From c29b044bf5cc874782b0fece6399768b66e5c43b Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Anton=20Luka=20=C5=A0ijanec?= <anton@sijanec.eu>
Date: Tue, 22 Mar 2022 23:00:17 +0100
Subject: pred testom matematike

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+% 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}
-- 
cgit v1.2.3

