# 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.  $$e^{x} = 1 + x + \dfrac{x^{2}}{2!} + ... + \dfrac{x^{n}}{n!} + ...$$ for all $$x$$
2.  $$\sin x = x - \dfrac{x^{3}}{3!} + \dfrac{x^{5}}{5!} - \dfrac{x^{7}}{7!} + ...$$ for all $$x$$
3.  $$\cos x = 1 - \dfrac{x^{2}}{2!} + \dfrac{x^{4}}{4!} - \dfrac{x^{6}}{6!} + ...$$ for all $$x$$
4.  $$\ln(1 + x) = x - \dfrac{x^{2}}{2} + \dfrac{x^{3}}{3} -... + (-1)^{n+1} \dfrac{x^{n}}{n} + ...$$ for $$(-1 < x \leq 1)$$
5.  $$\tan x = x + \dfrac{1}{3} x^{3} + \dfrac{2}{15} x^{5} + \dfrac{17}{315} x^{7} + ...$$ for $$(- \dfrac{\pi}{2} < x < \dfrac{\pi}{2})$$
6.  $$\arcsin x = x + \dfrac{1}{2} \dfrac{x^{3}}{3} + \dfrac{1 \cdot 3}{2 \cdot 4} \dfrac{x^{5}}{5} + \dfrac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \dfrac{x^{7}}{7} + ...$$ for $$(-1 < x < 1)$$
7.  $$\arctan x = x - \dfrac{x^{3}}{3} + \dfrac{x^{5}}{5} - ...$$ for $$(-1 < x < 1)$$
8.  $$\sinh x = x + \dfrac{x^{3}}{3!} + \dfrac{x^{5}}{5!} + \dfrac{x^{7}}{7!} + ...$$ for all $$x$$
9.  $$\cosh x = x + \dfrac{x^{2}}{2!} + \dfrac{x^{4}}{4!} + \dfrac{x^{6}}{6!} + ...$$ for all $$x$$
10.  $$\text{arcsinh } x = x - \dfrac{1}{2} \dfrac{x^{3}}{3} + \dfrac{1 \cdot 3}{2 \cdot 4} \dfrac{x^{5}} {5} - \dfrac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \dfrac{x^{7}}{7} + ...$$ for $$(-1 < x < 1)$$
11.  $$\dfrac{1}{1 - x} = 1 + x + x^{2} + x^{3} + ...$$ for $$(-1 < x < 1)$$

### Arithmetic Series.

12.  $$S_{n} = a + (a + d) + (a + 2d)+...+(a + [n -1] d) \\ = \dfrac{n}{2}[ \text{first term} + \text{last term} ] \\ = \dfrac{n}{2}[a + (a + [n - 1] d)] = n (a + [n - 1] d)$$

### Geometric Series.

13.  $$S_{n} = a + a r + a r^{2} + a r^{3} +...+ a r^{n-1} = a \dfrac{1 - r^{n}}{1 - r}$$

### Integer Series.

14.  $$1 + 2 + 3 + ... + n = \dfrac{1}{2} n (n + 1)$$
15.  $$1^{2} + 2^{2} + 3^{2} + ... + n^{2} = \dfrac{1}{6} n (n + 1)(2n + 1)$$
15.  $$1^{3} + 2^{3} + 3^{3} + ... + n^{3} = \left( \dfrac{1}{2} n (n + 1) \right)^{2}$$

## 3. Factorial, Permutations and Combinations.

1.  $$n \text{ factorial} = n ! = n.(n - 1).(n - 2)...2.1$$
2.  Permutations of $$n$$ objects taken $$r$$ at the time:
$$n \, ^{P} \, r = \dfrac{n !}{(n - r) !}$$

3.  Combinations of $$n$$ objects taken $$r$$ at the time:
$$n \, ^{C} \, r = \dfrac{n !}{r ! (n - r) !}$$

## 4. Binomial Expansion (Formula).

1. If $$n$$ is a positive integer, we can expand $$(x + y)^{n}$$ as follows
$$(x + y)^{n} = \binom{n}{0} x^{n} + \binom{n}{1} x^{n - 1} y + \binom{n}{2} x^{n - 2} y^{2} + ... + \binom{n}{n} y^{n}$$
The general term $$\binom{n}{r}$$ is given by
$$\binom{n}{r} = \dfrac{n !}{r ! (n - r) !}$$

## 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.  $$\tan(A + B) = \dfrac{\tan A + \tan B}{1 - \tan A \tan B}$$
6.  $$\tan(A - B) = \dfrac{\tan A - \tan B}{1 + \tan A \tan B}$$

### 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.  $$\sin^{2} A = \dfrac{1}{2} [ 1 - \cos 2A ]$$
19.  $$\cos^{2} A = \dfrac{1}{2} [ 1 + \cos 2A ]$$

## More Tables of Formulas

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