science