# 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 (-Pi/2 < x < Pi/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.      Permuatations 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 ]