User:Saul/algebra

Combinations and Permutations

Permutations/Pick

A permutation is a sequence where the order does matter.
This is often notated by n pick r where n is the data set - say 40 lotto balls and r is the selection count - say 8 balls.
The general formulae for permutations is:
P = n! / (n-k)!

Combinations/Choose

A combination is a sequence where the order does NOT matter.
This is often notated by n choose r - n and r mean the same thing as in pick.
The general formulae for combinations is:
C = n! / ((n-k)!k!)
Also note that:
n choose k is the same as n choose (n - k)

Binomial Theorem

(a + b)ⁿ = _k=0Σⁿ ( ( n choose k ) a^k b^n-k )

Approximation Using Binomial

Calculating a number to the power of another can be approximated by the following formulae:

(1 + x)ⁿ = 1 + nx + (n(n-1)x²)/2! + (n(n-1)(n-2)x³)/3! + (n(n-1)(n-2)(n-3)x⁴)/4! ...

Note: This is unending so the more steps you take the more accurate the number will be, however usually 3-4 steps is enough.

Trigonometry

Sin(30) = (√1) / 2 = Cos(60)
Sin(45) = (√2) / 2 = Cos(45)
Sin(60) = (√3) / 2 = Cos(30)

Various Formulae

Sin²(x) = Sin(x) * sin(x)
Sin(x²) = Sin(x * x)
Tan(x) = Sin(x) / Cos(x)
Sin²(x) + Cos²(x) = 1
Sin(x) = Cos(90 - x)
Cos(x) = Sin(90 - x)

A / Sin(a) = C / Sin(c) = C / Sin(c)
a² = b² + c² - 2bc * Cos (A)

Note:
± = ± → + = + → - = -
± = ∓; → + = - → - = +

Sin(a ± b) = Sin(a) * Cos(b) ± Cos(a) * Sin(b)
Sin(a ± b) = Cos(a) * Cos(b) ∓ Sin(a) * Sin(b)

Sin(2a) = 2Sin(a) * Cos(a)
Cos(2a) = Cos²(a) - Sin²(a)

Sin²(a) = ( 1 - Cos(2a) ) / 2
Cos²(a) = ( 1 + Cos(2a) ) / 2

Csc(x) = 1 / Sin(x)
Cot(x) = 1 / Tan(x) = Cos(x) / Sin(x)
Sec(x) = 1 / Cos(x)
Sec²(x) = 1 / Cos²(x)

Quadratics

Turning Point

The equation dor the turning point x value is: -b / 2a, after that y can be derived.

Complete the Square

The completed square looks like this:
y = a(x + dx)² + dy
a is the shape, a negative value indicates a n like shape where a positive value indicates a u like shape and if a is 0 the shape is a flat line.
dx is the negative displacement of x.
dy is the displacement of y.

Formulae

To convert a general quadratic to the completed square, use the following formulae:
y = ax² + bx + c → a(x + b / 2a)² - b² / 4a + c