User:Pedro Kakuse/Math

Nested radicals
$$\begin{align} &\sqrt{a\!\pm\!b\sqrt{c\,}\,}\\ &\quad=\sqrt{\tfrac{a+\sqrt{a^2-b^2 c\,}}{2}}\!\pm\!\sqrt{\tfrac{a-\sqrt{a^2-b^2 c\,}}{2}} \end{align}$$

Summation notation
$$\sum_{i=0}^n a_{_i}\!\!=\!a_{_0}\!\!+\!a_{_1}\!\!+\!a_{_2}\!+...\!+a_{_n}$$

Integration
$$\int_a^b\!\!f(\!x\!)\,dx\!=\!F(b)\!-\!F(a)$$

Root mean-square
$$ f(t)_{_\text{RMS}}\!=\!\sqrt{\tfrac{1}{T_2-T_1}\int_{T_1}^{T_2} {[f(t)]}^2\,dt\,}, $$

$${\text{A}\sin(2\pi ft)}_{_\text{RMS}}\!=\!\tfrac{\text{A}}{\sqrt2} $$

Angle-sum and difference identities
$$\cos(\alpha\!\pm\!\beta)\!=\!\cos\alpha\cos\beta\!\mp\!\sin\alpha\sin\beta$$

$$\sin(\alpha\!\pm\!\beta)\!=\!\sin\alpha\cos\beta\!\pm\!\cos\alpha\sin\beta$$

Sum-to-product identities
$$\begin{align} \cos&\,\theta\!+\!\cos\varphi\\ &\!=\!2\!\cos\left(\!\tfrac{\theta+\varphi}{2}\!\right)\cos\left(\!\tfrac{\theta-\varphi}{2}\!\right) \end{align}$$

$$\begin{align} \cos&\,\theta\!-\!\cos\varphi\\ &\!=\!\!-2\sin\left(\!\tfrac{\theta+\varphi}{2}\!\right)\sin\left(\!\tfrac{\theta-\varphi}{2}\!\right) \end{align}$$

$$\begin{align} \sin&\,\theta\!\pm\!\sin\varphi\\ &=\!2\sin\left(\!\tfrac{\theta\pm\varphi}{2}\!\right)\cos\left(\!\tfrac{\theta\mp\varphi}{2}\!\right) \end{align}$$

Product-to-sum identities
$$\cos\theta\cos\varphi\!=\!\tfrac{\!\cos(\theta+\varphi)+\cos(\theta-\varphi)}{\!2}$$

$$\sin\theta\cos\varphi\!=\!\tfrac{\sin(\theta+\varphi)+\sin(\theta-\varphi)}{2}$$

$$\cos\theta\sin\varphi\!=\!\tfrac{\sin(\theta+\varphi)-\sin(\theta-\varphi)}{2}$$

$$\sin\theta\sin\varphi\!=\!\tfrac{\!-\!\cos(\theta+\varphi)+\cos(\theta-\varphi)}{\!2}$$

Triple-angle identities
$$\begin{align} \cos (3\theta)&\!=\!4\cos^3\!\theta-3 \cos\theta\\ &\!=\!4\cos\theta\cos\left(\!\tfrac{\pi}{3}\!+\!\theta\right)\cos\left(\!\tfrac{\pi}{3}\!-\!\theta\right)\end{align}$$

$$\begin{align}\sin(3\theta) &\!=\!-4\sin^3\!\theta+3\sin\theta\\ &\!=\!4\sin\theta\sin\left(\!\tfrac{\pi}{3}\!+\!\theta\right)\sin\left(\!\tfrac{\pi}{3}\!-\!\theta\right) \end{align}$$

Double-angle identities
$$\cos(2\theta)\!=\!\cos^2\!\theta\!-\!\sin^2\!\theta$$ $$\sin(2\theta)\!=\!2\sin\theta\cos\theta$$

Half-angle identities
$$\cos(\tfrac{\theta}{2})\!=\!\sgn(\pi\!+\!\theta\!+\!4\pi\lfloor\!\tfrac{\pi-\theta}{4\pi}\!\rfloor)\sqrt{\tfrac{1+\cos\!\theta}{2}\,}$$

$$\sin(\tfrac{\theta}{2})\!=\!\sgn(2\pi\!-\!\theta\!+\!4\pi\lfloor\!\tfrac{\theta}{4\pi}\!\rfloor)\sqrt{\tfrac{1-\cos\!\theta}{2}\,} $$

$$\cos^2\!(\tfrac{\theta}{2})\!=\!\tfrac{1+\cos\!\theta}{2}$$

$$\sin^2\!(\tfrac{\theta}{2})\!=\!\tfrac{1-\cos\!\theta}{2}$$

Sign function
$$\sgn(x)\!=\! \begin{cases} \!-1,\ x\!<\!0\\ \!\ \ \ 0,\ x\!=\!0\\ \!\ \ \ 1,\ x\!>\!0 \end{cases}$$

Integration
$$\int\!\!\cos(ax)\,\text{d}x\!=\!\tfrac{1}{a}\sin(ax)\!+\!C$$

$$\int\!\!\cos^2\!(ax)\,\text{d}x\!=\!\tfrac{x}{2}\!+\!\tfrac{1}{4a}\sin(2ax)\!+\!C$$

$$\begin{align} \int\!\!\cos&^n\!(ax)\,\text{d}x\!=\!\tfrac{\cos^{n-1}(ax)\sin(ax)}{na}\\ &+\!\tfrac{n-1}{n}\!\!\int\!\!\cos^{n-2}\!(ax)\,\text{d}x,(n>0) \end{align}$$

Chord length
$$c=2\ r\sin\!\tfrac{\theta}{2}$$

Standard equation
$$\begin{align} &\tfrac{(x-h)^2}{a^2}\!+\!\tfrac{(y-k)^2}{b^2}\!=\!1,\\ &c\!=\!\sqrt{a^2\!-\!b^2\,},\\ &e\!=\!\frac{c}{a}\!=\!\sqrt{1\!-\!(\tfrac{b}{a})^2\,} \end{align}$$

General equation
$$\begin{align} A&x^2\!+\!Bxy\!+\!Cy^2\\ &+\!Dx\!+\!Ey\!+\!F\!=\!0,\\ B&^2\!\!-\!4AC\!<\!0,\\ A&\!=\!a^2\!\sin^2\!\theta\!+\!b^2\!\cos^2\!\theta,\\ B&\!=\!2(b^2\!\!-\!a^2\!)\sin\theta\cos\theta,\\ C&\!=\!a^2\!\cos^2\!\theta\!+\!b^2\!\sin^2\!\theta,\\ D&\!=\!-2Ah\!-\!Bk,\\ E&\!=\!-Bh\!-\!2Ck,\\ F&\!=\!Ah^2\!\!+\!Bhk\!+\!Ck^2\!\!-\!a^2b^2 \end{align}$$

Elliptic integral
$$f(x)\!=\!\!\int_{c}^{x}\!\!\!R \left(t,\sqrt{P(t)\,}\right) dt$$

Circumference
$$C\!=\!4a\!\!\int_0^{\pi/2}\!\!\!\!\!\sqrt{1\!-\!e^2\!\sin^2\!\theta\,}\, d\theta$$

Arc length
$$s\!=\!-b\!\int_{\cos\!\! ^{^{-1}}\!\left(\!\tfrac{x_{_1}}{a}\!\right)}^{\cos\!\!^{^{-1}}\!\left(\!\tfrac{x_{_2}}{a}\!\right)}\!\!\!\sqrt{1\!-\!e^2\!\sin^2\!\theta\,}\,d\theta$$

Parametric equation
$$\begin{align} (x,\,y)\!=&(a\cos t,\,b\sin t),\\ &\,0\!\le\!t\!<\!2\pi \end{align}$$

Polar equation
$$\begin{align} r&\!=\!\tfrac{ab}{\sqrt{\!(b \cos \theta)^2\!+(a\sin \theta)^2}}\\ &\!=\!\tfrac{b}{\sqrt{1 - (e\cos\theta)^2}}\\ &\!=\!\tfrac{a(1-e^2)}{1\pm e\cos\theta}\end{align}$$