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diff --git a/mat/trigonometrija.tex b/mat/trigonometrija.tex new file mode 100644 index 0000000..cb65720 --- /dev/null +++ b/mat/trigonometrija.tex @@ -0,0 +1,99 @@ +% 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} +% \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{Trigonometrične 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 +\begin{multicols}{2} + \begin{tabular}{|c|c|c|c|c|} + \hline + $\measuredangle$ & $\sin$ & $\cos$ & $\tan$ & $\cot$ \\ + \hline + $\ang{30}$ & 0 & 1 & 0 & ne obstaja \\ + \hline + $\ang{45}$ & $\frac{1}{2}$ & $\frac{\sqrt{3}}{2}$ & $\frac{\sqrt{3}}{2}$ & $\sqrt{3}$ \\ + \hline + $\ang{60}$ & $\frac{\sqrt{3}}{2}$ & $\frac{1}{2}$ & $\sqrt{3}$ & $\frac{\sqrt{3}}{3}$ \\ + \hline + $\ang{90}$ & 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. + $$\sin\left(-\alpha\right)=-\sin\alpha$$ + $$\cos\left(-\alpha\right)=\cos\alpha$$ + $$\sin\left(\frac{\pi}{2}-\alpha\right)=\cos\alpha$$ + $$\cos\left(\frac{\pi}{2}-\alpha\right)=\sin\alpha$$ + $$\tan\left(\frac{\pi}{2}-\alpha\right)=\cot\alpha$$ + $$\sin\left(\alpha\pm\beta\right)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta$$ + $$\cos\left(\alpha\pm\beta\right)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta$$ + $$\tan\left(\alpha\pm\beta\right)=\frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\tan\beta}$$ + $$\cot\left(\alpha\pm\beta\right)=\frac{\cot\alpha\cot\beta\mp1}{\cot\beta\pm\cot\alpha}$$ + $$\sin2\alpha=2\sin\alpha\cos\alpha$$ + $$\cos2\alpha=cos^2\alpha-\sin^2\alpha=2\cos^2\alpha-1=1-2\sin^2\alpha$$ + $$\tan2\alpha=\frac{2\tan\alpha}{1-\tan^2\alpha}$$ + $$\cot2\alpha=\frac{\cot^2\alpha-1}{2\cot\alpha}$$ + $$\sin3\alpha=3\sin\alpha-4\sin^3\alpha=4\sin\left(\frac{\pi}{3}-\alpha\right)\sin\left(\frac{\pi}{3}+\alpha\right)$$ + $$\cos3\alpha=4\cos^3\alpha-3\cos\alpha=4\cos\alpha\cos\left(\frac{\pi}{3}-\alpha\right)\cos\left(\frac{\pi}{3}+\alpha\right)$$ + $$\tan3\alpha=\frac{3\tan\alpha-\tan^3\alpha}{1-3\tan^2\alpha}=\tan\alpha\tan\left(\frac{\pi}{3}-\alpha\right)\tan\left(\frac{\pi}{3}+\alpha\right)$$ + $$\sin\frac{\alpha}{2}=\pm\sqrt{\frac{1-\cos\alpha}{2}}$$ + $$\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1+\cos\alpha}{2}}$$ + $$\tan\frac{\alpha}{2}=\pm\sqrt{\frac{1+\cos\alpha}{1-\cos\alpha}}=\frac{\sin\alpha}{1+\cos\alpha}$$ + $$2\cos\alpha\cos\beta=\cos\left(\alpha-\beta\right)+\cos\left(\alpha+\beta\right)$$ + $$2\sin\alpha\sin\beta=\pm\cos\left(\alpha\pm\beta\right)-\cos\left(\alpha\mp\beta\right)$$ + $$2\sin\alpha\cos\beta=\sin\left(\alpha+\beta\right)+\sin\left(\alpha-\beta\right)$$ + $$2\cos\alpha\sin\beta=\sin\left(\alpha+\beta\right)-\sin\left(\alpha-\beta\right)$$ + $$\tan\alpha\tan\beta=1-\frac{\tan\alpha+\tan\beta}{\tan\left(\alpha+\beta\right)}=\frac{\cos\left(\alpha-\beta\right)-\cos\left(\alpha+\beta\right)}{\cos\left(\alpha-\beta\right)+\cos\left(\alpha+\beta\right)}$$ + $$\sin\alpha\pm\sin\beta=2\sin\left(\frac{\alpha\pm\beta}{2}\right)\cos\left(\frac{\alpha\mp\beta}{2}\right)$$ + $$\cos\alpha+\cos\beta=2\cos\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)$$ + $$\cos\alpha-\cos\beta=-2\sin\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)$$ + $$\tan\alpha+\tan\beta=\frac{\sin\left(\alpha+\beta\right)}{\cos\alpha\cos\beta}$$ + $$\sin\alpha\cos\alpha=\frac{1}{2}\sin2\alpha$$ + $$2\cos^2\frac{\alpha}{2}=1+\cos\alpha$$ + $$2\sin^2\frac{\alpha}{2}=1-\cos\alpha$$ + $$\tan^2\frac{x}{2}=\frac{1-\cos\alpha}{1+\cos\alpha}$$ + $$\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1+\cos\alpha}{2}} \text{ in tako dalje}$$ +\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} |