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\usepackage{multicol}
\usepackage{amsmath}
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\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
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\section*{Razhroščevalne informacije}
Konec generiranja dokumenta \today\ ob \currenttime.
Dokument se je generiral R0qK1KR2 \SI{}{\second}. % aaasecgeninsaaa
\fi
\end{document}
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