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

$ax^{2}+bx+c=0,\quad a\neq 0$

$D=b^{2}-4ac,\qquad x_{1,2}={\frac {-b\pm {\sqrt {D}}}{2a}}$

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

$a^{\log _{a}b}=b\qquad \log _{a}a=1\qquad \log _{a}1=0$

$\log _{a}(m\cdot n)=\log _{a}m+\log _{a}n,\qquad \log _{a}(n^{k})=k\cdot \log _{a}n$

$\log _{a}(m:n)=\log _{a}m-\log _{a}n\qquad \log _{a^{k}}(n)={\frac {1}{k}}\cdot \log _{a}n$

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

$b_{n}=b_{1}\cdot q^{n-1}$

$S_{n}=b_{1}{\frac {q^{n}-1}{q-1}},\quad (q\neq 1)$

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

Main article: List of trigonometric identities

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

See also

• Binomial theorem
• Elementary arithmetic
• Long division
• Differentiation rules
• Lists of integrals