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