diff options
Diffstat (limited to 'mat/trigonometrija.tex')
| -rw-r--r-- | mat/trigonometrija.tex | 45 |
1 files changed, 37 insertions, 8 deletions
diff --git a/mat/trigonometrija.tex b/mat/trigonometrija.tex index cb65720..c9cfff5 100644 --- a/mat/trigonometrija.tex +++ b/mat/trigonometrija.tex @@ -11,6 +11,7 @@ \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 @@ -35,18 +36,21 @@ % Spisek izbranih trigonometričnih izrekov bom kot pripomoček imel na drugem testu pri matematiki v tretjem letniku. % \end{abstract} % \tableofcontents +\section{Drugi test} \begin{multicols}{2} - \begin{tabular}{|c|c|c|c|c|} + \begin{tabular}{|c|c|c|c|c|c|} \hline - $\measuredangle$ & $\sin$ & $\cos$ & $\tan$ & $\cot$ \\ + $\measuredangle$ & Rad & $\sin$ & $\cos$ & $\tan$ & $\cot$ \\ \hline - $\ang{30}$ & 0 & 1 & 0 & ne obstaja \\ + $\ang{0}$ & 0 & 0 & 1 & 0 & ne obstaja \\ \hline - $\ang{45}$ & $\frac{1}{2}$ & $\frac{\sqrt{3}}{2}$ & $\frac{\sqrt{3}}{2}$ & $\sqrt{3}$ \\ + $\ang{30}$ &$\frac{\pi}{6}$& $\frac{1}{2}$ & $\frac{\sqrt{3}}{2}$ & $\frac{\sqrt{3}}{2}$ & $\sqrt{3}$ \\ \hline - $\ang{60}$ & $\frac{\sqrt{3}}{2}$ & $\frac{1}{2}$ & $\sqrt{3}$ & $\frac{\sqrt{3}}{3}$ \\ + $\ang{45}$ & $\frac{\pi}{4}$& $\frac{\sqrt{2}}{2}$ & $\frac{\sqrt{2}}{2}$ & 1 & 1 \\ \hline - $\ang{90}$ & 1 & 0 & ne obstaja & 0 \\ + $\ang{60}$ & $\frac{\pi}{3}$& $\frac{\sqrt{3}}{2}$ & $\frac{1}{2}$ & $\sqrt{3}$ & $\frac{\sqrt{3}}{3}$ \\ + \hline + $\ang{90}$ & $\frac{\pi}{2}$& 1 & 0 & ne obstaja & 0 \\ \hline \end{tabular} $$\sin^2\alpha+\cos^2\alpha=1$$ @@ -80,12 +84,37 @@ $$\sin\alpha\pm\sin\beta=2\sin\left(\frac{\alpha\pm\beta}{2}\right)\cos\left(\frac{\alpha\mp\beta}{2}\right)$$ $$\cos\alpha+\cos\beta=2\cos\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)$$ $$\cos\alpha-\cos\beta=-2\sin\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)$$ - $$\tan\alpha+\tan\beta=\frac{\sin\left(\alpha+\beta\right)}{\cos\alpha\cos\beta}$$ + $$\tan\alpha\pm\tan\beta=\frac{\sin\left(\alpha\pm\beta\right)}{\cos\alpha\cos\beta}$$ $$\sin\alpha\cos\alpha=\frac{1}{2}\sin2\alpha$$ $$2\cos^2\frac{\alpha}{2}=1+\cos\alpha$$ $$2\sin^2\frac{\alpha}{2}=1-\cos\alpha$$ $$\tan^2\frac{x}{2}=\frac{1-\cos\alpha}{1+\cos\alpha}$$ - $$\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1+\cos\alpha}{2}} \text{ in tako dalje}$$ +\end{multicols} +\section{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{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}. |