### Teorija

 Saīsinātās reizināšanas formulas$\begin{array}{l}{\left(a±b\right)}^{3}={a}^{3}±3{a}^{2}b+3a{b}^{2}±{b}^{3}\\ {a}^{3}+{b}^{3}=\left(a+b\right)\left({a}^{2}-\mathit{ab}+{b}^{2}\right)\\ {a}^{3}-{b}^{3}=\left(a-b\right)\left({a}^{2}+\mathit{ab}+{b}^{2}\right)\end{array}$________________________________Kvadrāttrinoms$a{x}^{2}+\mathit{bx}+c=a\left(x-{x}_{1}\right)\left(x-{x}_{2}\right)$ Kvadrātvienādojums$\begin{array}{l}a{x}^{2}+\mathit{bx}+c=0\\ a\ne 0\\ \left\{\begin{array}{l}{x}_{1}+{x}_{2}=-\frac{b}{a}\\ {x}_{1}\cdot {x}_{2}=\frac{c}{a}\end{array}\right\\end{array}$ Modulis$\begin{array}{l}\left|a\right|=\left\{\begin{array}{l}a,\phantom{\rule{0.147em}{0ex}}\mathrm{ja}\phantom{\rule{0.147em}{0ex}}a\ge 0\\ -a,\phantom{\rule{0.147em}{0ex}}\mathrm{ja}\phantom{\rule{0.147em}{0ex}}a<0\end{array}\right\\\ \left|a\right|\ge 0\\ \left|a+b\right|\le \left|a\right|+\left|b\right|\end{array}$ Aritmētiskā progresija$\begin{array}{l}{a}_{n}={a}_{1}+\left(n-1\right)\cdot d\phantom{\rule{0.147em}{0ex}}\\ {S}_{n}=\frac{\left({a}_{1}+{a}_{n}\right)\cdot n}{2}\phantom{\rule{0.147em}{0ex}}\\ {a}_{k}=\frac{{a}_{k+1}+{a}_{k-1}}{2}\phantom{\rule{0.147em}{0ex}}\end{array}$ Ģeometriskā progresija$\begin{array}{l}{b}_{n}={b}_{1}\cdot {q}^{n-1}\phantom{\rule{0.147em}{0ex}}\\ {S}_{n}=\frac{{b}_{1}\cdot \left({q}^{n}-1\right)}{q-1}\phantom{\rule{0.147em}{0ex}}\\ {b}_{k}^{2}={b}_{k-1}\cdot {b}_{k+1\phantom{\rule{0.147em}{0ex}}}\end{array}$ Bezgalīgi dilstoša ģeometriskā progresija  $\begin{array}{l}\left|q\right|<1\\ \\ S=\frac{{b}_{1}}{1-q}\end{array}$ Sakņu īpašības$\begin{array}{l}\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}\\ \\ \frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}\\ \\ \sqrt[n\cdot m]{{a}^{k\cdot m}}=\sqrt[n]{{a}^{k}}\\ \\ \sqrt[n]{\sqrt[m]{a}}=\sqrt[n\cdot m]{a}\\ \\ \sqrt[n]{a}\cdot \sqrt[k]{b}=\sqrt[n\cdot k]{{a}^{k}\cdot {b}^{n}}\\ \\ \sqrt{{a}^{2}}=\left|a\right|\end{array}$ Pakāpju īpašības  $\begin{array}{l}{a}^{0}=1\\ \\ {a}^{-n}=\frac{1}{{a}^{n}}\\ {a}^{\frac{m}{n}}=\sqrt[n]{{a}^{m}}\\ \\ {a}^{m}\cdot {a}^{n}={a}^{m+n}\\ \\ {a}^{m}\cdot {b}^{m}={\left(\mathit{ab}\right)}^{m}\\ \\ {a}^{m}:{a}^{n}={a}^{m-n}\\ \\ {\left({a}^{m}\right)}^{n}={a}^{\mathit{mn}}\end{array}$ Logaritmu īpašības$\begin{array}{l}{a}^{{\mathrm{log}}_{a}b}=b\\ \\ {\mathrm{log}}_{a}\left(x\cdot y\right)={\mathrm{log}}_{a}x+{\mathrm{log}}_{a}y\\ \\ {\mathrm{log}}_{a}\frac{x}{y}={\mathrm{log}}_{a}x-{\mathrm{log}}_{a}y\phantom{\rule{0.147em}{0ex}}\\ \\ {\mathrm{log}}_{a}{x}^{k}=k\cdot {\mathrm{log}}_{a}x\phantom{\rule{0.147em}{0ex}}\\ \\ {\mathrm{log}}_{a}b=\frac{{\mathrm{log}}_{c}b}{{\mathrm{log}}_{c}a}\\ \\ {\mathrm{log}}_{{a}^{k}}x=\frac{1}{k}{\mathrm{log}}_{a}x\end{array}$ Kombinatorika$\begin{array}{l}{P}_{n}=n!\\ {\phantom{\rule{0.147em}{0ex}}A}_{n}^{k}=\frac{n!}{\left(n-k\right)!}\end{array}$ $\begin{array}{l}{C}_{n}^{k}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{n!}{k!\left(n-k\right)!}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \\ {C}_{n}^{m}={C}_{n}^{n-m}\\ \\ {C}_{n}^{0}+{C}_{n}^{1}+{C}_{n}^{2}+...+{C}_{n}^{n-1}+\phantom{\rule{0.147em}{0ex}}{C}_{n}^{n}={2}^{n}\end{array}$ Varbūtību teorija$P\left(A\right)=\frac{k}{n}$,$k$ - labvēlīgo notikumu skaits,$n$ - visu iespējamo notikumu skaits. $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)$,kur $A$, $B$ - nesavienojami notikumi. $P\left(A\cap B\right)=P\left(A\right)\cdot P\left(B\right)$,kur $A$, $B$ - neatkarīgi notikumi.

Trigonometrija
 $\begin{array}{l}{\mathrm{sin}}^{2}\mathrm{\alpha }+{\mathrm{cos}}^{2}\mathrm{\alpha }=1\\ \mathrm{tg}\mathrm{\alpha }=\frac{\mathrm{sin}\mathrm{\alpha }}{\mathrm{cos}\mathrm{\alpha }}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\mathrm{ctg}\mathrm{\alpha }=\frac{\mathrm{cos}\mathrm{\alpha }}{\mathrm{sin}\mathrm{\alpha }}\\ 1+{\mathrm{tg}}^{2}\mathrm{\alpha }=\frac{1}{{\mathrm{cos}}^{2}\mathrm{\alpha }}\\ 1+{\mathrm{ctg}}^{2}\mathrm{\alpha }=\frac{1}{{\mathrm{sin}}^{2}\mathrm{\alpha }}\\ \mathrm{tg}\mathrm{\alpha }\cdot \mathrm{ctg}\mathrm{\alpha }=1\\ \end{array}$ $\begin{array}{l}\mathrm{sin}\phantom{\rule{0.147em}{0ex}}2\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}2\phantom{\rule{0.147em}{0ex}}\mathrm{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\cdot \phantom{\rule{0.147em}{0ex}}\mathrm{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\\ \mathrm{cos}\phantom{\rule{0.147em}{0ex}}2\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}{\mathrm{cos}}^{2}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}{\mathrm{sin}}^{2}\mathrm{\alpha }\\ \mathrm{tg}\phantom{\rule{0.147em}{0ex}}2\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{2\phantom{\rule{0.147em}{0ex}}\mathrm{tg}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }}{1\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}{\mathrm{tg}}^{2}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }}\\ \mathrm{sin}\phantom{\rule{0.147em}{0ex}}\left(\mathrm{\alpha }±\mathrm{\beta }\right)\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\mathrm{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\mathrm{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }\phantom{\rule{0.147em}{0ex}}±\mathrm{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\mathrm{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }\\ \mathrm{cos}\phantom{\rule{0.147em}{0ex}}\left(\mathrm{\alpha }+\mathrm{\beta }\right)\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\mathrm{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\mathrm{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}\mathrm{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\mathrm{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }\\ \mathrm{cos}\phantom{\rule{0.147em}{0ex}}\left(\mathrm{\alpha }-\mathrm{\beta }\right)\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\mathrm{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\mathrm{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\mathrm{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\mathrm{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }\\ \mathrm{tg}\phantom{\rule{0.147em}{0ex}}\left(\mathrm{\alpha }+\mathrm{\beta }\right)\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{\mathrm{tg}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }+\phantom{\rule{0.147em}{0ex}}\mathrm{tg}\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }}{1-\left(\mathrm{tg}\mathrm{\alpha }\cdot \mathrm{tg}\mathrm{\beta }\right)}\\ \mathrm{tg}\phantom{\rule{0.147em}{0ex}}\left(\mathrm{\alpha }-\mathrm{\beta }\right)\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{\mathrm{tg}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }-\phantom{\rule{0.147em}{0ex}}\mathrm{tg}\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }}{1+\left(\mathrm{tg}\mathrm{\alpha }\cdot \mathrm{tg}\mathrm{\beta }\right)}\\ \mathrm{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}+\mathrm{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}2\phantom{\rule{0.147em}{0ex}}\mathrm{sin}\phantom{\rule{0.147em}{0ex}}\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\cdot \mathrm{cos}\phantom{\rule{0.147em}{0ex}}\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}\\ \mathrm{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}-\mathrm{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}2\phantom{\rule{0.147em}{0ex}}\mathrm{sin}\phantom{\rule{0.147em}{0ex}}\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}\cdot \mathrm{cos}\phantom{\rule{0.147em}{0ex}}\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\\ \\ \mathrm{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\mathrm{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}2\phantom{\rule{0.147em}{0ex}}\mathrm{cos}\phantom{\rule{0.147em}{0ex}}\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\phantom{\rule{0.147em}{0ex}}\cdot \mathrm{cos}\phantom{\rule{0.147em}{0ex}}\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}\\ \mathrm{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}\mathrm{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }\phantom{\rule{0.147em}{0ex}}=-\phantom{\rule{0.147em}{0ex}}2\phantom{\rule{0.147em}{0ex}}\mathrm{sin}\phantom{\rule{0.147em}{0ex}}\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\phantom{\rule{0.147em}{0ex}}\cdot \mathrm{sin}\phantom{\rule{0.147em}{0ex}}\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}\end{array}$