Tables of Mathematical Formulas
1. Decimal Multipliers
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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 + 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 (-π/2 < x < π/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. Permutations 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 ]
More Tables of Formulas
Table of Derivatives.
Table of Integrals.
Table of Laplace Transforms.
Table of Fourier Transforms.