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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}
% \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}
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