Tables of Mathematical Formulas

Reference Tables: Decimal Multipliers, Series, Factorials, Binomial Expansions, and Trigonometric Identities

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

Arithmetic Series

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

Geometric Series

  1. \( S_{n} = a + ar + ar^{2} + ar^{3} + \dots + ar^{n-1} = a \dfrac{1 - r^{n}}{1 - r} \)

Integer Series

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

3. Factorial, Permutations and Combinations

  1. \( n \text{ factorial} = n! = n \cdot (n - 1) \cdot (n - 2) \cdot \dots \cdot 2 \cdot 1 \)
  2. Permutations of \( n \) objects taken \( r \) at a time:
    \( nP_r = \dfrac{n!}{(n - r)!} \)
  3. Combinations of \( n \) objects taken \( r \) at a time:
    \( nC_r = \dfrac{n!}{r!(n - r)!} \)

4. Binomial Expansion (Formula)

  1. If \( n \) is a positive integer, we can expand \( (x + y)^{n} \) as follows: \[ (x + y)^{n} = \binom{n}{0} x^{n} + \binom{n}{1} x^{n - 1} y + \binom{n}{2} x^{n - 2} y^{2} + \dots + \binom{n}{n} y^{n} \] The general binomial coefficient \( \binom{n}{r} \) 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

  1. \( \sin A + \sin B = 2 \sin\left(\dfrac{A + B}{2}\right) \cos\left(\dfrac{A - B}{2}\right) \)
  2. \( \sin A - \sin B = 2 \cos\left(\dfrac{A + B}{2}\right) \sin\left(\dfrac{A - B}{2}\right) \)
  3. \( \cos A + \cos B = 2 \cos\left(\dfrac{A + B}{2}\right) \cos\left(\dfrac{A - B}{2}\right) \)
  4. \( \cos A - \cos B = -2 \sin\left(\dfrac{A + B}{2}\right) \sin\left(\dfrac{A - B}{2}\right) \)

Product of Trigonometric Functions Formulas

  1. \( 2 \sin A \cos B = \sin(A + B) + \sin(A - B) \)
  2. \( 2 \cos A \sin B = \sin(A + B) - \sin(A - B) \)
  3. \( 2 \cos A \cos B = \cos(A + B) + \cos(A - B) \)
  4. \( 2 \sin A \sin B = -\cos(A + B) + \cos(A - B) \)

Multiple Angles Formulas

  1. \( \sin 2A = 2 \sin A \cos A \)
  2. \( \cos 2A = \cos^{2} A - \sin^{2} A = 2 \cos^{2} A - 1 = 1 - 2 \sin^{2} A \)
  3. \( \sin 3A = 3 \sin A - 4 \sin^{3} A \)
  4. \( \cos 3A = 4 \cos^{3} A - 3 \cos A \)

Power Reducing Formulas

  1. \( \sin^{2} A = \dfrac{1}{2} [1 - \cos 2A] \)
  2. \( \cos^{2} A = \dfrac{1}{2} [1 + \cos 2A] \)

More Tables of Formulas