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.

privacy policy

{ezoic-ad-1}

{ez_footer_ads}