# Formulae

## Math

### Self-Computed

These may be prone to errors.
Ways to Organize a Secret Santa between n People: $$p=\frac{(n-1)(n-1)!}2$$ Volume of Minecraft: \begin{align} V &=128m\cdot(32000000m)^2 \\ &=2^{29}\cdot5^{12}m^3 \\ &=1.30172\cdot10^{17}m^3 \\ \end{align} Times n on average needed to achieve something with a probability p: $$n=\frac{-\ln{(2)}}{\ln{(1-p)}}$$

### Various

Apothem of an n-gon with side length s: $$a =\frac s{2\tan{\frac{\pi}n}}$$ Exterior Angle of an n-gon: $$\theta=\frac{2\pi}n$$ Interior Angle of an n-gon: $$\theta=\pi-\frac{2\pi}n$$ Sum of Angles of an n-gon: $$\theta=(n-2)\pi$$ Arc Length: $$l=r\theta$$ Circumradius of an n-gon with side length s: $$R =\frac s{2\sin{\frac{\pi}n}}$$ Chord Length: $$c=2r\sin (\frac \theta 2)$$ Unusual Trigonometric Functions: $$\operatorname{vers}{\theta}=1-\cos{\theta}$$ $$\operatorname{cvs}{\theta}=1-\sin{\theta}$$ $$\operatorname{exsec}{\theta}=\sec{\theta}-1$$ $$\operatorname{excsc}{\theta}=\csc{\theta}-1$$ Ellipse Eccentricity: $$e=\sqrt{1-\frac {b^2} {a^2}}$$ Quadratic: $$x=\frac {-b\pm\sqrt{b^2-4ac}} {2a}$$

### Area/Volume

Polygon
Polyhedron
Dimension Sides
Faces
Area Volume
Trigon 2 3 $$A=\frac {\sqrt 3} 4 a^2$$ $$n/a$$
Tetragon 2 4 $$A=a^2$$
Pentagon 2 5 $$A=\frac {\sqrt{25+10\sqrt 5}} 4a^2$$
Hexagon 2 6 $$A=\frac {3\sqrt 3} 2 a^2$$
Octagon 2 8 $$A=2(1+\sqrt 2)a^2$$
Circle 2 $$\infty$$ $$A=\pi r^2$$
Tetrahedron 3 4 $$A=\sqrt 3a^2$$ $$V=\frac{a^3}{6\sqrt{2}}$$
Hexahedron 3 6 $$A=6a^2$$ $$V=a^3$$
Octahedron 3 8 $$A=2\sqrt 3a^2$$ $$V=\frac{\sqrt 2a^3}3$$
Dodecahedron 3 12 $$A=3\sqrt{25+10\sqrt 5}a^2$$ $$V=\frac 1 4(15+7\sqrt 5)a^3$$
Icosahedron 3 20 $$A=5\sqrt 3a^2$$ $$V=\frac 5{12}(3+\sqrt 5)a^3$$
Sphere 3 $$\infty$$ $$A=4\pi r^2$$ $$V=\frac 4 3\pi r^3$$
Cone 3 $$n/a$$ $$A=\pi r(r+\sqrt{r^2+h^2})$$ $$V=\frac {\pi} 3 r^2h$$
Cylinder 3 $$A=2\pi r(r+h)$$ $$V=\pi r^2h$$
Ellipse Area: $$A=\pi ab$$ $$A=\pi a\sqrt{1-a^2e^2}$$ N-gon Area: $$A=\frac{nsa}2 = \frac {ns^2} {4\tan{(\frac {\pi}n)}}$$ Pyramid Volume: $$V=\frac {bh} 3$$ Triangle Area: $$A=\frac {bh} 2$$ $$A=\sqrt {s(s-a)(s-b)(s-c)}$$ $$A=\frac 1 2 ab\sin(\gamma)$$ $$A=\frac 1 2 ab\sin(\alpha+\beta)$$ $$A=\frac {b^2\sin(\alpha)\sin(\alpha+\beta)} {2\sin(\beta)}$$ $$A=\frac {a^2} {2(\cot(\beta)+\cot(\gamma))}$$

### Identities

Euler's Identity $$e^{i\pi}=-1$$ $$e^{2i\pi}=1$$ Trig $$\sin^2{\theta}+\cos^2{\theta}=1$$ $$\sin{2\theta}=2\sin{\theta}\cos{\theta}$$ $$\sin \frac{\theta}{2} = \operatorname{sgn} \left(2 \pi - \theta + 4 \pi \left\lfloor \frac{\theta}{4\pi} \right\rfloor \right) \sqrt{\frac{1 - \cos \theta}{2}}$$ Sine $$\sin {(x)} =\sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}$$ $$\sin{(x)}=\frac12ie^{-ix}-\frac12ie^{ix}$$ Cosine $$\cos{(x)}=\frac12e^{-ix}+\frac12e^{ix}$$ Tangent $$\tan{(x)}=i\frac{e^{-ix}-e^{ix}}{e^{-ix}+e^{ix}}$$ Cotangent $$\cot{(x)}=i\frac{e^{-ix}+e^{ix}}{e^{-ix}-e^{ix}}$$ Secant $$\sec{(x)}=\frac2{e^{-ix}+e^{ix}}$$ Cosecant $$\csc{(x)}=\frac{-2i}{e^{-ix}-e^{ix}}$$

### Integer Sequences and Constants

Bell number ~ Partitioning a set of n people ~ Genji chapter symbols: $$B_{n+1}=\sum_{k=0}^{n} \binom{n}{k} B_k.$$ E: $$e=\sum_{n=0}^{\infty}{\frac 1 {n!}}$$ Euler-Mascheroni: \begin{align} \gamma &= \lim_{n\to\infty}\left(-\ln n + \sum_{k=1}^n \frac1{k}\right)\\[5px] &=\int_1^\infty\left(\frac1{\lfloor x\rfloor}-\frac1{x}\right)\,dx. \end{align} Fibonacci Numbers: $$F_n=\frac{\phi^n-\psi^n}{\sqrt 5}$$ Golden Ratio: $$\phi=\frac{1+\sqrt 5}2$$ $$\psi=\frac{1-\sqrt 5}2=1-\phi=\frac{-1}{\phi}$$ Lucas Numbers: $$L_n=\phi^n+\psi^n$$ Pi: $$\pi=\sqrt{\sum_{n=1}^{\infty}{\frac 6 {n^2}}}$$ Triangular Numbers: $$T_n=\frac{n^2+n}2$$

## Science

Bi-elliptic Transfer:
• $$r_1$$ is the SMA of the initial orbit
• $$r_2$$ is the SMA of the final orbit
• $$r_b$$ is the common apoapsis radius of the two transfer ellipses, and is a free parameter of the manoeuvre
• $$a_1$$ and $$a_2$$ are the semimajor axes of the two elliptical transfer orbits, which are given by $$a_1=\frac{r_1+r_b}2$$ $$a_2=\frac{r_2+r_b}2$$
• The $$\Delta v$$ requirements are:
$$\Delta v_1=\sqrt{\frac{2\mu}{r_1}-\frac{\mu}{a_1}}-\sqrt{\frac{\mu}{r_1}}$$ $$\Delta v_2=\sqrt{\frac{2\mu}{r_b}-\frac{\mu}{a_2}}-\sqrt{\frac{2\mu}{r_b}-\frac{\mu}{a_1}}$$ $$\Delta v_3=\sqrt{\frac{2\mu}{r_2}-\frac{\mu}{a_2}}-\sqrt{\frac{\mu}{r_2}}$$ Drag: $$F_D=\frac 1 2\rho u^2C_DA$$ Fitts' Law: $$\text{ID} = \log_2 \Bigg(\frac{2D} {W}\Bigg)$$ Hohmann Transfer: $$\Delta v_1=\sqrt{\frac{\mu}{a_1}}(\sqrt{\frac{2a_2}{a_1+a_2}}-1)$$ $$\Delta v_2=\sqrt{\frac{\mu}{a_2}}(1-\sqrt{\frac{2a_1}{a_1+a_2}})$$ Kinematic Equations: $$d=v_it+\frac 1 2at^2$$ $$v_f=\sqrt{v_i^2+2ad}$$ $$v_f=v_i+at$$ $$d=t\frac{v_i+v_f}2$$ Orbital Inclination Change: $$\Delta v= {2\sin(\frac{\Delta{i}}{2})\sqrt{1-e^2}\cos(\omega+f)na \over {1+e\cos(f)}}$$ Orbit Period: $$P=2\pi\sqrt\frac{a^3}\mu$$ Orbit Velocity: $$v=\sqrt{\frac{\mu}a}$$ Standard Gravitational Parameter: $$\mu=GM$$ Synchronous Orbit: $$a=\sqrt[3]{\frac{\mu t^2}{4\pi^2}}$$ Tsiolkovsky Rocket Equation: $$\Delta v=v_e\ln{\frac{m_i}{m_f}}$$ $$\Delta v=gI_{sp}\ln{\frac{m_i}{m_f}}$$