% %ȉ̕RgƂ̂łC_ł͂܂ł܂B
% ̃^Cg
[Title]
405_sincos

% 蕶
% Ȃ΁C[Level1]ɏ̂̂܂ܖ蕶ƂȂB
[Problem]
̖₢ɓB
% tHg̑傫B1`10C܂TeX̃R}hw肷B
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[usepackage]
\usepackage{color,amssymb}
\usepackage{graphicx}
\usepackage{pxrubrica}

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% Level1̖BȉLevel7܂œlB
% 1sڂɂ͏ڍאݒ̃^CgB
% 2sڈȍ~ɖƂ̉𓚂B
% Ɖ𓚁C𓚂Ɩ͂PsďB
% vZߒꍇ́CƉ𓚂̊ԂɂPsԊuC
% ŏprocessƂsC̎̍svZߒĂB
[Level1]
Op
}̒pOp`ŁC$ \sin 30$C$ \cos 30$C$ \tan 30$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic23.tex}
\end{center}

$ \sin 30=\displaystyle \frac{1}{2}$C$ \cos 30=\displaystyle \frac{\sqrt{3}}{2}$C
$ \tan 30=\displaystyle \frac{1}{\sqrt{3}}$

}̒pOp`ŁC$ \sin 45$C$ \cos 45$C$ \tan 45$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic24.tex}
\end{center}

$ \sin 45=\displaystyle \frac{1}{\sqrt{2}}$C$ \cos 45=\displaystyle \frac{1}{\sqrt{2}}$C
$ \tan 45=1$

}̒pOp`ŁC$ \sin 60$C$ \cos 60$C$ \tan 60$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic25.tex}
\end{center}

$ \sin 60=\displaystyle \frac{\sqrt{3}}{2}$C$ \cos 60=\displaystyle \frac{1}{2}$C
$ \tan 60=\sqrt{3}$

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic02.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{1}{\sqrt{5}}$C$ \cos \theta=\displaystyle \frac{2}{\sqrt{5}}$C
$ \tan \theta=\displaystyle \frac{1}{2}$

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic03.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{2}{\sqrt{29}}$C$ \cos \theta=\displaystyle \frac{5}{\sqrt{29}}$C
$ \tan \theta=\displaystyle \frac{2}{5}$

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic06.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{4}{5}$C$ \cos \theta=\displaystyle \frac{3}{5}$C
$ \tan \theta=\displaystyle \frac{4}{3}$

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic09.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{8}{17}$C$ \cos \theta=\displaystyle \frac{15}{17}$C
$ \tan \theta=\displaystyle \frac{8}{15} $

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic10.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{2}{3}$C$ \cos \theta=\displaystyle \frac{\sqrt{5}}{3}$C
$ \tan \theta=\displaystyle \frac{2}{\sqrt{5}} $

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic11.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{5}{13}$C$ \cos \theta=\displaystyle \frac{12}{13}$C
$ \tan \theta=\displaystyle \frac{5}{12} $

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic12.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{3}{\sqrt{10}}$C$ \cos \theta=\displaystyle \frac{1}{\sqrt{10}}$C
$ \tan \theta=3 $

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic15.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{\sqrt{7}}{4}$C$ \cos \theta=\displaystyle \frac{3}{4}$C
$ \tan \theta=\displaystyle \frac{\sqrt{7}}{3} $

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic04.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{8}{17}$C$ \cos \theta=\displaystyle \frac{15}{17}$C
$ \tan \theta=\displaystyle \frac{8}{15}$

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic05.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{12}{13}$C$ \cos \theta=\displaystyle \frac{5}{13}$C
$ \tan \theta=\displaystyle \frac{12}{5}$

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic07.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{2\sqrt{2}}{3}$C$ \cos \theta=\displaystyle \frac{1}{3}$C
$ \tan \theta=2 \sqrt{2}$

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic08.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{2}{\sqrt{5}}$C$ \cos \theta=\displaystyle \frac{1}{\sqrt{5}}$C
$ \tan \theta=2 $

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic13.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{\sqrt{11}}{6}$C$ \cos \theta=\displaystyle \frac{5}{6}$C
$ \tan \theta=\displaystyle \frac{\sqrt{11}}{5} $

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic16.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{2\sqrt{6}}{7}$C$ \cos \theta=\displaystyle \frac{5}{7}$C
$ \tan \theta=\displaystyle \frac{2\sqrt{6}}{5} $

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic17.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{3}{\sqrt{13}}$C$ \cos \theta=\displaystyle \frac{2}{\sqrt{13}}$C
$ \tan \theta=\displaystyle \frac{3}{2} $

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic18.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{2}{\sqrt{13}}$C$ \cos \theta=\displaystyle \frac{3}{\sqrt{13}}$C
$ \tan \theta=\displaystyle \frac{2}{3} $

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic14.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{\sqrt{11}}{6}$C$ \cos \theta=\displaystyle \frac{5}{6}$C
$ \tan \theta=\displaystyle \frac{\sqrt{11}}{5} $

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic19.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{3}{5}$C$ \cos \theta=\displaystyle \frac{4}{5}$C
$ \tan \theta=\displaystyle \frac{3}{4}$

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic20.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{3}{\sqrt{34}}$C$ \cos \theta=\displaystyle \frac{5}{\sqrt{34}}$C
$ \tan \theta=\displaystyle \frac{3}{5} $

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic21.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{5}{6}$C$ \cos \theta=\displaystyle \frac{\sqrt{11}}{6}$C
$ \tan \theta=\displaystyle \frac{5}{\sqrt{11}} $

}̒pOp`ŁC$ \sin \theta$C$ \cos \theta$C$ \tan \theta$߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic22.tex}
\end{center}

$ \sin \theta=\displaystyle \frac{\sqrt{65}}{9}$C$ \cos \theta=\displaystyle \frac{4}{9}$C
$ \tan \theta=\displaystyle \frac{\sqrt{65}}{4}$








[Level2]
Oppĕ\
}̒pOp`ŁC̒$x$Cp$\theta$̎Opŕ\B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic301.tex}
\end{center}

process
$\displaystyle\frac{x}{2}=\sin \theta$\\
$\therefore x=2\sin\theta$

$2\sin\theta$

}̒pOp`ŁC̒$x$Cp$\theta$̎Opŕ\B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic302.tex}
\end{center}

process
$\displaystyle\frac{x}{3}=\tan \theta$\\
$\therefore x=3\tan\theta$

$3\tan\theta$

}̒pOp`ŁC̒$x$Cp$\theta$̎Opŕ\B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic303.tex}
\end{center}

process
$\displaystyle\frac{x}{2}=\cos \theta$\\
$\therefore x=2\cos\theta$

$2\cos\theta$

}̒pOp`ŁC̒$x$Cp$\theta$̎Opŕ\B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic304.tex}
\end{center}

process
$\displaystyle\frac{x}{3}=\cos \theta$\\
$\therefore x=3\cos\theta$

$3\cos\theta$

}̒pOp`ŁC̒$x$Cp$\theta$̎Opŕ\B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic305.tex}
\end{center}

process
$\displaystyle\frac{x}{1}=\sin \theta$\\
$\therefore x=1\sin\theta$

$\sin\theta$

}̒pOp`ŁC̒$x$Cp$\theta$̎Opŕ\B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic306.tex}
\end{center}

process
$\displaystyle\frac{x}{1}=\tan \theta$\\
$\therefore x=1\tan\theta$

$\tan\theta$

}̎Op`ŁC$\sin A=\displaystyle\frac{1}{3}$C$c=12$̂ƂCBC̒߂B\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic401.tex}
\end{center}

process
$\displaystyle\frac{\rm BC}{12}=\displaystyle\frac{1}{3}$\\
${\rm BC}=12 \times \displaystyle\frac{1}{3}=4$

$4$

}̎Op`ŁC$\tan A=\displaystyle\frac{1}{2}$C$b=4$̂ƂCBC̒߂B\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic402.tex}
\end{center}

process
$\displaystyle\frac{\rm BC}{4}=\displaystyle\frac{1}{2}$\\
${\rm BC}=4 \times \displaystyle\frac{1}{2}=2$

$2$

}̎Op`ŁC$\cos A=\displaystyle\frac{1}{\sqrt{2}}$C$c=5$̂ƂCAC̒߂B\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic403.tex}
\end{center}

process
$\displaystyle\frac{\rm AC}{5}=\displaystyle\frac{1}{2}$\\
${\rm AC}=5 \times \displaystyle\frac{1}{\sqrt{2}}=\displaystyle\frac{5}{\sqrt{2}}=\displaystyle\frac{5\sqrt{2}}{2}$

$\displaystyle\frac{5\sqrt{2}}{2}$













[Level3]
{IȎOp̒li180܂Łj
̕\̋󗓂𖄂߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic_basic180s.tex}
\end{center}
\begin{tabular}{|c|c|c|c|c|c|} 
\hline
@\textcircled{\small 6}@& @\textcircled{\small 5}@ & @\textcircled{\small 4}@ & @\textcircled{\small 3}@ & @\textcircled{\small 2}@ & @\textcircled{\small 1}@ \\ \hline
   &    &    &    &    & \\
   &    &    &    &    & \\
   &    &    &    &    & \\
\hline
\end{tabular}

\begin{tabular}{|c|c|c|c|c|c|} 
\hline
@\textcircled{\small 6}@& @\textcircled{\small 5}@ & @\textcircled{\small 4}@ & @\textcircled{\small 3}@ & @\textcircled{\small 2}@ & @\textcircled{\small 1}@ \\ \hline
 $\displaystyle \frac{1}{2}$  & $\displaystyle \frac{\sqrt{2}}{2}$ & $\displaystyle \frac{\sqrt{3}}{2}$ & $\displaystyle \frac{\sqrt{3}}{2}$ &  $\displaystyle \frac{\sqrt{2}}{2}$ & $\displaystyle \frac{1}{2}$\\
   &    &    &    &    & \\
\hline
\end{tabular}

̕\̋󗓂𖄂߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic_basic180c.tex}
\end{center}
\begin{tabular}{|c|c|c|c|c|c|} 
\hline
@\textcircled{\scriptsize 12}@& @\textcircled{\scriptsize 11}@ & @\textcircled{\scriptsize 10}@ & @\textcircled{\small 9}@ & @\textcircled{\small 8}@ & @\textcircled{\small 7}@ \\ \hline
   &    &    &    &    & \\
   &    &    &    &    & \\
   &    &    &    &    & \\
\hline
\end{tabular}

\begin{tabular}{|c|c|c|c|c|c|} 
\hline
@\textcircled{\scriptsize 12}@& @\textcircled{\scriptsize 11}@ & @\textcircled{\scriptsize 10}@ & @\textcircled{\small 9}@ & @\textcircled{\small 8}@ & @\textcircled{\small 7}@ \\ \hline
 $-\displaystyle \frac{\sqrt{3}}{2}$  & $-\displaystyle \frac{\sqrt{2}}{2}$ & $-\displaystyle \frac{1}{2}$ & $\displaystyle \frac{1}{2}$ &  $\displaystyle \frac{\sqrt{2}}{2}$ & $\displaystyle \frac{\sqrt{3}}{2}$\\
   &    &    &    &    & \\
\hline
\end{tabular}














[Level4]
30C45C60
$\cos ^2 30^\circ\times\tan ^2 30^\circ$̒l$\displaystyle\frac{\fbox{E}}{\fbox{\,G\,}}$łB

process
$\left( \displaystyle\frac{\sqrt{3}}{2} \right)^2\times\left( \displaystyle\frac{1}{\sqrt{3}} \right)^2=\displaystyle\frac{3}{4}\times\displaystyle\frac{1}{3} =\displaystyle\frac{1}{4}$

$\displaystyle\frac{1}{4}$

̎̒l߂ȂB\\
$\sin 30-\cos 60+\tan 45$

process
$\displaystyle \frac{1}{2}-\displaystyle \frac{1}{2}+1$

$1$

̎̒l߂ȂB\\
$\sin 60\times \cos 30\times \tan 45$

$\displaystyle \frac{3}{4}$

̎̒l߂ȂB\\
$\sin 30+ \cos 60$

process
$\displaystyle \frac{1}{2}+\displaystyle \frac{1}{2}$

$1$

̎̒l߂ȂB\\
$\tan 45- \cos 60$

process
$1-\displaystyle \frac{1}{2}$

$\displaystyle \frac{1}{2}$

̎̒l߂ȂB\\
$\sin 30+ \cos 60+\tan 45$

process
$\displaystyle \frac{1}{2}+\displaystyle \frac{1}{2}+1$

$2$

̎̒l߂ȂB\\
$\tan 45\sin 60$

process
$1 \times \displaystyle \frac{\sqrt{3}}{2}$

$\displaystyle \frac{\sqrt{3}}{2}$

̎̒l߂ȂB\\
$4(\sin 30)(\cos 45)$

process
$4 \times \displaystyle \frac{1}{2} \times \displaystyle \frac{\sqrt{2}}{2}$

$\sqrt{2}$

̎̒l߂ȂB\\
$(1+\sin 45+\sin 30)(1-\cos 45+\cos 60)$

process
$\left( 1+\displaystyle \frac{1}{\sqrt{2}}+\displaystyle \frac{1}{2} \right)
\left( 1-\displaystyle \frac{1}{\sqrt{2}}+\displaystyle \frac{1}{2} \right)
=\left( \displaystyle \frac{3}{2}+\displaystyle \frac{1}{\sqrt{2}} \right)
\left( \displaystyle \frac{3}{2}-\displaystyle \frac{1}{\sqrt{2}} \right)$

$\displaystyle \frac{7}{4}$

}̎Op`ABCɂāC\\ $\angle A=45,\,\angle B=30,\,$ BC$=6cm$\\ łB
̂ƂAC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic237.tex}
\end{center}

process
_CAB։̑HƂƁC\\
CH:CB$=1:2$CCH=3 cmCCH:AC=$1:\sqrt{2}$CCH=$3 \sqrt{2}$\\
$\displaystyle\frac{\rm AC}{\sin 30^\circ}=\displaystyle\frac{6}{\sin 45^\circ}$\\
$\displaystyle\frac{\rm AC}{1/2}=\displaystyle\frac{6}{\sqrt{2}/2}$\\
${\rm AC}= \displaystyle\frac{6}{\sqrt{2}}=3\sqrt{2}$

$3 \sqrt{2} cm$

}̎Op`ABCɂāC\\ $\angle A=30,\,\angle B=45,\,$ BC$=4cm$\\ łB
̂ƂAC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic205.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic205p.tex}
\end{center}
$\displaystyle \frac{4}{\sqrt{2}} \times 2=4 \sqrt{2}$

$4 \sqrt{2} cm$

̎̒l߂ȂB\\
$\sin 30^\circ  \cos 30 ^\circ$

process
$\displaystyle\frac{1}{2}\times \displaystyle\frac{\sqrt{3}}{2}$

$\displaystyle\frac{\sqrt{3}}{4}$

̎̒l߂ȂB\\
$\left( \sin 60^\circ + \cos 45 ^\circ\right)\left( \cos 30^\circ - \sin 45 ^\circ\right)$

process
$\left( \displaystyle\frac{\sqrt{3}}{2} + \displaystyle\frac{1}{\sqrt{2}}\right) \left( \displaystyle\frac{\sqrt{3}}{2} - \displaystyle\frac{1}{\sqrt{2}}\right)$\\
$=\left( \displaystyle\frac{\sqrt{3}}{2} \right)^2 - \left( \displaystyle\frac{1}{\sqrt{2}}\right)^2=\displaystyle\frac{1}{4}$

$\displaystyle\frac{1}{4}$

$\cos 45^\circ\times\tan 45^\circ$̒l
$\displaystyle\frac{1}{\sqrt{\fbox{\,E\,}}}$łB

process
$\cos 45^\circ=\displaystyle\frac{1}{\sqrt{2}}$C
$\tan 45^\circ=1$ł邩C\\
$\cos 45^\circ\times\tan 45^\circ =\displaystyle\frac{1}{\sqrt{2}} \times 1=\displaystyle\frac{1}{\sqrt{2}}$

2

̎̒l߂B\\
$\sin {30}^\circ \cos {60}^\circ +\sin {60}^\circ\cos {30}^\circ $

process
$\displaystyle\frac{1}{2}\times\displaystyle\frac{1}{2}+\displaystyle\frac{\sqrt{3}}{2}\times \displaystyle\frac{\sqrt{3}}{2}=1$

$1$

̎̒l߂B\\
$\sin 30+ \cos 60-\tan 45$

process
$\displaystyle \frac{1}{2}+\displaystyle \frac{1}{2}-1=0$

$0$

̎̒l߂B\\
$\sin 45\times \cos 45-\tan 60\times \tan 30$

process
$\displaystyle \frac{1}{\sqrt{2}} \times \displaystyle \frac{1}{\sqrt{2}} 
-\sqrt{3}\times \displaystyle \frac{1}{\sqrt{3}}=-\displaystyle \frac{1}{2} $

$-\displaystyle \frac{1}{2} $

$\sin{30}^{\circ} \cos{60}^{\circ}+\cos{30}^{\circ} \sin{60}^{\circ}$
̒l\fbox{\bf \hspace{2pt} E \hspace{2pt}}łB

process
$\displaystyle\frac{1}{2} \times \displaystyle\frac{1}{2}+\displaystyle\frac{\sqrt{3}}{2}\times \displaystyle\frac{\sqrt{3}}{2}=1$

$1$





[Level5]
Op̗p
}̂悤ɁCXΊp20ŐAB$10m$̃GXJ[^[B
̂ƂC̃GXJ[^[ŏ鍂BC͂悻$m$B
1ʂ܂ŋ߂B
Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 20=0.3420,\, \cos 20=0.9397,\, \\ \tan 20=0.3640$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic203.tex}
\end{center}

process
$\tan 20=\displaystyle \frac{x}{10}$,\\
$x=10\times \tan 20=10 \times 0.3640=3.64 \fallingdotseq 3.6$

$3.6m$

}̂悤Ȑ쌴Bh̎Ζʂ̊px40ŁCΖʂ̒$5m$łƂC
͐~i񂵂jy܂ł̍AB͂悻$m$B1ʂ܂ŋ߂B
Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 40=0.6428,\, \cos 40=0.7660,\, \\ \tan 40=0.8391$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic211.tex}
\end{center}

process
$\sin 40=\displaystyle \frac{y}{5}$,\\
$y=5\times \sin 40=5 \times 0.6428=3.214 \fallingdotseq 3.2$

$3.2m$

}̂悤ȁCr̒$2m$̋riႽjB
Jr̊Ԋu$1.2m$łƂCrƒnʂ̂p$\theta$̑傫́C
\textcircled{\small 1}`\textcircled{\small 4}̂ǂ͈̔͂ɂ邩B
łK؂Ȃ̂IׁB\\
\begin{tabbing}
@@@@@@@@@@@\=@@@@@@@@@@@\= \\ 
\textcircled{\small 1} $71ȏ72$ \>\textcircled{\small 2} $72ȏ73$\\
\textcircled{\small 3} $73ȏ74$ \>\textcircled{\small 4} $74ȏ75$
\end{tabbing}
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic210.tex}
\end{center}
Kvł΁C̎Op̕\𗘗p邱ƁB\\
\begin{table}[!h]
\begin{center}
\begin{tabular}{c|ccc}
 \hline
 p & ($\sin$) & ]($\cos$) & ($\tan$) \\
 \hline
 71 & 0.9455 & 0.3256 & 2.9042 \\
 \hline
 72 & 0.9511 & 0.3090 & 3.0777 \\
 \hline
 73 & 0.9563 & 0.2924 & 3.2709 \\
 \hline
 74 & 0.9613 & 0.2756 & 3.4874 \\
 \hline
 75 & 0.9659 & 0.2588 & 3.7321 \\
 \hline
\end{tabular}
\end{center}
\end{table}

process
$\cos \theta =\displaystyle \frac{0.6}{2}=0.3$\\
\$\cos \theta$̗ɂC$72ȏ73$

\textcircled{\small 2}

ɐ}̂悤ȃuRCx̍AB$3.5m$C
ij̒AC$3.0m$łB
̃uRɏĐ悭Ƃ
$\angle $BAC$33$ɂȂB
̂ƂCݔ̒nʂ̍CD͂悻$m$B
1ʂ܂ŋ߂BKvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 33=0.5446, \ \cos 33=0.8387,\\
\tan 33=0.6494$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic218.tex}
\end{center}

process
$\cos 33=\displaystyle \frac{3.5-x}{3}$\\
$3.5-x=3 \cos 33=3 \times 0.8387=2.5161$\\
$x=3.5-2.516 \fallingdotseq 1.0$

$1.0$

鏭N\ruby{}{}\ruby{g}{}ƂC}̂悤ɁC̒AB$30 $mCƐ̂Ȃp$41$łB
N̎肪Cnʂ$1.4$m̍ɂƂC̒nʂ̍BC͂悻mBlœB
Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 41=0.6561,\, \cos 41=0.7547,\\
\tan 41=0.8693$\\
Cɂ݂͂ȂCɒĂ̂ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic224.tex}
\end{center}

process
BC̒$x$ƂƁC$\sin 41=\displaystyle \frac{x-1.4}{30}$\\
$x-1.4=30 \times \sin 41=30 \times 0.6561=19.683$\\
$x=21.083 \fallingdotseq 21$

$21$m

}̂悤ȍ⓹B
̍⓹̌XΊp$\theta$̑傫͎\textcircled{\small 1}`\textcircled{\small 4}̂ǂ͈̔͂ɂ邩B
łK؂Ȃ̂IׁB\\
\begin{tabbing}
@@@@@@@@@@@\=@@@@@@@@@@@\= \\ 
\textcircled{\small 1} $15ȏ16$ \>\textcircled{\small 2} $16ȏ17$\\
\textcircled{\small 3} $17ȏ18$ \>\textcircled{\small 4} $18ȏ19$
\end{tabbing}
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic230.tex}
\end{center}
Kvł΁C̎Op̕\𗘗p邱ƁB\\
\begin{table}[!h]
\begin{center}
\begin{tabular}{c|ccc}
 \hline
 p & ($\sin$) & ]($\cos$) & ($\tan$) \\
 \hline
 15 & 0.2588 & 0.9659 & 0.2679 \\
 \hline
 16 & 0.2756 & 0.9613 & 0.2867 \\
 \hline
 17 & 0.2924 & 0.9563 & 0.3057 \\
 \hline
 18 & 0.3090 & 0.9511 & 0.3249 \\
 \hline
 19 & 0.3256 & 0.9455 & 0.3443 \\
 \hline
\end{tabular}
\end{center}
\end{table}

process
$\cos \theta =\displaystyle \frac{19}{20}=0.95$\\
\C$18ȏ19$

\textcircled{\small 4}

̖₢ɓB
Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 35=0.5736$C$\cos 35=0.8192$C\\
$\tan 35=0.7002$\\
@̐}̂悤ɁC錚̍𑪂邽߁Č琅$20m$ꂽn_ŁC
̏[グp𑪂Ƃ35łB\\
@ڂ̍1.7m̂ƂC̍AB͂悻mB
\textcircled{\small 1}`\textcircled{\small 4}̂łKȂ̂IׁB\\
\textcircled{\small 1}13.2mC@\textcircled{\small 2}14.0mC@\textcircled{\small 3}15.7mC@\textcircled{\small 4}18.1m\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic235.tex}
\end{center}

process
$AB-1.7=20 \times \tan 35\\
=20 \times 0.7002=14.004\\
\,AB=14.001+1.7=15.701$

\textcircled{\small 3}

̂悤Ȃ苴B
̂苴̒̎x璣ꂽC[ABCAC̒͂ꂼ70młC
̃C[Ƌ̓Hʂ̂p40łB
̂ƂC̒BC͎\textcircled{\small 1}`\textcircled{\small 4}̂ǂ͈̔͂ɂ邩BłKȂ̂IׁB
CC[͒ɒĂ̂ƂB
Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 40=0.6428$C$\cos 40=0.7660$C\\
$\tan 40=0.8391$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic213.tex}
\end{center}
\textcircled{\small 1}$80$mȏ$90$m\\
\textcircled{\small 2}$90$mȏ$100$m\\
\textcircled{\small 3}$100$mȏ$110$m\\
\textcircled{\small 4}$110$mȏ$120$m\\

process
BC$=2 \times 70 \cos 40=2 \times 70 \times 0.7660\\
=107.24$\\

\textcircled{\small 3}

̐}̂悤ȃTbJ[ŁCR[i[̒n_A{[RƂC
{[S[C20̕40m񂾁B
̂ƂC{[̗n_BS[C܂ł̍ŒZBC͂悻mB
\textcircled{\small 1}`\textcircled{\small 4}̂łK؂Ȃ̂IׁB\\
Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 20=0.3420,\,\cos 20=0.9397,$\\
$\tan 20=0.3640$\\
\textcircled{\small 1} 13.7m~~~
\textcircled{\small 2} 14.6m~~~
\textcircled{\small 3} 18.8m~~~
\textcircled{\small 4} 37.6m
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic221.tex}
\end{center}

process
$\displaystyle\frac{\rm{BC}}{40}=\sin 20$\\
$\rm{BC}=40 \times 0.3420=13.68$

\textcircled{\small 1}

̐}́Cr̉ɐݒuĂ鑾zdplC^猩̂łB
pl̒AB5mŁC$\angle \rm{BAC}$̑傫37łƂCBC͂悻mB
\textcircled{\small 1}`\textcircled{\small 4}̂łK؂Ȃ̂IׁB\\
Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 37=0.6018,\,\cos 37=0.7986,$\\
$\tan 37=0.7536$\\
\textcircled{\small 1} 3.0m~~~
\textcircled{\small 2} 3.4m~~~
\textcircled{\small 3} 3.8m~~~
\textcircled{\small 4} 4.0m
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic223.tex}
\end{center}

process
$\displaystyle\frac{\rm{BC}}{5}=\sin 37$C$\rm{BC}=5 \sin 37=5\times 0.6018=3.009$

\textcircled{\small 1}

̐}̂悤ɁCCnʂ10m̃[vłȂꂽԂŐÎ~ĂB
[v̒[AC̉eB܂ł̋13młƂCzx$\theta$͂悻xB
Kvł΁C̎Op̕\𗘗p邱ƁB\\
\begin{table}[!h]
\begin{center}
\begin{tabular}{c|ccc}
 \hline
 p & ($\sin$) & ]($\cos$) & ($\tan$) \\
 \hline
 36 & 0.5878 & 0.8090 & 0.7265 \\
 \hline
 37 & 0.6018 & 0.7986 & 0.7536 \\
 \hline
 38 & 0.6157 & 0.7880 & 0.7813 \\
 \hline
 39 & 0.6293 & 0.7771 & 0.8098 \\
 \hline
 40 & 0.6428 & 0.7660 & 0.8391 \\
 \hline
\end{tabular}
\end{center}
\end{table}
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic227.tex}
\end{center}
\textcircled{\small 1} 36ȏ37\\
\textcircled{\small 2} 37ȏ38\\
\textcircled{\small 3} 38ȏ39\\
\textcircled{\small 4} 39ȏ40

process
$\tan \theta=\displaystyle\frac{10}{13}=0.7692$\\
$0.7536<0.7692<0.7813$

\textcircled{\small 2}

\hspace{8pt} R̃P[uJ[̋O͐ʂ$22^\circ $̌XɂȂĂB
̃P[uJ[\ruby{[}{ӂ}̉wAƎRwBƂ̒500młƂC
RwB̒n_͘[̉wA̒n_mB\\
C$\sin 22 ^\circ =0.3746$C$\tan 22^\circ=0.4040$ƂB

process
$\displaystyle\frac{h}{500}=\sin 22^\circ$\\
$h=500 \times 0.3746=187.3$

187.3m

\hspace{8pt} 
}̂悤ȃh[ꂪB
n_A璸BグƂC
Ƃ̂Ȃp31$^\circ$ɂȂB
B̐^̒n_Cn_A܂
AC88młB
̂ƂCh[̍BC͂悻mB
\textcircled{\small 1}`\textcircled{\small 4}̂łK؂Ȃ̂IׁB\\
\textcircled{\small 1} 27.3~~~
\textcircled{\small 2} 45.3~~~
\textcircled{\small 3} 52.9~~~
\textcircled{\small 4} 75.4\\
CKvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 31^\circ = 0.5150$C$\cos 31^\circ = 0.8572$C$\tan 31^\circ = 0.6009$
\begin{figure}[!hbt]\begin{center}
  \includegraphics[width=70mm]{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic261.eps}
 \end{center}\end{figure} 

process
$\displaystyle\frac{x}{88}=\tan 31^\circ$\\
$\therefore x=88 \times 0.6009 = 52.8792$

\textcircled{\small 3}

}̂悤An_ŃejX̃T[ułƂC
{[Bn_ɒnB
T[ȗœ_P̍͒nʂ2.8młCABԂ̋16mB
̂ƂC$\angle $PBȂ傫\fbox{\,A\,}łB
C$\angle $PAB$=90^\circ$ƂB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂PIׁB\\
\hspace{8pt}CKvł΁C̎Op̕\𗘗p邱ƁB\\
\begin{table}[htb]
  \begin{tabular}{|c||c|c|c|} \hline
    p & (sin) & ](cos) & (tan) \\ \hline \hline
    $9 ^\circ $ & 0.1564 & 0.9877 & 0.1584 \\ \hline
    $10 ^\circ $ & 0.1736 & 0.9848 & 0.1763 \\ \hline
    $11 ^\circ $ & 0.1908 & 0.9816 & 0.1944 \\ \hline
    $12 ^\circ $ & 0.2079 & 0.9781 & 0.2126 \\ \hline
    $13 ^\circ $ & 0.2250 & 0.9744 & 0.2309 \\ \hline
  \end{tabular}
\end{table} \\
\textcircled{\small 1}\,$9^\circ$ȏ$10^\circ$\\
\textcircled{\small 2}\,$10^\circ$ȏ$11^\circ$\\
\textcircled{\small 3}\,$11^\circ$ȏ$12^\circ$\\
\textcircled{\small 4}\,$12^\circ$ȏ$13^\circ$\\
\begin{figure}[!hbt]\begin{center}
  \includegraphics[width=40mm]{D:/texlive/2018/bin/win32/mathtex/405_sincos/h2702m0501.eps}
 \end{center}\end{figure} 

process
$\tan \angle {\rm PBA} =\displaystyle\frac{2.8}{16}=0.175$\\
$\tan 9^\circ =0.1584 < \tan \angle {\rm PBA} < 0.1763=\tan 10 ^\circ$

\textcircled{\small 1}

}̂悤ȓH̋rB
H̒[̒n_A狴r̐[BグƂC
Ƃ̂Ȃp$85^\circ$łB
r̐[B̐^̒n_Cn_A܂ł̋AC17młƂCH
r̐[܂ł̍BC͂悻\fbox{\,A\,}młB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂łK؂Ȃ̂IׁB\\
\textcircled{\small 1}\,194~~~
\textcircled{\small 2}\,207~~~
\textcircled{\small 3}\,388~~~
\textcircled{\small 4}\,414\\
\hspace{8pt} Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 85^\circ=0.9962$C$\cos 85^\circ=0.0872$C\\
$\tan 85^\circ=11.4301$\\
\begin{figure}[!hbt]\begin{center}
  \includegraphics[width=60mm]{D:/texlive/2018/bin/win32/mathtex/405_sincos/h2701m0501.eps}
\end{center}\end{figure} 

process
$\displaystyle\frac{\rm BC}{17}=\tan 85^\circ=11.4301$\\
${\rm BC}=17\times 11.4301=194.\cdots$

\textcircled{\small 1}

̐}́C̖ɓ̋ŋPƂ́CzƋƒn̈ʒu\}łBzƒn̋AB1VPʂƂƂC
zƋ̋AC\fbox{\,A\,}VPʂɂȂB\\
\hspace{8pt}C$\angle {\rm ABC}=46^\circ$C$\angle {\rm ACB}=90^\circ$ƂB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂łK؂Ȃ̂IׁB\\
\hspace{8pt}Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 46^\circ=0.7193$C~~~
$\cos 46^\circ=0.6947$C\\
$\tan 46^\circ=1.0355$\\
\textcircled{\small 1}\,0.6947~~~
\textcircled{\small 2}\,0.7193~~~
\textcircled{\small 3}\,0.8165~~~
\textcircled{\small 4}\,0.9657\\
\begin{figure}[!hbt]\begin{center}
  \includegraphics[width=50mm]{D:/texlive/2018/bin/win32/mathtex/405_sincos/h2601m0501.eps}
 \end{center}\end{figure} 

process
$\displaystyle\frac{\rm AC}{\rm AB}=\sin 46^\circ$\\
${\rm AC=AB}\times \sin 46^\circ=1 \times 0.7193=0.7193$

\textcircled{\small 2}

̐}̂悤Ƀh[n_A΂B
n_Bh[グƂCƂȂp${43}^\circ$
ɂȂBh[̐^̒n_An_B܂ł̐AB400młB
̂ƂCh[̍AC͂悻\fbox{\bf\,A\,}\,$\rm m$łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂łK؂Ȃ̂IׁB\\
\textcircled{\small 1}\,273~~~
\textcircled{\small 2}\,293~~~
\textcircled{\small 3}\,373~~~
\textcircled{\small 4}\,429\\
\hspace{8pt} Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 43^\circ=0.6820$C$\cos 43^\circ=0.7314$C\\
$\tan 43^\circ=0.9325$\\
\begin{figure}[!hbt]\begin{center}
  \includegraphics[width=60mm]{D:/texlive/2018/bin/win32/mathtex/405_sincos/h2801501.eps}
\end{center}\end{figure} 

process
$\displaystyle\frac{\rm AC}{400}=\tan 43^\circ=0.9325$\\
${\rm AC}=400\times 0.9325=373$

\textcircled{\small 3}

}̂悤ɁCV䂩${\rm 50cm}$̂ЂABŋ\ruby{}{}艺ĂB
eƂCЂ${52}^\circ$̊pȂʒu܂ŁCUꂽB
}ŁCŏ̈ʒu֓BC͂悻\fbox{\,\bf A\,}cmłB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂łK؂Ȃ̂IׁB\\
\hspace{8pt}ȂKvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin{52}^{\circ}=0.7880$C$\cos{52}^{\circ}=0.6157$C$\tan{52}^{\circ}=1.2799$\\
\textcircled{\small 1}\,$10.6$~~~
\textcircled{\small 2}\,$19.2$~~~
\textcircled{\small 3}\,$30.8$~~~
\textcircled{\small 4}\,$39.4$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/h29020501.tex}
\end{center}

process
${\rm AC}=50\times \cos{52}^{\circ}=30.785$\\
${\rm AC}=50-30.785=19.215$

\textcircled{\small 2}

Kvł΁C̎Op̒l𗘗p邱ƁB\\
\hspace{8pt} R̎ΖʂCɋ삯オoR[XsꂽB
X^[gn_AS[n_B܂ł̒${\rm AB}$$10000{\rm m}$
CR̕WBC$2100{\rm m}$łB
̂Ƃ$\angle {\rm BAC}$\fbox{\bf A}łB
C$\angle {\rm ACB}={90}^{\circ}$ƂB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,${11}^{\circ}$ȏ${12}^{\circ}$~~~
\textcircled{\small 2}\,${12}^{\circ}$ȏ${13}^{\circ}$\\
\textcircled{\small 3}\,${13}^{\circ}$ȏ${14}^{\circ}$~~~
\textcircled{\small 4}\,${14}^{\circ}$ȏ${15}^{\circ}$
\begin{figure}[!hbt]\begin{center}
  \includegraphics[width=40mm]{D:/texlive/2018/bin/win32/mathtex/405_sincos/h2802m501.eps}
 \end{center}\end{figure} 
 \begin{table}[!htb]
\begin{center}\begin{tabular}{|c||c|c|c|}
\hline
 p & (sin) & ](cos) & (tan) \\ \hline
 ${11}^{\circ}$ & $0.1908$ & $0.9816$ & $0.1944$ \\ \hline
  ${12}^{\circ}$ & $0.2079$ & $0.9781$ & $0.2126$ \\ \hline
   ${13}^{\circ}$ & $0.2250$ & $0.9744$ & $0.2309$ \\ \hline
  ${14}^{\circ}$ & $0.2419$ & $0.9703$ & $0.2493$ \\ \hline
  ${15}^{\circ}$ & $0.2588$ & $0.9659$ & $0.2679$ \\ \hline
\end{tabular}\end{center}\end{table}

process
$0.2079<\sin {\rm A}=\displaystyle\frac{2100}{10000}=0.21<0.2250$

\textcircled{\small 2}

Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin {75}^{\circ}=0.9659$C$\cos {75}^{\circ}=0.2588$C$\tan {75}^{\circ}=3.7321$\\
\hspace{8pt} }́C
݂̊̒n_A甽Α݂̊̒n_BɑDŌlqłB
̌CD͗Ēn_CɓB2_BCCԂ̋${\rm 40m}$
$\angle {\rm ACB}={75}^{\circ}$C$\angle {\rm ABC}={90}^{\circ}$łƂC
2_ACBԂ̋͂悻\fbox{\bf\hspace{2pt} A \hspace{2pt}}\, młB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂łK؂Ȃ̂IׁB\\
\textcircled{\small 1}\,$10.4$~~~
\textcircled{\small 2}\,$38.6$~~~
\textcircled{\small 3}\,$149.3$~~~
\textcircled{\small 4}\,$154.6$
\begin{figure}[!hbt]\begin{center}
  \includegraphics[width=40mm]{D:/texlive/2018/bin/win32/mathtex/405_sincos/h2801m501.eps}
 \end{center}\end{figure}

process
$\displaystyle\frac{\rm AB}{40}=\tan {75}^{\circ}$\\
${\rm AB}=40\times 3.7321=149.284 \fallingdotseq  149.3$

\textcircled{\small 3}

}́CXL[̃Wv猩}łB
I肪ؒn_AŃWvCA狗${\rm 110m}$C
ʂ\ruby{p}{|}${35}^\circ$
̒n_BɒnB
̂ƂCn_Ɠؒn_Ƃ̍፷BĆC悻\fbox{\bf\, A \,}\,${\rm m}$łB
C$\angle {\rm ACB}={90}^\circ$ƂB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂łK؂Ȃ̂IׁB\\
\textcircled{\small 1}\,$63.1$~~~
\textcircled{\small 2}\,$77.0$~~~
\textcircled{\small 3}\,$90.1$~~~
\textcircled{\small 4}\,$191.8$\\
\hspace{8pt}
CKvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin {35}^\circ=0.5736$~~~
$\cos {35}^\circ=0.8192$\\
$\tan {35}^\circ=0.7002$
\begin{figure}[!hbt]\begin{center}
  \includegraphics[width=70mm]{D:/texlive/2018/bin/win32/mathtex/405_sincos/h3001501.eps}
 \end{center}\end{figure} 

process
${\rm BC}=110\times \sin{35}^\circ=110\times 0.5736\fallingdotseq 63.1$

\textcircled{\small 1}

Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin {57}^\circ=0.8387$C$\cos {57}^\circ=0.5446$C$\tan {57}^\circ=1.5399$\\
\hspace{8pt} ̐}̂悤ȊϗԂB
ϗԂ̍łʒuɂShA̐^̒n_B$20{\rm m}$ꂽn_CƂB\\
$\angle {\rm ACB}={57}^\circ$
C$\angle {\rm ABC}={90}^\circ$łƂCAB͂悻\fbox{\bf\hspace{4pt} A \hspace{4pt}}\,młB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂łK؂Ȃ̂IׁB\\
\textcircled{\small 1}\,$10.9$~~~
\textcircled{\small 2}\,$16.8$~~~
\textcircled{\small 3}\,$30.8$~~~
\textcircled{\small 4}\,$36.7$
\begin{figure}[!hbt]\begin{center}
  \includegraphics[width=50mm]{D:/texlive/2018/bin/win32/mathtex/405_sincos/h31m501.eps}
 \end{center}\end{figure}

process
$20 \times \tan{57}^\circ=20\times 1.5399=30.7980$

\textcircled{\small 3}

̐}̂悤Ȗؑ̌B
n_AB܂ł͊KiC̍CDɂāC${\rm CD=2BD}$łB
ABԂ̋$100{\rm m}$C$\angle {\rm BAD}={14}^\circ$C
$\angle {\rm ADB}={90}^\circ$łB
̂ƂC̍CD͂悻\fbox{\bf\, A \,}młB
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̂łK؂Ȃ̂IׁB
Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin {14}^\circ=0.2419$C$\cos {14}^\circ=0.9703$C
$\tan {14}^\circ=0.2493$\\
\textcircled{\small 1}\,$24.2$~~~
\textcircled{\small 2}\,$48.4$~~~
\textcircled{\small 3}\,$49.9$~~~
\textcircled{\small 4}\,$97.0$
\begin{figure}[!hbt]\begin{center}
  \includegraphics[width=50mm]{D:/texlive/2018/bin/win32/mathtex/405_sincos/h3101m501.eps}
 \end{center}\end{figure} 

process
$\sin {14}^\circ=\displaystyle\frac{BD}{100}$~~~
$BD=100\times 0.2419=24.19$

\textcircled{\small 2}





[Level6]
Op̑݊֌WiƂ͉spj
AspŁC$\sin A=\displaystyle \frac{2}{5}$̂ƂC
$\cos A$̒l߂B

process
$x^2+2^2=5^2$C$x=\sqrt{5^2-2^2}=\sqrt{25-4}=\sqrt{21}$

$\cos A=\displaystyle \frac{\sqrt{21}}{5}$

$\cos \theta =\displaystyle \frac{3}{4}$̂ƂC$\sin \theta$C$\tan \theta$̒l߂B
C$\theta$͉spƂB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic41.tex}
\end{center}
$x=\sqrt{4^2-3^2}=\sqrt{16-9}=\sqrt{7}$B

$\sin \theta=\displaystyle \frac{\sqrt{7}}{4}$C$\tan \theta=\displaystyle \frac{\sqrt{7}}{3}$

$\sin \theta =\displaystyle \frac{3}{5}$̂ƂC$\cos \theta$C$\tan \theta$̒l߂B
C$\theta$͉spƂB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic42.tex}
\end{center}
$3:4:5$̔䂾C$x=4$B

$\cos \theta=\displaystyle \frac{4}{5}$C$\tan \theta=\displaystyle \frac{3}{4}$

$\sin \theta =\displaystyle \frac{1}{3}$i0$< \theta <$ 90ĵƂC$\cos \theta$C$\tan \theta$̒l߂B

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic43.tex}
\end{center}
$x=\sqrt{3^2-1^2}=\sqrt{9-1}=\sqrt{8}=\sqrt{2^2 \times 2}\\
=2 \sqrt{2}$B

$\cos \theta=\displaystyle \frac{2\sqrt{2}}{3}$C$\tan \theta=\displaystyle \frac{1}{2\sqrt{2}}$

$\tan \theta =\displaystyle \frac{4}{3}$̂ƂC$\sin \theta$C$\cos \theta$̒l߂B
C$\theta$͉spƂB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic44.tex}
\end{center}
$3:4:5$̔䂾C$x=5$B

$\sin \theta=\displaystyle \frac{4}{5}$C$\cos \theta=\displaystyle \frac{3}{5}$

$\tan \theta =3$̂ƂC$\sin \theta$C$\cos \theta$̒l߂B
C$\theta$͉spƂB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic45.tex}
\end{center}
$x=\sqrt{1^2+3^2}=\sqrt{1+9}=\sqrt{10}$B

$\sin \theta=\displaystyle \frac{3}{\sqrt{10}}$C$\cos \theta=\displaystyle \frac{1}{\sqrt{10}}$

$\tan \theta =\displaystyle \frac{3}{2}$̂ƂC$\sin \theta$C$\cos \theta$̒l߂B
C$\theta$͉spƂB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic46.tex}
\end{center}
$x=\sqrt{2^2+3^2}=\sqrt{4+9}=\sqrt{13}$B

$\sin \theta=\displaystyle \frac{3}{\sqrt{13}}$C$\cos \theta=\displaystyle \frac{2}{\sqrt{13}}$

$\sin \theta =\displaystyle \frac{\sqrt{5}}{3}$̂ƂC$\cos \theta$C$\tan \theta$̒l߂B
C$\theta$́i0$< \theta <$ 90jƂB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic47.tex}
\end{center}
$x=\sqrt{3^2-5}=\sqrt{9-5}=\sqrt{4}=2$B

$\cos \theta=\displaystyle \frac{2}{3}$C$\tan \theta=\displaystyle \frac{\sqrt{5}}{2}$

$\sin \theta =\displaystyle \frac{4}{5}$̂ƂC$\cos \theta$C$\tan \theta$̒l߂B
C$\theta$͉spƂB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic48.tex}
\end{center}
$3:4:5$̔䂾C$x=3$B

$\cos \theta=\displaystyle \frac{3}{5}$C$\tan \theta=\displaystyle \frac{4}{3}$

$\cos \theta =\displaystyle \frac{5}{6}$̂ƂC$\sin \theta$C$\tan \theta$̒l߂B
C$\theta$͉spƂB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic49.tex}
\end{center}
$x=\sqrt{6^2-5^2}=\sqrt{36-25}=\sqrt{11}$B

$\sin \theta=\displaystyle \frac{\sqrt{11}}{6}$C$\tan \theta=\displaystyle \frac{\sqrt{11}}{5}$

$\sin \theta =\displaystyle \frac{2\sqrt{2}}{3}$̂ƂC$\cos \theta$C$\tan \theta$̒l߂B
C$\theta$͉spƂB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic50.tex}
\end{center}
$x=\sqrt{3^2-2^2 \times 2}=\sqrt{9-8}=1$B

$\cos \theta=\displaystyle \frac{1}{3}$C$\tan \theta=2\sqrt{2}$

$\cos \theta =\displaystyle \frac{2}{\sqrt{5}}$̂ƂC$\sin \theta$C$\tan \theta$̒l߂B
C$\theta$͉spƂB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic51.tex}
\end{center}
$x=\sqrt{5-2^2}=\sqrt{5-4}=1$B

$\sin \theta=\displaystyle \frac{1}{\sqrt{5}}$C$\tan \theta=\displaystyle \frac{1}{2}$

$\sin A=\displaystyle \frac{15}{17}$C$\cos A=\displaystyle \frac{8}{17}$̂ƂC$\tan A$̒l߂B

process
$\tan A=\displaystyle \frac{\sin A}{\cos A}=\displaystyle \frac{15}{17} \div \displaystyle \frac{8}{17}
=\displaystyle \frac{15}{17} \times \displaystyle \frac{17}{8}=\displaystyle \frac{15}{8}$

$\displaystyle \frac{15}{8}$

$\sin A=\displaystyle \frac{3 \sqrt{5}}{7}$C$\cos A =\displaystyle \frac{2}{7}$̂ƂC
$\tan A$̒l߂B

process
$\tan A=\displaystyle \frac{\sin A}{\cos A}=\displaystyle \frac{3 \sqrt{5}}{7} \div \displaystyle \frac{2}{7}
=\displaystyle \frac{3 \sqrt{5}}{7} \times \displaystyle \frac{7}{2}$

$\displaystyle \frac{3 \sqrt{5}}{2}$

$\sin A=\displaystyle \frac{2\sqrt{2}}{3},\, \cos A=\displaystyle \frac{1}{3}$̂ƂC$\tan A$̒l߂B

process
$\tan A=\displaystyle \frac{\sin A}{\cos A}=\displaystyle \frac{2\sqrt{2}}{3}\div \displaystyle \frac{1}{3}=2\sqrt{2}$

$2\sqrt{2}$

$\sin A=\displaystyle\frac{\sqrt{15}}{4},\,\cos A=\displaystyle\frac{1}{4}$̂ƂC
$\tan A$̒l߂B

process
$\tan A=\displaystyle\frac{\sin A}{\cos A}=\displaystyle\frac{\displaystyle\frac{\sqrt{15}}{4}}{\displaystyle\frac{1}{4}}
=\sqrt{15}$

$\sqrt{15}$

$A$spŁC$\sin A=\displaystyle\frac{3}{4}$̂ƂC$\cos A$̒l߂B

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic229.tex}
\end{center}
$x^2+3^2=4^2$C$x=\sqrt{7}$\\
$\cos A=\displaystyle\frac{\sqrt{7}}{4}$

$\displaystyle\frac{\sqrt{7}}{4}$

$\cos A=\displaystyle\frac{4}{5}$̂ƂC$\sin AC\tan A$̒l߂B
C$A$͉spƂB

$\sin A=\displaystyle\frac{3}{5}$C$\tan A=\displaystyle\frac{3}{4}$

$\sin A=\displaystyle\frac{12}{13}$̂ƂC$\cos AC\tan A$̒l߂B
C$A$͉spƂB

$\cos A=\displaystyle\frac{5}{13}$C$\tan A=\displaystyle\frac{12}{5}$

$\sin A=\displaystyle\frac{4}{5}$C$\cos A=\displaystyle\frac{3}{5}$̂ƂC
$\tan A=$ \fbox{\,E\,} łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$\displaystyle\frac{4}{3}$~~~
\textcircled{\small 2}\,$\displaystyle\frac{3}{4}$~~~
\textcircled{\small 3}\,$\displaystyle\frac{7}{5}$~~~
\textcircled{\small 2}\,$\displaystyle\frac{12}{25}$

process
$\tan A=\displaystyle\frac{\sin A}{\cos A}=\displaystyle\frac{4}{5}\div \displaystyle\frac{3}{5}=\displaystyle\frac{4}{3}$

\textcircled{\small 1}

$\sin 44^\circ$̒l\fbox{\,C\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂łK؂Ȃ̂IׁB\\
\textcircled{\small 1}\,0.6947~~~
\textcircled{\small 2}\,$-0.6947$~~~
\textcircled{\small 3}\,$0.7193$~~~
\textcircled{\small 4}\,$-0.7193$\\
Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 46^\circ=0.7193$C~~~
$\cos 46^\circ=0.6947$C\\
$\tan 46^\circ=1.0355$

process
$\sin 44^\circ=\sin(90^\circ-46^\circ)\\
=\cos 46^\circ=0.6947$

\textcircled{\small 1}

$\cos 35$ƓlɂȂ̂͂ǂꂩB
\textcircled{\small 1}`\textcircled{\small 4}̂łKȂ̂IׁB\\
\textcircled{\small 1}$\sin 55$C@\textcircled{\small 2}$\cos 55$C@\textcircled{\small 3}$\sin 145$C@\textcircled{\small 4}$\cos 145 $\\

process
$\cos 35=\cos(90-55)=\sin 55$

\textcircled{\small 1}

$\cos 15$̒l߂B\\
Kvł΁C̎Op̕\𗘗p邱ƁB\\
\begin{table}[!h]
\begin{center}
\begin{tabular}{c|ccc}
 \hline
 p & ($\sin$) & ]($\cos$) & ($\tan$) \\
 \hline
 71 & 0.9455 & 0.3256 & 2.9042 \\
 \hline
 72 & 0.9511 & 0.3090 & 3.0777 \\
 \hline
 73 & 0.9563 & 0.2924 & 3.2709 \\
 \hline
 74 & 0.9613 & 0.2756 & 3.4874 \\
 \hline
 75 & 0.9659 & 0.2588 & 3.7321 \\
 \hline
\end{tabular}
\end{center}
\end{table}

process
$\cos 15^{\circ}=\cos (90^{\circ}-75^{\circ})=\sin 75^{\circ}=0.9659$

$0.9659$

$\cos 57$̒l߂B
Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 33=0.5446, \ \cos 33=0.8387, \\
\tan 33=0.6494$\\

process
$\cos 57=\cos (90-33)=\sin 33=0.5446 $

$0.5446$

Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin {57}^\circ=0.8387$C$\cos {57}^\circ=0.5446$C$\tan {57}^\circ=1.5399$\\
$\sin {33}^\circ$̒l\fbox{\bf\hspace{4pt} C \hspace{4pt}}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂łK؂Ȃ̂IׁB\\
\textcircled{\small 1}\,$0.8387$~~~
\textcircled{\small 2}\,$0.5446$~~~
\textcircled{\small 3}\,$1.5399$~~~
\textcircled{\small 4}\,$-0.8387$

process
$\sin {33}^\circ=\sin ({90}^\circ-{57}^\circ)=\cos{57}^\circ=0.5446$

\textcircled{\small 2}

$\sin 49$̒l߂BKvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 41=0.6561,\, \cos 41=0.7547,\\
\tan 41=0.8693$

process
$\sin 49=\sin(90-41)=\cos 41=0.7547$

$0.7547$

$\sin 72$̒l߂B
Kvł΁C̎Op̕\𗘗p邱ƁB\\
\begin{table}[!h]
\begin{center}
\begin{tabular}{c|ccc}
 \hline
 p & ($\sin$) & ]($\cos$) & ($\tan$) \\
 \hline
 15 & 0.2588 & 0.9659 & 0.2679 \\
 \hline
 16 & 0.2756 & 0.9613 & 0.2867 \\
 \hline
 17 & 0.2924 & 0.9563 & 0.3057 \\
 \hline
 18 & 0.3090 & 0.9511 & 0.3249 \\
 \hline
 19 & 0.3256 & 0.9455 & 0.3443 \\
 \hline
\end{tabular}
\end{center}
\end{table}

process
$\sin 72^{\circ}=\sin(90^{\circ}-18^{\circ})=\cos 18^{\circ}=0.9511$

$0.9511$

$\sin 5^\circ$̒l\fbox{\,C\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂łK؂Ȃ̂IׁB\\
\textcircled{\small 1}\,0.9962~~~
\textcircled{\small 2}\,0.0872~~~
\textcircled{\small 3}\,$-0.9962$~~~
\textcircled{\small 4}\,$-0.0872$\\
\hspace{8pt} Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 85^\circ=0.9962$C$\cos 85^\circ=0.0872$C\\
$\tan 85^\circ=11.4301$

process
$\sin 5^\circ=\sin(90^\circ-85^\circ)=\cos 85^\circ=0.0872$

\textcircled{\small 2}

$A$spŁC
$\sin A=a$C$\cos A=b$łƂC
$\tan A=$\fbox{\,E\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂łKȂ̂IׁB\\
\textcircled{\small 1}\,$ab$~~~
\textcircled{\small 2}\,$\displaystyle\frac{b}{a}$~~~
\textcircled{\small 3}\,$\displaystyle\frac{a}{b}$~~~
\textcircled{\small 4}\,$a^2+b^2$

\textcircled{\small 3}

$\cos 15$̒ĺC\textcircled{\small 1}`\textcircled{\small 4}̂ǂꂩB
łKȂ̂IׁB
Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 75=0.9659$C$\cos 75=0.2588$\\
$\tan 75=3.7321$\\
\begin{tabbing}
@@@@@\=@@@@@@\=@@@@@\=@@@@@\\ 
\textcircled{\small 1} $0.9659$ \>\textcircled{\small 2} $-0.9659$ \> \textcircled{\small 3} $0.2588$ \> \textcircled{\small 4} $-0.2588$
\end{tabbing}

process
$\cos 15=\cos(90-75)=\sin 75=0.9659$

\textcircled{\small 1}

$\sin A=\displaystyle\frac{12}{13}$C$\cos A=\displaystyle\frac{5}{13}$̂ƂC
$\tan A$̒l\fbox{\hspace{2pt} {\bf E} \hspace{2pt}}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$-\displaystyle\frac{5}{12}$~~~
\textcircled{\small 2}\,$-\displaystyle\frac{12}{5}$~~~
\textcircled{\small 3}\,$\displaystyle\frac{5}{12}$~~~
\textcircled{\small 4}\,$\displaystyle\frac{12}{5}$

process
$\tan A=\displaystyle\frac{\sin A}{\cos A}=\displaystyle\frac{12}{5}$

\textcircled{\small 4}

$A$͉spƂB$\sin A=\displaystyle\frac{2}{3}$̂ƂC$\cos A=$\fbox{\bf\hspace{2pt} E \hspace{2pt}}łB\\
\textcircled{\small 1}\,$\displaystyle\frac{2}{3}$~~~
\textcircled{\small 2}\,$\displaystyle\frac{2\sqrt{2}}{3}$~~~
\textcircled{\small 3}\,$\displaystyle\frac{\sqrt{5}}{3}$~~~
\textcircled{\small 4}\,$\displaystyle\frac{2\sqrt{5}}{3}$

\textcircled{\small 3}

$0^\circ \leqq \theta \leqq {180}^\circ$ŁC$\tan \theta =2\sqrt{2}$
̂ƂC$\cos \theta =$
\fbox{\hspace{8pt} 14 \hspace{8pt}}łB\\
\textcircled{\small 1}\,$-\displaystyle\frac{2\sqrt{2}}{3}$~~~
\textcircled{\small 2}\,$-\displaystyle\frac{1}{3}$~~~
\textcircled{\small 3}\,$\displaystyle\frac{1}{3}$~~~
\textcircled{\small 4}\,$\displaystyle\frac{2\sqrt{2}}{3}$

process
$\tan \theta =2\sqrt{2}$胈R$=1$C
^e$=2\sqrt{2}$ƒ߂ƁCii$=3$\\
āC$\cos \theta =\displaystyle\frac{1}{3}$

\textcircled{\small 3}

$\cos A=\displaystyle\frac{3}{\sqrt{13}}$C$\tan A=\displaystyle\frac{2}{3}$
̂ƂC$\sin A$̒l\fbox{\bf\, E \,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$-\displaystyle\frac{2}{\sqrt{13}}$~~~
\textcircled{\small 2}\,$\displaystyle\frac{2}{\sqrt{13}}$~~~
\textcircled{\small 3}\,$-\displaystyle\frac{9}{2\sqrt{13}}$~~~
\textcircled{\small 4}\,$\displaystyle\frac{9}{2\sqrt{13}}$

process
$\tan A=\displaystyle\frac{\sin A}{\cos A}$C
$\sin A=\tan A \times \cos A=\displaystyle\frac{3}{\sqrt{13}} \times \displaystyle\frac{2}{3}=\displaystyle\frac{2}{\sqrt{13}}$

\textcircled{\small 2}








[Level7]
Op̑݊֌WiƂ͓݊pj
$\cos{166}^\circ$̒l\fbox{\bf\, C \,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̂łK؂Ȃ̂IׁB\\
Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin {14}^\circ=0.2419$C$\cos {14}^\circ=0.9703$C
$\tan {14}^\circ=0.2493$\\
\textcircled{\small 1}\,$0.2419$~~~
\textcircled{\small 2}\,$-0.2419$~~~
\textcircled{\small 3}\,$0.9703$~~~
\textcircled{\small 4}\,$-0.9703$

process
$\cos {160}^\circ=\cos({180}^\circ-{14}^\circ)=-\cos{14}^\circ=-0.9703$

\textcircled{\small 4}

$A$݊pŁC$\sin A=\displaystyle \frac{4}{5}$̂ƂC$\cos A$̒l߂B

process
}ɂC\\
$\cos A=\displaystyle \frac{-3}{5}$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic215.tex}
\end{center}

$\cos A=-\displaystyle \frac{3}{5}$

$\sin \theta=\displaystyle \frac{4}{5}$̂ƂC$\cos \theta$$\tan \theta$̒l߂B
C$90<\theta<180$ƂB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic61.tex}
\end{center}
$\cos \theta=\displaystyle \frac{-3}{5}$C$\tan \theta=\displaystyle \frac{4}{-3}$

$\cos \theta=-\displaystyle \frac{3}{5}$C$\tan \theta=-\displaystyle \frac{4}{3}$

$\sin \theta=\displaystyle \frac{3}{5}$̂ƂC$\cos \theta$$\tan \theta$̒l߂B
C$90<\theta<180$ƂB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic62.tex}
\end{center}
$\cos \theta=\displaystyle \frac{-4}{5}$C$\tan \theta=\displaystyle \frac{3}{-4}$

$\cos \theta=-\displaystyle \frac{4}{5}$C$\tan \theta=-\displaystyle \frac{3}{4}$

$\sin \theta=\displaystyle \frac{1}{3}$̂ƂC$\cos \theta$$\tan \theta$̒l߂B
C$90<\theta<180$ƂB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic63.tex}
\end{center}
$x^2=3^2-1^2=9-1=8$B\\
$x=-\sqrt{8}=-\sqrt{2^2 \times 2}=-2\sqrt{2}$B\\
$\cos \theta=\displaystyle \frac{-2\sqrt{2}}{3}$C$\tan \theta=\displaystyle \frac{1}{-2\sqrt{2}}$

$\cos \theta=-\displaystyle \frac{2\sqrt{2}}{3}$C$\tan \theta=-\displaystyle \frac{1}{2\sqrt{2}}$

$\cos \theta=-\displaystyle \frac{3}{4}$̂ƂC$\sin \theta$$\tan \theta$̒l߂B
C$90<\theta<180$ƂB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic64.tex}
\end{center}
$y^2=4^2-(-3)^2=16-9=7$B$y=\sqrt{7}$B\\
$\sin \theta=\displaystyle \frac{\sqrt{7}}{4}$C$\tan \theta=\displaystyle \frac{\sqrt{7}}{-3}$

$\sin \theta=\displaystyle \frac{\sqrt{7}}{4}$C$\tan \theta=-\displaystyle \frac{\sqrt{7}}{3}$

$\cos \theta=-\displaystyle \frac{5}{13}$̂ƂC$\sin \theta$$\tan \theta$̒l߂B
C$90<\theta<180$ƂB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic65.tex}
\end{center}
$y^2=13^2-5^2=169-25=144$B$y=12$B\\
$\sin \theta=\displaystyle \frac{12}{13}$C$\tan \theta=\displaystyle \frac{12}{-5}$

$\sin \theta=\displaystyle \frac{12}{13}$C$\tan \theta=-\displaystyle \frac{12}{5}$

$\sin \theta=\displaystyle \frac{2}{3}$̂ƂC$\cos \theta$$\tan \theta$̒l߂B
C$90<\theta<180$ƂB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic66.tex}
\end{center}
$x^2=3^2-2^2=9-4=5$B$x=-\sqrt{5}$B\\
$\cos \theta=\displaystyle \frac{-\sqrt{5}}{3}$C$\tan \theta=\displaystyle \frac{2}{-\sqrt{5}}$

$\cos \theta=-\displaystyle \frac{\sqrt{5}}{3}$C$\tan \theta=-\displaystyle \frac{2}{\sqrt{5}}$

$\cos \theta=-\displaystyle \frac{1}{3}$̂ƂC$\sin \theta$$\tan \theta$̒l߂B
C$90<\theta<180$ƂB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic67.tex}
\end{center}
$y^2=3^2-(-1)^2=9-1=8$B\\
$y=\sqrt{2^2 \times 2}=2 \sqrt{2}$B\\
$\sin \theta=\displaystyle \frac{2\sqrt{2}}{3}$C$\tan \theta=\displaystyle \frac{2\sqrt{2}}{-1}$

$\sin \theta=\displaystyle \frac{2\sqrt{2}}{3}$C$\tan \theta=-2\sqrt{2}$

$\cos \theta=-\displaystyle \frac{1}{\sqrt{5}}$̂ƂC$\sin \theta$$\tan \theta$̒l߂B
C$90<\theta<180$ƂB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic68.tex}
\end{center}
$y^2=5-(-1)^2=5-1=4$B$y=2$B\\
$\sin \theta=\displaystyle \frac{2}{\sqrt{5}}$C$\tan \theta=\displaystyle \frac{2}{-1}$

$\sin \theta=\displaystyle \frac{2}{\sqrt{5}}$C$\tan \theta=-2$

$0\le \theta \le 180$̂ƂĈ悤$\theta$߂B\\
$\sin \theta =\displaystyle \frac{\sqrt{3}}{2}$

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic71.tex}
\end{center}

$\theta =60C120$

$0\le \theta \le 180$̂ƂĈ悤$\theta$߂B\\
$\cos \theta =-\displaystyle \frac{1}{2}$

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic72.tex}
\end{center}

$\theta =120$

$0\le \theta \le 180$̂ƂĈ悤$\theta$߂B\\
$\tan \theta =1$

process
$1=\displaystyle \frac{1}{1}$ł邩C߂$\theta$͐}̂悤łB\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic73.tex}
\end{center}

$\theta =45$

$0\le \theta \le 180$̂ƂĈ悤$\theta$߂B\\
$\tan \theta =-\displaystyle \frac{1}{\sqrt{3}}$

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic74.tex}
\end{center}

$\theta =150$

$0\le \theta \le 180$̂ƂĈ悤$\theta$߂B\\
$\tan \theta =\displaystyle \frac{1}{\sqrt{3}}$

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic75.tex}
\end{center}

$\theta =30$

$0\le \theta \le 180$̂ƂĈ悤$\theta$߂B\\
$\tan \theta =-1$

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic76.tex}
\end{center}

$\theta =135$

$\sin 160$̒l߂B
ȂCKvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 20=0.3420,\, \cos 20=0.9397,\\
\tan 20=0.3640$

process
$\sin 160=\sin(180-20)=\sin 20=0.3420$

0.3420

$\sin ^2 150+\cos ^2 150$̒l߂B

process
$\sin 150=\sin(180-30)=\sin 30 =\displaystyle \frac{1}{2}$\\
$\cos 150=\cos(180-30)=-\sin 30 =-\displaystyle \frac{\sqrt{3}}{2}$\\
$\sin ^2 150+\cos ^2 150
=\left( \displaystyle \frac{1}{2}\right)^2+\left( -\displaystyle \frac{\sqrt{3}}{2} \right)^2
=1$

$1$

\textcircled{\small 1}`\textcircled{\small 4}̂ĈIׁB\\
\begin{tabbing}
@@@@@@@@@@@\=@@@@@@@@@@@\= \\ 
\textcircled{\small 1}@$\sin 80=\sin 10$ \>\textcircled{\small 2}@$\cos 80=\cos 10$\\
\textcircled{\small 3}@$\sin 100=\sin 80$ \>\textcircled{\small 4}@$\cos 100=\cos 80$
\end{tabbing}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic212.tex}
\end{center}
}C$\sin 100=\sin 80$

\textcircled{\small 3}

$\cos 120$̒l߂B

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic72.tex}
\end{center}
$\displaystyle \frac{-1}{2}$

$-\displaystyle \frac{1}{2}$

$A$݊pŁC$\sin A=\displaystyle \frac{3}{5}$̂ƂC$\cos A$̒l߂B

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic62.tex}
\end{center}
}C$\cos A=\displaystyle \frac{-4}{5}=-\displaystyle \frac{4}{5}$

$-\displaystyle \frac{4}{5}$

$\tan 150$̒l߂B

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic225.tex}
\end{center}
}C$\displaystyle \frac{1}{-\sqrt{3}}=-\displaystyle \frac{1}{\sqrt{3}}$

$\displaystyle \frac{1}{\sqrt{3}}$

$0\le \theta \le 180$ƂB$\cos \theta =\displaystyle \frac{2}{3}$̂ƂC
$\sin \theta$̒l߂B

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic226.tex}
\end{center}

$\displaystyle \frac{\sqrt{5}}{3}$

$0<A<180$̂ƂC$\cos A=-\displaystyle \frac{1}{\sqrt{2}}$𖞂$A$߂B

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic231.tex}
\end{center}

$135$

$ \sin 150$̒l߂B

process
$\sin 150=\sin(180-30)=\sin 30=\displaystyle \frac{1}{2}$

$\displaystyle \frac{1}{2}$

$\tan 140$̒ĺC\textcircled{\small 1}`\textcircled{\small 4}̂ǂꂩB
łK؂Ȃ̂IׁB
CKvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 40=0.6428$C$\cos 40=0.7660$C\\
$\tan 40=0.8391$\\\\
\textcircled{\small 1}$-0.8391$\ \textcircled{\small 2}\ $0.8391$\ \textcircled{\small 3}$-0.7660$\ \textcircled{\small 4}$0.6428$

process
}ɂC\\
$\tan 140=\tan(180-40)=-\tan 40=-0.8391$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic213p.tex}
\end{center}

\textcircled{\small 1}

$\cos ^2 45+\sin ^2 135$̒l߂B

process
$\left( \displaystyle\frac{1}{\sqrt{2}} \right)^2+\left( \displaystyle\frac{1}{\sqrt{2}} \right)^2=1$

$1$

$A$݊pŁC$\sin A=\displaystyle\frac{\sqrt{5}}{3}$̂ƂC
$\cos A$̒l߂B

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/pic228.tex}
\end{center}
$x^2+(\sqrt{5})^2=3^2$C$x=-2$

$-\displaystyle\frac{2}{3}$

$\sin 168^\circ$̒l\fbox{\,C\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂PIׁB
ȂCKvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 12^\circ=0.2079$~~~
$\cos 12^\circ=0.9781$\\
$\tan 12^\circ=0.2126$\\
\textcircled{\small 1}\,$0.9781$~~~
\textcircled{\small 2}\,$0.2079$~~~
\textcircled{\small 3}\,$-0.2079$~~~
\textcircled{\small 4}\,$-0.9781$

process
$\sin 168^\circ =\sin (180^\circ -12^\circ)=\sin 12^\circ =0.2079$

\textcircled{\small 2}

$\cos 149^\circ$̒l\fbox{\,C\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂łKȂ̂IׁB
ȂCKvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 31^\circ=0.5150$~~~
$\cos 31^\circ=0.8572$\\
$\tan 31^\circ=0.6009$\\
\textcircled{\small 1}\,0.5150~~~
\textcircled{\small 2}\,$-0.5150$~~~
\textcircled{\small 3}\,0.8572~~~
\textcircled{\small 4}\,$-0.8572$

process
$\cos 149^\circ=\cos (180^\circ-31^\circ)
=-\cos 31^\circ\\ =-0.8572$

\textcircled{\small 4}

$\sin 137$̒ĺC\textcircled{\small 1}`\textcircled{\small 4}̂ǂꂩB
łK؂Ȃ̂IׁB
CKvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin 43^\circ=0.6820$C$\cos 43^\circ=0.7314$C\\
$\tan 43^\circ=0.9325$\\
\textcircled{\small 1}\ $0.6820$~~~\textcircled{\small 2}\ $-0.6820$~~~\ \textcircled{\small 3}\ $0.7314$~~~\textcircled{\small 4}\ $-0.7314$

process
}ɂC\\
$\sin 137=\sin(180-43)=\sin 43=0.6820$

\textcircled{\small 1}

̎̒l߂B\\
$\sin {45}^\circ \cos {45}^\circ -\sin {120}^\circ\cos {150}^\circ $

process
$\displaystyle\frac{1}{\sqrt{2}}\times\displaystyle\frac{1}{\sqrt{2}}-\displaystyle\frac{\sqrt{3}}{2}\times \left( -\displaystyle\frac{\sqrt{3}}{2} \right)=\displaystyle\frac{5}{4}$

$\displaystyle\frac{5}{4}$

̎̒l߂B\\
$\sin {150}^\circ +\tan {150}^\circ\sin {120}^\circ $

process
$\displaystyle\frac{1}{2} +\left(-\displaystyle\frac{1}{\sqrt{3}} \right)\times \left( \displaystyle\frac{\sqrt{3}}{2} \right)=0$

$0$

̎̒l߂B\\
$\sin {150}^\circ -\tan {150}^\circ\sin {120}^\circ $

process
$\displaystyle\frac{1}{2} -\left(-\displaystyle\frac{1}{\sqrt{3}} \right)\times \left( \displaystyle\frac{\sqrt{3}}{2} \right)=1$

$1$

̒l߂B\\
$\cos 60 \sin 150+\cos 150\tan 30$

process
$\displaystyle \frac{1}{2} \cdot \displaystyle \frac{1}{2}-\displaystyle \frac{\sqrt{3}}{2} \cdot \displaystyle \frac{1}{\sqrt{3}}
=-\displaystyle \frac{1}{4}$

$-\displaystyle \frac{1}{4}$

$\sin{128}^{\circ}$̒l\fbox{\,\bf C\,}cmłB
ȂCKvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin{52}^{\circ}=0.7880$C$\cos{52}^{\circ}=0.6157$\\
C$\tan{52}^{\circ}=1.2799$\\
\textcircled{\small 1}\,$0.7880$~~~
\textcircled{\small 2}\,$-0.7880$~~~
\textcircled{\small 3}\,$0.6157$~~~
\textcircled{\small 4}\,$-0.6157$

process
$\sin({180}^{\circ}-{52}^{\circ})=\sin{52}^{\circ}=0.7880$

\textcircled{\small 1}

$\tan {165}^{\circ}$̒l\fbox{\hspace{2pt} C \hspace{2pt}}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB
Kvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin {15}^{\circ}=0.2588$C$\cos {15}^{\circ}=0.9659$C$\tan {15}^{\circ}=0.2679$\\
\textcircled{\small 1}\,$0.2679$~~~
\textcircled{\small 2}\,$-0.2588$~~~
\textcircled{\small 3}\,$-0.2679$~~~
\textcircled{\small 4}\,$0.9659$

process
$\tan {165}^{\circ}=\tan (180-15)^{\circ}=-\tan {15}^{\circ}=-0.2679$

\textcircled{\small 3}

$\cos {105}^{\circ}$̒l\fbox{\bf\hspace{2pt} C \hspace{2pt}}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂łK؂Ȃ̂IׁB
ȂCKvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin {75}^{\circ}=0.9659$C$\cos {75}^{\circ}=0.2588$C$\tan {75}^{\circ}=3.7321$\\
\textcircled{\small 1}\,$0.2588$~~~
\textcircled{\small 2}\,$-0.2588$~~~
\textcircled{\small 3}\,$0.9659$~~~
\textcircled{\small 4}\,$-0.9659$

process
$\cos {105}^{\circ}=\cos ({180}^{\circ}-{75}^{\circ})=-\cos {75}^{\circ}\\ =-0.2588$

\textcircled{\small 2}

$\cos {145}^{\circ}$̒l\fbox{\bf\, C\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̂łK؂Ȃ̂IׁB
ȂCKvł΁C̎Op̒l𗘗p邱ƁB\\
$\sin {35}^{\circ}=0.5736, \ \cos {35}^{\circ}=0.8192, \\
\tan {35}^{\circ}=0.7002$\\
\textcircled{\small 1}\,$0.5736$~~~
\textcircled{\small 2}\,$-0.5736$~~~
\textcircled{\small 3}\,$0.8192$~~~
\textcircled{\small 4}\,$-0.8192$

process
$\cos {145}^{\circ}=\cos ({180}^{\circ}-{35}^{\circ})=-\cos {35}^{\circ}=-0.8192 $

\textcircled{\small 4}

$A$݊płƂC$A$̎Op̑̕g
\fbox{\bf\, E\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̂琳̂IׁB
\begin{table}[!htb]
\begin{center}\begin{tabular}{|c|c|c|c|}
\hline
  & $\sin A$ & $\cos A$ & $\tan A$  \\ \hline
  \textcircled{\small 1} & $+$ & $+$ & $+$ \\ \hline
 \textcircled{\small 2} & $+$ & $-$ & $+$ \\ \hline
 \textcircled{\small 3} & $+$ & $-$ & $-$ \\ \hline
 \textcircled{\small 4} & $+$ & $+$ & $-$ \\ \hline
\end{tabular}\end{center}\end{table}

\textcircled{\small 3}

̎ȒPɂȂB\\
$\sin {70}^\circ+\cos {60}^\circ+\cos{120}^\circ-\sin{110}^\circ$

process
$\sin{110}^\circ=\sin({180}^\circ-{70}^\circ)=\sin {70}^\circ$ł邩C\\
^$=\cos {60}^\circ+\cos{120}^\circ=\displaystyle\frac{1}{2}-\displaystyle\frac{1}{2}=0$

$0$

̎ȒPɂȂB\\
$\sin ({180}^\circ-x)+\cos ({90}^\circ-x)$

process
^$=\sin x+\sin x=2\sin x$

$2\sin x$

${\sin}^2 {120}^\circ+{\cos}^2{30}^\circ$̒l߂B

process
$\left( \displaystyle\frac{\sqrt{3}}{2} \right)^2+\left( \displaystyle\frac{\sqrt{3}}{2} \right)^2=\displaystyle\frac{3}{2}$

$\displaystyle\frac{3}{2}$

${\sin}^2 {120}^\circ+{\cos}^2{60}^\circ$̒l߂B

process
$\left( \displaystyle\frac{\sqrt{3}}{2} \right)^2+\left( \displaystyle\frac{1}{2} \right)^2=1$

$1$

$\sin {120}^\circ$̒l\fbox{\bf\, E \,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$\displaystyle\frac{1}{2}$~~~
\textcircled{\small 2}\,$\displaystyle\frac{\sqrt{3}}{2}$~~~
\textcircled{\small 3}\,$-\displaystyle\frac{1}{2}$~~~
\textcircled{\small 4}\,$-\displaystyle\frac{\sqrt{3}}{2}$

\textcircled{\small 2}





[EOF]
% t@C̍Ōɂ
