Difference between revisions of "User:Saul/algebra"
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== Trigonometry == | == Trigonometry == | ||
| + | '''Sin(30) = (√1) / 2 = Cos(60)'''<br> | ||
| + | '''Sin(45) = (√2) / 2 = Cos(45)'''<br> | ||
| + | '''Sin(60) = (√3) / 2 = Cos(30)'''<br> | ||
=== Various Formulae === | === Various Formulae === | ||
| − | ''' | + | '''Sin<sup>2</sup>(x) = Sin(x) * sin(x)'''<br> |
| − | ''' | + | '''Sin(x<sup>2</sup>) = Sin(x * x)'''<br> |
| − | ''' | + | '''Tan(x) = Sin(x) / Cos(x)'''<br> |
| − | ''' | + | '''Sin<sup>2</sup>(x) + Cos<sup>2</sup>(x) = 1'''<br> |
| − | ''' | + | '''Sin(x) = Cos(90 - x)'''<br> |
| + | '''Cos(x) = Sin(90 - x)'''<br> | ||
<br> | <br> | ||
| − | '''A / | + | '''A / Sin(a) = C / Sin(c) = C / Sin(c)'''<br> |
| − | '''a<sup>2</sup> = b<sup>2</sup> + c<sup>2</sup> - 2bc * | + | '''a<sup>2</sup> = b<sup>2</sup> + c<sup>2</sup> - 2bc * Cos (A)'''<br> |
<br> | <br> | ||
Note:<br> | Note:<br> | ||
| Line 38: | Line 42: | ||
'''± = ∓; → + = - → - = +'''<br> | '''± = ∓; → + = - → - = +'''<br> | ||
<br> | <br> | ||
| − | ''' | + | '''Sin(a ± b) = Sin(a) * Cos(b) ± Cos(a) * Sin(b)'''<br> |
| − | ''' | + | '''Sin(a ± b) = Cos(a) * Cos(b) ∓ Sin(a) * Sin(b)'''<br> |
<br> | <br> | ||
| − | ''' | + | '''Sin(2a) = 2Sin(a) * Cos(a)'''<br> |
| − | ''' | + | '''Cos(2a) = Cos<sup>2</sup>(a) - Sin<sup>2</sup>(a)'''<br> |
<br> | <br> | ||
| − | ''' | + | '''Sin<sup>2</sup>(a) = ( 1 - Cos(2a) ) / 2'''<br> |
| − | ''' | + | '''Cos<sup>2</sup>(a) = ( 1 + Cos(2a) ) / 2'''<br> |
| + | <br> | ||
| + | '''Csc(x) = 1 / Sin(x)'''<br> | ||
| + | '''Cot(x) = 1 / Tan(x) = Cos(x) / Sin(x)'''<br> | ||
| + | '''Sec(x) = 1 / Cos(x)'''<br> | ||
| + | '''Sec<sup>2</sup>(x) = 1 / Cos<sup>2</sup>(x)'''<br> | ||
| + | |||
| + | == 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:<br> | ||
| + | '''y = a(x + dx)<sup>2</sup> + dy'''<br> | ||
| + | '''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.<br> | ||
| + | '''dx''' is the '''negative''' displacement of '''x'''.<br> | ||
| + | '''dy''' is the displacement of '''y'''. | ||
| + | |||
| + | ==== Formulae ==== | ||
| + | To convert a general quadratic to the completed square, use the following formulae:<br> | ||
| + | '''y = ax<sup>2</sup> + bx + c → a(x + b / 2a)<sup>2</sup> - b<sup>2</sup> / 4a + c''' | ||
Latest revision as of 01:32, 17 March 2020
Contents
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 ) ak bn-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
