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diff --git a/mat/formule.tex b/mat/formule.tex new file mode 100644 index 0000000..1995684 --- /dev/null +++ b/mat/formule.tex @@ -0,0 +1,136 @@ +% do-vimlatex-onwrite +\documentclass[]{article} +\usepackage[utf8]{inputenc} +\usepackage{siunitx} +\usepackage[slovene]{babel} +\usepackage[inline]{enumitem} +\usepackage[a4paper, total={7in, 10in}]{geometry} +\usepackage{hologo} +\usepackage[hidelinks,unicode]{hyperref} +\usepackage{datetime} +\usepackage{tkz-euclide} +\usepackage{amssymb} +\usepackage{multicol} +\usepackage{amsmath} +% \sisetup{output-decimal-marker = {,}, quotient-mode=fraction, per-mode=fraction} % frac način +% \sisetup{output-decimal-marker = {,}, quotient-mode=fraction, per-mode=symbol} % poševnica način +\sisetup{output-decimal-marker = {,}, quotient-mode=fraction} % na -1 način +\settimeformat{hhmmsstime} +\newcommand{\razhroscevanje}{0} +\newcommand{\razhroscevanjeg}{0} % grafično razhroščevanje +\makeatletter +\newcommand{\xslalph}[1]{\expandafter\@xslalph\csname c@#1\endcsname} +\newcommand{\@xslalph}[1]{% + \ifcase#1\or a\or b\or c\or \v{c}\or d\or e\or f\or g\or h\or i% + \or j\or k\or l\or m\or n\or o\or p\or r\or s\or \v{s}% + \or t\or u\or v\or z\or \v{z} + \else\@ctrerr\fi% +} +\AddEnumerateCounter{\xslalph}{\@xslalph}{m} +\makeatother +\title{Formule} +\author{Anton Luka Šijanec, 3. a} +\begin{document} +\maketitle +% \begin{abstract} +% Spisek izbranih trigonometričnih izrekov bom kot pripomoček imel na drugem testu pri matematiki v tretjem letniku. +% \end{abstract} +% \tableofcontents +\section{Trigonometrija: Drugi test} +\begin{multicols}{2} + \begin{tabular}{|c|c|c|c|c|c|} + \hline + $\measuredangle$ & Rad & $\sin$ & $\cos$ & $\tan$ & $\cot$ \\ + \hline + $\ang{0}$ & 0 & 0 & 1 & 0 & ne obstaja \\ + \hline + $\ang{30}$ &$\frac{\pi}{6}$& $\frac{1}{2}$ & $\frac{\sqrt{3}}{2}$ & $\frac{\sqrt{3}}{2}$ & $\sqrt{3}$ \\ + \hline + $\ang{45}$ & $\frac{\pi}{4}$& $\frac{\sqrt{2}}{2}$ & $\frac{\sqrt{2}}{2}$ & 1 & 1 \\ + \hline + $\ang{60}$ & $\frac{\pi}{3}$& $\frac{\sqrt{3}}{2}$ & $\frac{1}{2}$ & $\sqrt{3}$ & $\frac{\sqrt{3}}{3}$ \\ + \hline + $\ang{90}$ & $\frac{\pi}{2}$& 1 & 0 & ne obstaja & 0 \\ + \hline + \end{tabular} + $$\sin^2\alpha+\cos^2\alpha=1$$ + $$\sin\alpha=\pm\sqrt{1-\cos^2\alpha}$$ + $$\cos\alpha=\pm\sqrt{1-\sin^2\alpha}$$ + $\sin, \tan, \cot$ so lihe, $\cos$ je soda. + $$\sin\left(-\alpha\right)=-\sin\alpha$$ + $$\cos\left(-\alpha\right)=\cos\alpha$$ + $$\sin\left(\frac{\pi}{2}-\alpha\right)=\cos\alpha$$ + $$\cos\left(\frac{\pi}{2}-\alpha\right)=\sin\alpha$$ + $$\tan\left(\frac{\pi}{2}-\alpha\right)=\cot\alpha$$ + $$\sin\left(\alpha\pm\beta\right)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta$$ + $$\cos\left(\alpha\pm\beta\right)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta$$ + $$\tan\left(\alpha\pm\beta\right)=\frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\tan\beta}$$ + $$\cot\left(\alpha\pm\beta\right)=\frac{\cot\alpha\cot\beta\mp1}{\cot\beta\pm\cot\alpha}$$ + $$\sin2\alpha=2\sin\alpha\cos\alpha$$ + $$\cos2\alpha=cos^2\alpha-\sin^2\alpha=2\cos^2\alpha-1=1-2\sin^2\alpha$$ + $$\tan2\alpha=\frac{2\tan\alpha}{1-\tan^2\alpha}$$ + $$\cot2\alpha=\frac{\cot^2\alpha-1}{2\cot\alpha}$$ + $$\sin3\alpha=3\sin\alpha-4\sin^3\alpha=4\sin\left(\frac{\pi}{3}-\alpha\right)\sin\left(\frac{\pi}{3}+\alpha\right)$$ + $$\cos3\alpha=4\cos^3\alpha-3\cos\alpha=4\cos\alpha\cos\left(\frac{\pi}{3}-\alpha\right)\cos\left(\frac{\pi}{3}+\alpha\right)$$ + $$\tan3\alpha=\frac{3\tan\alpha-\tan^3\alpha}{1-3\tan^2\alpha}=\tan\alpha\tan\left(\frac{\pi}{3}-\alpha\right)\tan\left(\frac{\pi}{3}+\alpha\right)$$ + $$\sin\frac{\alpha}{2}=\pm\sqrt{\frac{1-\cos\alpha}{2}}$$ + $$\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1+\cos\alpha}{2}}$$ + $$\tan\frac{\alpha}{2}=\pm\sqrt{\frac{1+\cos\alpha}{1-\cos\alpha}}=\frac{\sin\alpha}{1+\cos\alpha}$$ + $$2\cos\alpha\cos\beta=\cos\left(\alpha-\beta\right)+\cos\left(\alpha+\beta\right)$$ + $$2\sin\alpha\sin\beta=\pm\cos\left(\alpha\pm\beta\right)-\cos\left(\alpha\mp\beta\right)$$ + $$2\sin\alpha\cos\beta=\sin\left(\alpha+\beta\right)+\sin\left(\alpha-\beta\right)$$ + $$2\cos\alpha\sin\beta=\sin\left(\alpha+\beta\right)-\sin\left(\alpha-\beta\right)$$ + $$\tan\alpha\tan\beta=1-\frac{\tan\alpha+\tan\beta}{\tan\left(\alpha+\beta\right)}=\frac{\cos\left(\alpha-\beta\right)-\cos\left(\alpha+\beta\right)}{\cos\left(\alpha-\beta\right)+\cos\left(\alpha+\beta\right)}$$ + $$\sin\alpha\pm\sin\beta=2\sin\left(\frac{\alpha\pm\beta}{2}\right)\cos\left(\frac{\alpha\mp\beta}{2}\right)$$ + $$\cos\alpha+\cos\beta=2\cos\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)$$ + $$\cos\alpha-\cos\beta=-2\sin\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)$$ + $$\tan\alpha\pm\tan\beta=\frac{\sin\left(\alpha\pm\beta\right)}{\cos\alpha\cos\beta}$$ + $$\sin\alpha\cos\alpha=\frac{1}{2}\sin2\alpha$$ + $$2\cos^2\frac{\alpha}{2}=1+\cos\alpha$$ + $$2\sin^2\frac{\alpha}{2}=1-\cos\alpha$$ + $$\tan^2\frac{x}{2}=\frac{1-\cos\alpha}{1+\cos\alpha}$$ +\end{multicols} +\section{Trikotniki in krogi: Tretji test} +\begin{multicols}{2} + $$s=\frac{a+b+c}{2} \wedge S=\sqrt{s(s-a)(s-b)(s-c)}$$ + $$S_\text{trikotnika v izseku}=\frac{r^2\sin\alpha}{2}$$ + $$\frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R$$ + $$a^2=b^2+c^2-2bc\cos\alpha$$ + $$S_\text{paralelograma}=av_a=ab\sin\alpha=\frac{ef}{2}\sin\omega$$ + $$S_\text{romba}=av=a^2\sin\alpha=\frac{ef}{2}$$ + $$S_\text{trapeza}=\frac{v(a+c)}{2}$$ + $$S_\text{deltoida}=\frac{ef}{2}$$ + $$S_\text{trikotnika}=\frac{ab\sin\gamma}{2}=\frac{av_a}{2}$$ + $$S_\text{enakostraničnega}=\frac{a^2\sqrt{3}}{4}$$ + $$\arcsin x+\arccos x=\frac{\pi}{2}$$ + $$S_\text{trikotnika}=\frac{abc}{4R}=2R^2\sin\alpha\sin\beta\sin\gamma=rs\text{, kjer je } s=\frac{a+b+c}{2}$$ + $$Diagonal_\text{pravilnega mnogokotnika}=\frac{n(n-3)}{2}$$ + $$\alpha_\text{pravilnega mnogokotnika}=\frac{n-2}{n}\ang{180}$$ + $$S_\text{pravilnega mnogokotnika}=\frac{n}{2}R^2\sin\frac{\ang{360}}{n}= + na^2\tan\frac{\alpha}{2}\frac{1}{2}=\frac{na^2}{4\tan\frac{\ang{180}}{n}}$$ + $$\alpha_\text{ene premice}=\arctan k_p$$ + $$\alpha_\text{med dvema premicama}=\arctan\lvert\frac{k_q-k_p}{1+k_p-k_q}\rvert$$ + $$D_\text{arcsin}=D_\text{arccos}=[-1; 1] \wedge V_\text{arcsin}=[\ang{-90}; \ang{90}] \wedge V_\text{arccos}=[\ang{0}; \ang{180}]$$ + $$D_\text{arctan}=D_\text{arccot}=\mathbb{R} \wedge V_\text{arctan}=(\ang{-90}; \ang{90}) \wedge V_\text{arccot}=(\ang{0}; \ang{180})$$ + $$soda(x)=-soda(x) \wedge liha(-x)=-liha(x)$$ + $$f(x)\neq-f(x)\nLeftrightarrow f(-x)=-f(x) \text{ in obratno}$$ + $$f(x)=-f(x) \wedge f(-x)=-f(x) \Leftrightarrow f(x)=0$$ +\end{multicols} +\section{Trorazsežnostna geometrijska telesa: Četrti test} % todo: funkcije na likih - notranji kot, prisekana piramida, prisekan stožec, kuboktaeder, tetraeder, včartavanje teles v druga telesa +\begin{multicols}{2} + $$S_\text{odseka}=r^2\pi\frac{\alpha}{\ang{360}}-\frac{r^2\sin\alpha}{2}$$ + $$V_\text{piramide}=\frac{P_\text{osnovna}v}{3}$$ + $$P_\text{stožca}=\frac{s\cdot2\pi r}{2}\text{(špornova fora)}+r^2\pi=r\pi\left(r+s\right)$$ + $$V_\text{stožca}=\frac{r^2v\pi}{3}$$ + $$P_\text{enakostraničnega trikotnika}=\frac{a^2\sqrt{3}}{4}$$ +\end{multicols} +\section{Zaključek} +\hologo{LaTeX} izvorna koda dokumenta je objavljena na \url{https://git.sijanec.eu/sijanec/sola-gimb-3}. Za izdelavo dokumenta je potreben \texttt{TeXLive 2020}. +\if\razhroscevanje1 +\vfill +\section*{Razhroščevalne informacije} +Konec generiranja dokumenta \today\ ob \currenttime. + +Dokument se je generiral R0qK1KR2 \SI{}{\second}. % aaasecgeninsaaa +\fi +\end{document} |