Math

Math formulas &c.

$$\tan(x)=\frac{\sin(x)}{\cos(x)}$$
$$x(°)$$	0	30	45	60	90
$$\sin(x)$$	$$\frac{\sqrt[2]{0}}{2}$$	$$\frac{\sqrt[2]{1}}{2}$$	$$\frac{\sqrt[2]{2}}{2}$$	$$\frac{\sqrt[2]{3}}{2}$$	$$\frac{\sqrt[2]{4}}{2}$$
$$\cos(x)$$	$$\frac{\sqrt[2]{4}}{2}$$	$$\frac{\sqrt[2]{3}}{2}$$	$$\frac{\sqrt[2]{2}}{2}$$	$$\frac{\sqrt[2]{1}}{2}$$	$$\frac{\sqrt[2]{0}}{2}$$

$\emptyset\subset\mathbb{B}\subset\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{A}\subset\mathbb{R}\subset\mathbb{C}\subset\mathbb{H}\subset\mathbb{O}\subset\mathbb{S}$
$$\eqalign{\sum_{i=0}^{n}{i}&=\frac{n(n+1)}{2} \\ \sum_{i=0}^{n}{i^2}&=\frac{n(n+1)(2n+1)}{6}=\frac{n^3}{4}+\frac{n^3}{2}+\frac{n^2}{4} \\ \sum_{i=0}^{n}{i^3}&=\left(\sum_{i=0}^{n}{i}\right)^2 \\ \sum_{i=0}^{n}{i^a}&=\frac{n^{a+1}}{a+1}+\frac{n^a}{2}+\sum_{j=0}^{a}\frac{B_j}{j!} \frac{a! n^{a-j+1}}{a-j+1}\quad B_j\; \text{is the}\; j^\text{th}\; \text{Bernoulli number} }$$
$$\sum_{i=1}^{n}\frac {1}{i^{k}}=H_{n}^{k}$$
$$D_x=\left\{a\in\mathbb{Z}\mid x\vdots a\right\}$$
$$M_x=\left\{a\mid a\vdots x\right\}$$
$$\left(\frac{a}{b}\right)^{-1}=\frac{b}{a}$$
$$\eqalign{ (a+b)(a-b)&=a^2-b^2 \\ (a\pm b)^2&=a^2\pm 2ab+b^2 \\ (a\pm b)^3&=a^3 \pm 3a^2 b+3ab^2 \pm b^2 \\ a^3 \pm b^3&=(a\pm b)(a^2 \mp ab +b^2) }$$
$$\eqalign{ n!^{(a)}&=a^{\frac{n-1}{a}}\frac{\Gamma(\frac{n}{a}+1,0)}{\Gamma(\frac{1}{a}+1,0)} \\ !n&=n!\sum_{i=0}^{n}\frac{(-1)^i}{i!}=\frac{\Gamma(n+1,-1)}{e} \\ n\# &= \prod_{i=1}^{\pi(n)}p_i \\ n\$ &= G(n+2) \\ H(n)&= \frac{(\Gamma(n,0))^{n-1}}{G(n)} }$$
$$\eqalign{ M_b(x_1,\dots,x_n)&=\sqrt[b]{\frac{\sum_{i=1}^{n}x_i^b w_i}{\sum_{i=1}^{n} w_i}} \\ M_0(x_1,\dots,x_n)&=\sqrt[\sum _{i=1}^{n}w_{i}]{\prod_{i=1}^{n}x_{i}^{w_{i}}} \\ L_{b}(x_1,\dots,x_n)&={\frac {\sum _{i=1}^{n}w_{i} x_{i}^{b}}{\sum _{i=1}^{n}w_{i} x_{i}^{b-1}}} }$$
$$a^{\left(b^{-1}\right)}=\sqrt[b]{a}$$
$$\sqrt[2]{a\pm\sqrt[2]{b}}=\sqrt[2]{\frac{a+\sqrt[2]{a^2+b}}{2}}\pm\sqrt[2]{\frac{a+\sqrt[2]{a^2-b}}{2}}$$
$$\frac{a-b}{a*b}=\frac{\not a}{\not a*b}-\frac{\not b}{a*\not b}=\frac{1}{b}-\frac{1}{a}$$
$$\text{for a regular polygon: } \eqalign{ l&=r*2*\sin(\frac{\pi}{n}) \\ a&=r*\cos(\frac{\pi}{n}) \\ A&=\frac{r^2*n*\sin\left(\frac{2\pi}{n}\right)}{2} \\ \measuredangle&=\frac{\pi(n-2)}{n} \\ n_d&=\frac{n(n-3)}{2} }$$
$$a^2=b^2+c^2-2a*b*\cos(\alpha)$$
$$\left.\begin{align} AO&\perp \alpha \\ OX&\perp d \\ O,X,d&\subset\alpha \\ a&\not\subset\alpha \end{align} \right \} \implies AX \perp d$$