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
See also here.
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:
$$\Delta v=v_e\ln{\frac{m_i}{m_f}}$$
$$\Delta v=gI_{sp}\ln{\frac{m_i}{m_f}}$$