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