Tables of Mathematical Formulas

# Tables of Mathematical Formulas

## 1. Decimal Multipliers

 10 1 deka (da) 10 -1 deci (d) 10 2 hecto (h) 10 -2 centi (c) 10 3 kilo (k) 10 -3 milli (m) 10 6 mega (M) 10 -6 micro (u) 10 9 giga (G) 10 -9 nano (n) 10 12 tera (T) 10 -12 pico (p) 10 15 peta (P) 10 -15 femto (f) 10 18 exa (E) 10 -18 atto (a)

## 2. Series

### Maclaurin Series.

1.  e x = 1 + x + x 2 / 2! + ... + x n / n! + ... for all x
2.   sin x = x - x
3 / 3! + x 5 / 5! - x 7 / 7! + ... for all x
3.   cos x = 1 - x
2 / 2! + x 4 / 4! - x 6 / 6! + ... for all x
4.   ln(1 + x) = x - x
2 / 2 + x 3 / 3 -... + (-1) n+1 x n / n + ... for (-1 < x ≤ 1)
5.   tan x = x + (1/3) x
3 + (2/15) x 5 + (17/315) x 7 + ... for (-π/2 < x < π/2)
6.   arcsin x = x + (1/2) x
3 / 3 + (1.3/2.4) x 5 / 5 + (1.3.5/2.4.6) x 7 / 7 + ... for (-1 < x < 1)
7.   arctan x = x - x
3 / 3 + x 5 / 5 - ... for (-1 < x < 1)
8.   sinh x = x + x
3 / 3! + x 5 / 5! + x 7 / 7! + ... for all x
9.   cosh x = x + x
2 / 2! + x 4 / 4! + x 6 / 6! + ... for all x
10.   arcsinh x = x - (1/2) x
3 / 3 + (1.3/2.4) x 5 / 5 - (1.3.5/2.4.6) x 7 / 7 + ... for (-1 < x < 1)
11.   1 / (1 - x) = 1 + x + x
2 + x 3 + ... for (-1 < x < 1)

### Arithmetic Series.

12.   Sn = a + (a + d) + (a + 2d)+...+(a + [n -1] d) = (n / 2)[ first term + last term ] = (n / 2)[a + (a + [n - 1] d) = n (a + [n - 1] d)

### Geometric Series.

13.   Sn = a + a r + a r 2 + a r 3 +...+ a r n-1 = a (1 - r n )/(1 - r)

### Integer Series.

14.   1 + 2 + 3 + ... + n = (1 / 2) n (n + 1)
15.   1
2 + 2 2 + 3 2 + ... + n 2 = (1 / 6) n (n + 1)(2n + 1)
15.   1
3 + 2 3 + 3 3 + ... + n 3 = [ (1 / 2) n (n + 1) ] 2

## 3. Factorial, Permutations and Combinations.

1.   n factorial = n ! = n.(n - 1).(n - 2)...2.1
2.  Permutations of n objects taken r at the time:
n P r = n ! / [ (n - r) ! ]

3.  Combinations of n objects taken r at the time:
n C r = n ! / [ r ! (n - r) ! ]

## 4. Binomial Expansion (Formula).

1. If n is a positive integer, we can expand (x + y) n as follows
(x + y)
n = n C 0 x n + n C 1 x n - 1 y + n C 2 x n - 2 y 2 + ... + n C n y n
The general term n
C r is given by
n C r = n ! / [ r ! (n - r) ! ]

## 5. Trigonometric Formulas.

### Sum / Difference of Angles Formulas.

1.   cos(A + B) = cos A cos B - sin A sin B
2.   cos(A - B) = cos A cos B + sin A sin B
3.   sin(A + B) = sin A cos B + cos A sin B
4.   sin(A - B) = sin A cos B - cos A sin B
5.   tan(A + B) = [ tan A + tan B ] / [ 1 - tan A tan B]
6.   tan(A - B) = [ tan A - tan B ] / [ 1 + tan A tan B]

### Sum / Difference of Trigonometric Functions Formulas.

7.   sin A + sin B = 2 sin [ (A + B) / 2 ] cos [ (A - B) / 2 ]
8.   sin A - sin B = 2 cos [ (A + B) / 2 ] sin [ (A - B) / 2 ]
9.   cos A + cos B = 2 cos [ (A + B) / 2 ] cos [ (A - B) / 2 ]
10.   cos A - cos B = - 2 sin [ (A + B) / 2 ] sin [ (A - B) / 2 ]

### Product of Trigonometric Functions Formulas.

11.   2 sin A cos B = sin (A + B) + sin (A - B)
12.   2 cos A sin B = sin (A + B) - sin (A - B)
13.   2 cos A cos B = cos (A + B) + cos (A - B)
14.   2 sin A sin B = - cos (A + B) + cos (A - B)

### Multiple Angles Formulas.

15.   sin 2A = 2 sin A cos A
16.   cos 2A = cos
2 A - sin 2 A = 2 cos 2 A - 1 = 1 - 2 sin 2 A
17.   sin 3A = 3 sin A - 4 sin
3 A
18.   cos 3A = 4 cos
3 A - 3 cos A

### Power Reducing Formulas.

19.   sin 2 A = (1/2) [ 1 - cos 2A ]
19.   cos
2 A = (1/2) [ 1 + cos 2A ]

## More Tables of Formulas

Table of Derivatives.
Table of Integrals.
Table of Laplace Transforms.
Table of Fourier Transforms.