Draft:Basic math formulas

The list of the simplest elementary mathematics formulas presented in a concise form.

Distributive property

 * $$\ a \cdot (b + c) = (a \cdot b) + (a \cdot c)$$

Sophie Germain's identity

 * $$(a\pm b)^2=a^2\pm 2ab+b^2$$
 * $$a^2-b^2=(a+b)(a-b)$$

Quadratic equation

 * $$ax^2 + bx + c = 0, \quad a\ne 0$$
 * $$D = b^2 - 4ac, \qquad x_{1,2} = \frac{-b \pm \sqrt{D}}{2a}$$


 * Vieta's formulas
 * $$a(x-x_1)(x-x_2) = 0, \qquad x_1 + x_2= \frac{-b}{a}, \qquad x_1 \cdot x_2 = \frac{c}{a}$$.

Exponentiation

 * $$a^{-n} =\frac{1}{a^n} \qquad\qquad\quad \sqrt[n]{a^m} = a^\frac{m}{n}$$
 * $$a^m \cdot a^n = a^{m+n}, \qquad (a^m)^n = a^{m\cdot n}$$
 * $$a^m : a^n = a^{m-n}, \qquad (a\cdot b)^n = a^n \cdot b^n$$

Logarithm

 * $$a^{\log_{a}b} = b \qquad \log_{a}a = 1 \qquad \log_{a}1 = 0$$
 * $$\log_{a} (m\cdot n) = \log_{a} m + \log_{a} n, \qquad \log_{a} (n^k) = k\cdot \log_{a} n$$
 * $$\log_{a} (m : n)=\log_{a} m - \log_{a} n \qquad \log_{a^k} (n) = \frac1k\cdot \log_{a} n$$

Arithmetic progression

 * $$a_n=a_1+d(n-1)$$
 * $$S_n=a_1 + \ldots + a_n ={a_1+a_n \over 2}n={2a_1 + d(n-1) \over 2}n$$

Geometric progression

 * $$b_n=b_1\cdot q^{n-1}$$
 * $$S_n=b_1 \frac{q^n - 1}{q-1}, \quad (q\ne 1)$$

Combinatorics

 * $$P_n=1\cdot 2 \cdot 3 \cdot \ldots \cdot n = n!

\qquad C_n^k = \frac{n!}{k!(n-k)!} \qquad A_n^k = \frac{n!}{(n-k)!}$$

Trigonometry

 * $$\sin \alpha = \cos(90^\circ - \alpha)$$
 * $$\operatorname{tg} \alpha = \operatorname{ctg}(90^\circ - \alpha)$$
 * $$\sin^2 \alpha + \cos^2 \alpha = 1$$
 * $$\operatorname{tg} \alpha = \frac {\sin \alpha} {\cos \alpha}$$
 * $$\operatorname{ctg} \alpha = \frac {\cos\alpha} {\sin \alpha}$$
 * $$\operatorname{tg}\alpha \cdot \operatorname{ctg}\alpha=1$$
 * $$1+\operatorname{tg}^2\alpha = \frac{1}{\cos^2\alpha}$$
 * $$1+\operatorname{ctg}^2\alpha = \frac{1}{\sin^2\alpha}$$
 * $$\sin2\alpha = 2\sin\alpha\cos\alpha$$
 * $$\cos2\alpha = \cos^2\alpha - \sin^2\alpha$$
 * $$\sin3\alpha = 3\sin\alpha - 4\sin^3\alpha$$
 * $$\cos3\alpha = 4\cos^3\alpha - 3\cos\alpha$$