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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.

$e^{x} = 1 + x + \dfrac{x^{2}}{2!} + ... + \dfrac{x^{n}}{n!} + ...$ 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.

$n \text{ factorial} = n ! = n.(n - 1).(n - 2)...2.1$
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.

$\cos(A + B) = \cos A \cos B - \sin A \sin B$
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.

$\sin A + \sin B = 2 \sin [ (A + B) / 2 ] \cos [ (A - B) / 2 ]$
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.