% Author: Till Tantau % Source: The PGF/TikZ manual \documentclass{minimal} \usepackage{tikz} \usepackage{tkz-euclide} \usetkzobj{all} \usetikzlibrary{positioning} \usetikzlibrary{calc,through,intersections} \usepackage{verbatim} \usepackage{xepersian} \setlength{\hoffset}{0.2cm}%1 \setlength{ \voffset}{-30pt}%2 \setlength{\oddsidemargin }{-6pt}%3 \setlength{\topmargin }{10pt}%4 \setlength{\headheight}{20pt}%5 \setlength{\headsep}{15pt}%6 \setlength{\textheight}{700pt}%7 \setlength{\textwidth}{440pt}%8 \setlength{\marginparsep }{+3pt}%9 \setlength{\marginparwidth}{-2pt}%10 \setlength{\footskip}{10pt}%11 \linespread{1.9} \setlength{\parindent}{1.3em} \setlength{\paperwidth}{100pt} \setlength{\paperheight}{150pt} \settextfont[Scale=1]{Yas} \setlatintextfont[Scale=1]{Yas} \setdigitfont[Scale=1]{Yas} %\usetikzlibrary{trees,snakes} \begin{document} \pagestyle{empty} \begin{comment} :Title: A picture for Karl's students :Slug: tutorial :Tags: Manual This example is from the tutorial: A picture for Karl's students. | Author: Till Tantau | Source: The PGF/TikZ manual \end{comment} \begin{center} یافتن نسبت‌های مثلثاتی مجموع دو زاویه مانند $ \alpha $ و $ \beta $ بر اساس روش بوزجانی \end{center} \begin{tikzpicture}[scale=2.7,cap=round] % Local definitions \def\costhirty{0.8660256} % Colors \colorlet{anglecolor}{green!50!black} \colorlet{sincolor}{red} \colorlet{tancolor}{orange!80!black} \colorlet{coscolor}{blue} % Styles \tikzstyle{axes}=[] \tikzstyle{important line}=[very thick] \tikzstyle{information text}=[rounded corners,fill=red!10,inner sep=1ex] % The graphic \draw[style=help lines,step=0.5cm] (-1.4,-1.4) grid (1.4,1.4); \draw (0,0) circle (1cm); \coordinate (O) at(0,0); \coordinate (A) at(50:1cm); \coordinate (F) at(20:1cm); \coordinate (B) at(80:1cm); \coordinate (C) at(120:1cm); \coordinate (E) at(160:1cm); \draw[style=important line,thin](0,0)node[ below] {$O$}--(50:1cm)node[ above right]{$A$} ; \draw[style=important line,thin](0,0)--(80:1cm)node[ above]{$B$} ; \draw[style=important line,thin](0,0)--(120:1cm)node[ above ]{$C$} ; \draw[fill=green!30] (0,0) -- (50:.1cm) arc (50:80:.1cm); \draw (65:2mm) node[anglecolor] {$\alpha$}; \draw[fill=green!30] (0,0) -- (80:.15cm) arc (80:120:.15cm); \draw (100:2mm) node[anglecolor] {$\beta$}; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \tkzRightAngle[color=blue](B/HA/O); % \draw[style=important line,sincolor] (30:1cm) -- node[left=1pt,fill=white] {$\sin \alpha$} +(0,-.5); % \draw[style=important line,coscolor] (0,0) -- node[below=2pt,fill=white] {$\cos \alpha$} (\costhirty,0); \draw[xshift=1.85cm] node [right,text width=8cm,style=information text] { \begin{flushright} \rl{در دایره واحد دو زاویه $ \alpha $ و $ \beta $ رابه مرکز مبدأ پهلوی هم رسم می‌کنیم. حالتی را در نظر می‌گیریم که $ \alpha + \beta $ حاده باشد . } \end{flushright} }; \end{tikzpicture} %%%%%%%%%%%%%%%%%%222222222222222222222%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{tikzpicture}[scale=2.7,cap=round] % Local definitions \def\costhirty{0.8660256} % Colors \colorlet{anglecolor}{green!50!black} \colorlet{sincolor}{red} \colorlet{tancolor}{orange!80!black} \colorlet{coscolor}{blue} % Styles \tikzstyle{axes}=[] \tikzstyle{important line}=[very thick] \tikzstyle{information text}=[rounded corners,fill=red!10,inner sep=1ex] % The graphic \draw[style=help lines,step=0.5cm] (-1.4,-1.4) grid (1.4,1.4); \draw (0,0) circle (1cm); \coordinate (O) at(0,0); \coordinate (A) at(50:1cm); \coordinate (F) at(20:1cm); \coordinate (B) at(80:1cm); \coordinate (C) at(120:1cm); \coordinate (E) at(160:1cm); \draw[style=important line,thin](0,0)node[ below] {$O$}--(50:1cm)node[ above right]{$A$} ; \draw[style=important line,thin](0,0)--(80:1cm)node[ above]{$B$} ; \draw[style=important line,thin](0,0)--(120:1cm)node[ above ]{$C$} ; \draw[style=important line,thin,tancolor](0,0)--(20:1cm)node[ right ]{$F$} ; \draw[style=important line,thin,tancolor](0,0)--(160:1cm)node[above left ]{$E$} ; \draw[fill=green!30] (0,0) -- (50:.1cm) arc (50:80:.1cm); \draw (65:2mm) node[anglecolor] {$\alpha$}; \draw[fill=green!30] (0,0) -- (80:.15cm) arc (80:120:.15cm); \draw (100:2mm) node[anglecolor] {$\beta$}; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \draw[style=important line,sincolor,thin] (F) --(B)--(E) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \draw[fill=black] ($(B)!.5!(F)$) circle (.2pt) node[anchor=north ] {$H$}; \draw[fill=black] ($(B)!.5!(E)$) circle (.2pt) node[anchor= north ] {$K$}; \coordinate (H) at (intersection of B--F and A--O); \coordinate (K) at (intersection of B--E and C--O); \tkzMarkRightAngles[scale=.25](B,H,O O,K,B); \tkzMarkSegments[mark=||,color=red](H,F H,B) \tkzMarkSegments[mark=|,color=red](E,K K,B) % \tkzRightAngle[color=blue](B/HA/O); % \draw[style=important line,sincolor] (30:1cm) -- node[left=1pt,fill=white] {$\sin \alpha$} +(0,-.5); % \draw[style=important line,coscolor] (0,0) -- node[below=2pt,fill=white] {$\cos \alpha$} (\costhirty,0); \draw[xshift=1.85cm] node [right,text width=8cm,style=information text] { \begin{flushright} \rl{ از $ B $ به $ OC $ و $ OA $ عمودی رسم می‌کنیم تا دایره را به ترتیب در نقاط $ E $ و $ F $ قطع کنند. چون $ OB=OF=OE=R=1 $ پس مثلث‌های $ \triangle OBF $ و $\triangle OBE $ متساوی الساقین هستند. در مثلث قائم الزاویه $ \triangle OBH $ داریم که }\end{flushright} $ \displaystyle \sin \alpha = \frac{BH}{OB}\Rightarrow BH=\sin \alpha $ \\ $ \displaystyle \sin \beta = \frac{BK}{OB} \Rightarrow BK=\sin \beta $ }; \end{tikzpicture} %٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪33333333333333333333333333333 \begin{tikzpicture}[scale=2.7,cap=round] % Local definitions \def\costhirty{0.8660256} % Colors \colorlet{anglecolor}{green!50!black} \colorlet{sincolor}{red} \colorlet{tancolor}{orange!80!black} \colorlet{coscolor}{blue} % Styles \tikzstyle{axes}=[] \tikzstyle{important line}=[very thick] \tikzstyle{information text}=[rounded corners,fill=red!10,inner sep=1ex] % The graphic \draw[style=help lines,step=0.5cm] (-1.4,-1.4) grid (1.4,1.4); \draw (0,0) circle (1cm); \coordinate (O) at(0,0); \coordinate (A) at(50:1cm); \coordinate (F) at(20:1cm); \coordinate (B) at(80:1cm); \coordinate (C) at(120:1cm); \coordinate (E) at(160:1cm); \draw[style=important line,thin](0,0)node[ below] {$O$}--(50:1cm)node[ above right]{$A$} ; \draw[style=important line,thin](0,0)--(80:1cm)node[ above]{$B$} ; \draw[style=important line,thin](0,0)--(120:1cm)node[ above ]{$C$} ; \draw[style=important line,thin,tancolor](0,0)--(20:1cm)node[ right ]{$F$} ; \draw[style=important line,thin,tancolor](0,0)--(160:1cm)node[above left ]{$E$} ; \draw[fill=green!30] (0,0) -- (50:.1cm) arc (50:80:.1cm); \draw (65:2mm) node[anglecolor] {$\alpha$}; \draw[fill=green!30] (0,0) -- (80:.15cm) arc (80:120:.15cm); \draw (100:2mm) node[anglecolor] {$\beta$}; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \draw[style=important line,sincolor,thin] (F) --(B)--(E) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \draw[fill=black] ($(B)!.5!(F)$) circle (.2pt) node[anchor=north ] {$H$}; \draw[fill=black] ($(B)!.5!(E)$) circle (.2pt) node[anchor= north ] {$K$}; \coordinate (H) at (intersection of B--F and A--O); \coordinate (K) at (intersection of B--E and C--O); \coordinate[label=200:$M$] (M) at (200:1cm); \draw [style=dashed, color=black] (K)--(H); \draw [style=dashed, color=black] (E)--(F); \tkzMarkRightAngles[scale=.25](B,H,O O,K,B); \tkzMarkSegments[mark=||,color=red](H,F H,B) \tkzMarkSegments[mark=|,color=red](E,K K,B) % \tkzRightAngle[color=blue](B/HA/O); % \draw[style=important line,sincolor] (30:1cm) -- node[left=1pt,fill=white] {$\sin \alpha$} +(0,-.5); % \draw[style=important line,coscolor] (0,0) -- node[below=2pt,fill=white] {$\cos \alpha$} (\costhirty,0); \draw[xshift=1.85cm] node [right,text width=8cm,style=information text] { \begin{flushright} \rl{ با رسم پاره خط‌های $ EF $ و $ KH $ با استفاده از قضیه تالس نتیجه می‌گیریم این دو پاره‌خط موازیند و $ KH $ نصف $ EF $ است. }\end{flushright} }; \end{tikzpicture} %٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪44444444444444444444444444444444 \begin{tikzpicture}[scale=2.7,cap=round] % Local definitions \def\costhirty{0.8660256} % Colors \colorlet{anglecolor}{green!50!black} \colorlet{sincolor}{red} \colorlet{tancolor}{orange!80!black} \colorlet{coscolor}{blue} % Styles \tikzstyle{axes}=[] \tikzstyle{important line}=[very thick] \tikzstyle{information text}=[rounded corners,fill=red!10,inner sep=1ex] % The graphic \draw[style=help lines,step=0.5cm] (-1.4,-1.4) grid (1.4,1.4); \draw (0,0) circle (1cm); \coordinate (O) at(0,0); \coordinate (A) at(50:1cm); \coordinate (F) at(20:1cm); \coordinate (B) at(80:1cm); \coordinate (C) at(120:1cm); \coordinate (E) at(160:1cm); \draw[style=important line,thin](0,0)node[ below] {$O$}--(50:1cm)node[ above right]{$A$} ; \draw[style=important line,thin](0,0)--(80:1cm)node[ above]{$B$} ; \draw[style=important line,thin](0,0)--(120:1cm)node[ above ]{$C$} ; \draw[style=important line,thin,tancolor](0,0)--(20:1cm)node[ right ]{$F$} ; \draw[style=important line,thin,tancolor](0,0)--(160:1cm)node[above left ]{$E$} ; \draw[fill=green!30] (0,0) -- (50:.1cm) arc (50:80:.1cm); \draw (65:2mm) node[anglecolor] {$\alpha$}; \draw[fill=green!30] (0,0) -- (80:.15cm) arc (80:120:.15cm); \draw (100:2mm) node[anglecolor] {$\beta$}; \draw[fill=green!30] (0,0) -- (20:.1cm) arc (20:50:.1cm); \draw (35:2mm) node[anglecolor] {$\alpha$}; \draw[fill=green!30] (0,0) -- (120:.15cm) arc (120:160:.15cm); \draw (140:2mm) node[anglecolor] {$\beta$}; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \draw[style=important line,sincolor,thin] (F) --(B)--(E) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \draw[fill=black] ($(B)!.5!(F)$) circle (.2pt) node[anchor=north ] {$\scriptstyle H$}; \draw[fill=black] ($(B)!.5!(E)$) circle (.2pt) node[anchor= north ] {$\scriptstyle K$}; \coordinate (H) at (intersection of B--F and A--O); \coordinate (K) at (intersection of B--E and C--O); \coordinate[label=200:$M$] (M) at (200:1cm); \draw [style=dashed, color=black] (O)--(M); \draw [style=dashed, color=black] (E)--(M); \draw [style=dashed, color=black] (E)--(F); \tkzMarkRightAngles[scale=.25](B,H,O O,K,B); \tkzMarkRightAngles[scale=.25](M,E,F); \tkzMarkAngle[fill=green!25,mkpos=.2, size=0.2](O,M,E) \tkzLabelAngle[pos=.3](O,M,E){$\scriptstyle \alpha + \beta$} \tkzMarkSegments[mark=||,color=red](H,F H,B) \tkzMarkSegments[mark=|,color=red](E,K K,B) % \tkzRightAngle[color=blue](B/HA/O); % \draw[style=important line,sincolor] (30:1cm) -- node[left=1pt,fill=white] {$\sin \alpha$} +(0,-.5); % \draw[style=important line,coscolor] (0,0) -- node[below=2pt,fill=white] {$\cos \alpha$} (\costhirty,0); \draw[xshift=1.85cm] node [right,text width=8cm,style=information text] { \begin{flushright} \rl{ $ FM $ قطر دایره است بنابراین زاویه $\widehat{FEM} = 90\,^{\circ} $ است . طبق زاویه‌ی محاطی داریم $ \widehat{OME}=\alpha + \beta $ بنابراین }\end{flushright} $ \displaystyle \Rightarrow \sin\, \left(\alpha + \beta \right)= \frac{EF}{MF}\Rightarrow EF=2\, \sin \left( \alpha + \beta \right)$ \\ $ \displaystyle \Rightarrow KH=\frac{EF}{2}=\sin \left( \alpha + \beta \right) $ }; \end{tikzpicture} %٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪٪555555555555555555555555555555555 \begin{tikzpicture}[scale=2.7,cap=round] % Local definitions \def\costhirty{0.8660256} % Colors \colorlet{anglecolor}{green!50!black} \colorlet{sincolor}{red} \colorlet{tancolor}{orange!80!black} \colorlet{coscolor}{blue} % Styles \tikzstyle{axes}=[] \tikzstyle{important line}=[very thick] \tikzstyle{information text}=[rounded corners,fill=red!10,inner sep=1ex] % The graphic \draw[style=help lines,step=0.5cm] (-1.4,-1.4) grid (1.4,1.4); \coordinate (F) at(20:1cm); \coordinate (B) at(80:1cm); \coordinate (E) at(160:1cm); \tkzDrawSegments[color=magenta, line width=3pt, opacity=0.4](F,B E,F B,E) \tkzLabelSegment[below=3ex,pos=.55](F,E){$\sin \left( \alpha + \beta \right)$} \tkzLabelSegment[above=2ex,pos=.5](B,F){$\sin \alpha $} \tkzLabelSegment[above=1ex,pos=.5](B,E){$\sin \beta $} \tkzLabelPoint[above](B){$B$} \tkzLabelPoint[left](E){$K$} \tkzLabelPoint[right](F){$H$} \tkzDefPointBy[projection=onto E--F](B) \tkzGetPoint{P} \tkzDrawSegment[style=dashed,color=orange](B,P) \tkzLabelPoint[below](P){$P$} \tkzMarkRightAngles[scale=.25](E,P,B B,P,F) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \draw[style=important line,thin] (F) --(B)--(E) --cycle ; \tkzMarkAngle[fill=green!25,mkpos=.2, size=0.2](F,E,B) \tkzLabelAngle[pos=.3](F,E,B){$ \alpha$} \tkzMarkAngle[fill=green!25,mkpos=.2, size=0.2](B,F,E) \tkzLabelAngle[pos=.3](B,F,E){$ \beta$} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \draw[xshift=1.85cm] node [right,text width=8cm,style=information text] { $ KP= \cos \alpha \times\sin \beta \,, \, PH= \cos\beta \times \sin \alpha $\\ $KH=KP+PH $ \\ $ \sin \left(\, \alpha + \beta \, \right) = \sin \alpha \times \cos\beta + \cos \alpha \times\sin \beta$ }; \end{tikzpicture} \end{document}