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authorAnton Luka Šijanec <anton@sijanec.eu>2022-02-02 22:39:19 +0100
committerAnton Luka Šijanec <anton@sijanec.eu>2022-02-02 22:39:19 +0100
commit3c3c3a6b1ab5a98ef4f400d5aa615ddabf3b94c5 (patch)
tree3e7821a6bfca04de0ecf851a41ae00e3b7caa648 /mat
parentdnmat (diff)
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-rw-r--r--mat/trigonometrija.tex45
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}.