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[Title]
406_ʐρCE]藝

% 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}

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% Level1̖BȉLevel7܂œlB
% 1sڂɂ͏ڍאݒ̃^CgB
% 2sڈȍ~ɖƂ̉𓚂B
% Ɖ𓚁C𓚂Ɩ͂PsďB
% vZߒꍇ́CƉ𓚂̊ԂɂPsԊuC
% ŏprocessƂsC̎̍svZߒĂB
[Level1]
Op`̖ʐρispj
}̎Op`ABCɂāC\\
AB$=10 cm$,\ AC $=6cm$, $\angle A=30^\circ$
łB̂ƂCOp`ABC̖ʐς͉$cm^2$B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic207.tex}
\end{center}

process
$S=\displaystyle \frac{1}{2} bc \sin A \\
=\displaystyle \frac{1}{2} \times 6 \times 10 \times \sin 30^\circ
=\displaystyle \frac{1}{2} \times 6 \times 10 \times \displaystyle \frac{1}{2}=15$

$15 cm^2 $

}̕slӌ`ABCDɂāC\\
AB$=6 cm$CAD $=5 cm$C$\angle $DAB $=45^\circ$\\
łB̂ƂCslӌ`ABCD̖ʐς͉$cm^2$B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic216.tex}
\end{center}

process
Op`̖ʐς2{ɂ΂悢B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic216p.tex}
\end{center}
$S=2 \times \displaystyle \frac{1}{2} \times 5 \times 6 \times \sin 45^\circ\\
=2 \times \displaystyle \frac{1}{2} \times 5 \times 6 \times \displaystyle \frac{\sqrt{2}}{2}\\
=15 \sqrt{2}$

$15 \sqrt{2}$

}̎Op`ABCɂāC\\
AB$=4$\,cmCBC$=6$\,cmC$\angle $B$=60^\circ$\\
łB̂ƂCOp`ABC̖ʐς͉$cm^2$B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic208.tex}
\end{center}

process
$S=\displaystyle \frac{1}{2} ca \sin B \\
=\displaystyle \frac{1}{2} \times 4 \times 6 \times \sin 60^\circ
=\displaystyle \frac{1}{2} \times 4 \times 6 \times \displaystyle \frac{\sqrt{3}}{2}\\
=6\sqrt{3}$

$6\sqrt{3}$

}̐p`ɂāCSOƒ_񂾐OA̒$2\,cm$łB
̐p`̖ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic218.tex}
\end{center}

process
$\left( \displaystyle \frac{1}{2} \times 2 \times 2 \times \sin 45^\circ \right) \times 8\\
=\left( \displaystyle \frac{1}{2} \times 2 \times 2 \times \displaystyle \frac{\sqrt{2}}{2} \right) \times 8\\
=8\sqrt{2}$

$8\sqrt{2} \, cm^2$

$b=4,\,c=5,\,A=60^\circ$ł$\triangle$ABC̖ʐ$S$߂B

process
$S=\displaystyle \frac{1}{2} bc \sin A
=\displaystyle \frac{1}{2} \cdot 4 \times 5 \times \displaystyle \frac{\sqrt{3}}{2}
=5\sqrt{3}$

$5\sqrt{3}$

̎Op`̖ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic109.tex}
\end{center}

process
$S=\displaystyle \frac{1}{2} \times 8 \times 10 \times \sin 60^\circ\\
=\displaystyle \frac{1}{2} \times 8 \times 10 \times\displaystyle \frac{\sqrt{3}}{2}$ 

$20\sqrt{3}$

$\triangle$ABC$A=30^\circ,\,b=17,\,c=15$̖ʐς߂B

process
$S=\displaystyle \frac{1}{2} \times 17 \times 15 \times \sin 30^\circ\\
=\displaystyle \frac{1}{2} \times 17 \times 15 \times\displaystyle \frac{1}{2}$ 

$\displaystyle \frac{255}{4}$

}$\triangle$ABCɂāC$\triangle$ABC̖ʐςBC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic112.tex}
\end{center}

process
$S=\displaystyle \frac{1}{2} \times 8 \times 5 \times \sin 60^\circ\\
=\displaystyle \frac{1}{2} \times 8 \times 5 \times\displaystyle \frac{\sqrt{3}}{2}=10\sqrt{3}$\\
$a^2=b^2+c^2-2bc \cos A=8^2+5^2-2 \times 8 \times 5 \cos 60^\circ\\
=8^2+5^2-2 \times 8 \times 5 \displaystyle \frac{1}{2}=49$\\
$a>0$$a=7$

$S=10\sqrt{3},\, a=7$

$\triangle$ABC̖ʐς߂B\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic301.tex}
\end{center}

process
$\displaystyle\frac{1}{2} \cdot  4 \cdot 3 \cdot \sin 30^\circ\\
=\displaystyle\frac{1}{2}\cdot  4 \cdot 3 \cdot \displaystyle\frac{1}{2}=3$

3

}$\triangle$ABCɂāC$\triangle$ABC̖ʐςBC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic111.tex}
\end{center}

process
$S=\displaystyle \frac{1}{2} \times 4 \times 3 \times \sin 60^\circ\\
=\displaystyle \frac{1}{2} \times 4 \times 3 \times\displaystyle \frac{\sqrt{3}}{2}=3\sqrt{3}$\\
$a^2=b^2+c^2-2bc \cos A=4^2+3^2-2 \times 4 \times 3 \cos 60^\circ\\
=4^2+3^2-2 \times 4 \times 3 \displaystyle \frac{1}{2}=13$\\
$a>0$$a=\sqrt{13}$

$S=3\sqrt{3},\, a=\sqrt{13}$

$\triangle$ABC̖ʐς߂B\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic302.tex}
\end{center}

process
$\displaystyle\frac{1}{2} \cdot  5 \cdot 2\sqrt{2} \cdot \sin 45^\circ\\
=\displaystyle\frac{1}{2}\cdot  5 \cdot 2\sqrt{2} \cdot \displaystyle\frac{1}{\sqrt{2}}=5$

5

$\triangle$ABC̖ʐς߂B\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic303.tex}
\end{center}

process
$\displaystyle\frac{1}{2} \cdot  4 \cdot \sqrt{3} \cdot \sin 30^\circ\\
=\displaystyle\frac{1}{2}\cdot  4 \cdot \sqrt{3} \cdot \displaystyle\frac{1}{2}=\sqrt{3}$

$\sqrt{3}$

$\triangle$ABC̖ʐς߂B\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic304.tex}
\end{center}

process
$\displaystyle\frac{1}{2} \cdot  4 \cdot 2\sqrt{2} \cdot \sin 60^\circ\\
=\displaystyle\frac{1}{2}\cdot  4 \cdot 2\sqrt{2} \cdot \displaystyle\frac{\sqrt{3}}{2}=2\sqrt{6}$

$2\sqrt{6}$

$\triangle$ ABCɂāC\\
AB$=5$cmCAC$=2$cmC$\angle${\rm BAC}$=60^\circ$łB\\
$\triangle$ ABC̖ʐς߂ȂB

process
$S=\displaystyle\frac{1}{2}\times 5 \times 2\times \sin 60^\circ$\\
$=\displaystyle\frac{1}{2}\times 5 \times 2\times \displaystyle\frac{\sqrt{3}}{2}=\displaystyle\frac{5\sqrt{3}}{2}$

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

$\triangle$ABC̖ʐς߂B\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic306.tex}
\end{center}

process
$\displaystyle\frac{1}{2} \cdot  2 \cdot (1+\sqrt{3}) \cdot \sin 60^\circ\\
=\displaystyle\frac{1}{2}\cdot  2 \cdot (1+\sqrt{3}) \cdot \displaystyle\frac{\sqrt{3}}{2}=\displaystyle\frac{3+\sqrt{3}}{2}$

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

}̎Op`ABCɂāC\\
${\rm AB=7 cm}$C${\rm AC=7 cm}$C$\sin A=\displaystyle\frac{2}{7}$
łB\\
\hspace{8pt} ̂ƂCOp`ABC̖ʐς\fbox{\,J\,}$cm^2$łB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic401.tex}
\end{center}

process
$S=\displaystyle\frac{1}{2}\times 7 \times 7 \times \sin A\\
=\displaystyle\frac{1}{2}\times 7 \times 7 \times\displaystyle\frac{2}{7}=7$

7

ӂ̒3 cm̐Op`̖ʐς\\
$\displaystyle\frac{\fbox{\,L\,}\sqrt{\fbox{\,N\,}}}{\fbox{\,P\,}}\,{\rm cm}^2$\\
łB\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic402.tex}
\end{center}

process
$\displaystyle\frac{1}{2}\cdot 3 \cdot 3 \cdot\sin 60^\circ\\
=\displaystyle\frac{1}{2}\cdot 3 \cdot  3\cdot\displaystyle\frac{\sqrt{3}}{2}=\displaystyle\frac{9\sqrt{3}}{4} $

$\displaystyle\frac{9\sqrt{3}}{4} $

a2cm̉~Oɓڂ鐳p`̖ʐς\fbox{IJ}$cm^2$łB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/h29m020503.tex}
\end{center}

process
$12\times \displaystyle\frac{1}{2}\times 2 \times 2 \times \displaystyle\frac{1}{2}=12$

12

}̕slӌ`ABCDɂāC
${\rm AB=3cm}$C${\rm AD=2cm}$C${\rm \angle A=60^\circ}$
łB\\
\hspace{8pt} ̂ƂCslӌ`ABCD̖ʐς$\fbox{\, I \,}\sqrt{\fbox{\, J \,}}{\rm cm^2}$
łB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/h2801m0504.tex}
\end{center}

process
$S=2\times \displaystyle\frac{1}{2} \cdot 2\cdot 3 \sin 60^\circ=3\sqrt{3}$

$3\sqrt{3}$

}̎Op`ABCɂāC
${\rm AB=6 cm}$C${\rm AC=4 cm}$
C$\angle {\rm A}={30}^\circ$łB
̂ƂCOp`ABC̖ʐς\fbox{\bf\, I \,}${\rm cm}^2$łB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/picA05.tex}
\end{center}

process
$S=\displaystyle\frac{1}{2} \times 4\times 6 \times \sin{30}^\circ=6$

$6$

















[Level2]
Op`̖ʐρi݊pj
}̎Op`ABCɂāC\\
AB $=5 cm$CAC $=4$cmC$\angle $A $=135^\circ$\\
łB̂ƂCOp`ABC̖ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic223.tex}
\end{center}

process
$S=\displaystyle \frac{1}{2} bc \sin A \\
=\displaystyle \frac{1}{2} \times 4 \times 5 \times \sin 135^\circ
=\displaystyle \frac{1}{2} \times 4 \times 5 \times \displaystyle \frac{1}{\sqrt{2}}\\
=\displaystyle \frac{1}{2} \times 4 \times 5 \times \displaystyle \frac{\sqrt{2}}{2}
=5\sqrt{2}$

$5\sqrt{2}$

̎Op`̖ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic110.tex}
\end{center}

process
$S=\displaystyle \frac{1}{2} \times 12 \times 16 \times \sin 135^\circ\\
=\displaystyle \frac{1}{2} \times 12 \times 16 \times\displaystyle \frac{\sqrt{2}}{2}$ 

$48\sqrt{2}$

}̎Op`ABCɂāC\\
$\rm{AB}=4\,cm,\,\rm{AC}=5\,cm,\,\angle \rm{A}=120^\circ$\\
łB̂ƂCOp`ABC̖ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic241.tex}
\end{center}

process
$S=\displaystyle\frac{1}{2}\times 4 \times 5 \sin 120^\circ
=\displaystyle\frac{1}{2}\times 4 \times 5 \times\displaystyle\frac{\sqrt{3}}{2}=5\sqrt{3}$

$5\sqrt{3}$

$\triangle$ABC̖ʐς߂B\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic305.tex}
\end{center}

process
$\displaystyle\frac{1}{2} \cdot  7 \cdot 8 \cdot \sin 120^\circ \\
=\displaystyle\frac{1}{2}\cdot  7 \cdot 8 \cdot \displaystyle\frac{\sqrt{3}}{2}=14\sqrt{3}$

$14\sqrt{3}$

$\triangle$ABC̖ʐς߂B\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic506.tex}
\end{center}

process
$\displaystyle\frac{1}{2} \cdot  10 \cdot 6 \cdot \sin 120^\circ\\
=\displaystyle\frac{1}{2}\cdot  10 \cdot 6 \cdot \displaystyle\frac{\sqrt{3}}{2}=15\sqrt{3}$

$15\sqrt{3}$







[Level3]
藝ipx͉spj
$\triangle $ABCŁC$\angle A=60^\circ,\angle C=75^\circ,\,BC=\sqrt{6}$̂ƂCAC̒߂B

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic104.tex}
\end{center}
$\angle B=180^\circ-(60^\circ+75^\circ)=45^\circ$\\
$\displaystyle \frac{b}{\sin 45^\circ}=\displaystyle \frac{\sqrt{6}}{\sin 60^\circ}$\\
$b=\displaystyle \frac{\sin 45^\circ}{\sin 60^\circ}
=\displaystyle \frac{\displaystyle \frac{\sqrt{2}}{2}}{\displaystyle \frac{\sqrt{3}}{2}}
=2$

$2$

}$\triangle$ABCɂāCAB̒ƁCOډ~̔a߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic105.tex}
\end{center}

process
$\displaystyle \frac{c}{\sin 45^\circ}=\displaystyle \frac{8}{\sin 30^\circ}=2R$\\
$8 \times \displaystyle \frac{\sin 45^\circ}{\sin 30^\circ}
=8 \times \displaystyle \frac{\displaystyle \frac{\sqrt{2}}{2}}{\displaystyle \frac{1}{2}}
8\sqrt{2}$\\
$R=\displaystyle \frac{8}{2 \sin 30^\circ}
=\displaystyle \frac{8}{2 \times \displaystyle \frac{1}{2}}=8$

$AB=8\sqrt{2},\, R=8$

}$\triangle$ABCŁCBC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic106.tex}
\end{center}

process
$\displaystyle \frac{a}{\sin 45^\circ}=\displaystyle \frac{3\sqrt{2}}{\sin 60^\circ}$\\
$a=3\sqrt{2} \times \displaystyle \frac{\sin 45^\circ}{\sin 60^\circ}
=3\sqrt{2} \times \displaystyle \frac{\displaystyle \frac{\sqrt{2}}{2}}{\displaystyle \frac{\sqrt{3}}{2}}
=2\sqrt{3}$

$2\sqrt{3}$

}$\triangle$ABCɂāCAC̒ƊOډ~̔a߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic107.tex}
\end{center}

process
$\displaystyle \frac{b}{\sin 30^\circ}=\displaystyle \frac{8\sqrt{2}}{\sin 45^\circ}=2R$\\
$b=8\sqrt{2} \times \displaystyle \frac{\sin 30^\circ}{\sin 45^\circ}
=8\sqrt{2} \times \displaystyle \frac{\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{2}}{2}}=8$\\
$R=\displaystyle \frac{4 \sqrt{2}}{\sin 45^\circ}=\displaystyle \frac{4 \sqrt{2}}{\displaystyle \frac{\sqrt{2}}{2}}=8$

$AC=8,\, R=8$

}$\triangle$ABCƂɊOڂ~OɂĕAB̒ƁCAC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic108.tex}
\end{center}

process
$\displaystyle \frac{AB}{\sin 45^\circ}=2\sqrt{3}$C\\
$AB=2\sqrt{3} \times \displaystyle \frac{1}{\sqrt{2}}=\sqrt{6}$\\
$\displaystyle \frac{AC}{\sin 60^\circ}=2\sqrt{3}$C
$AC=2\sqrt{3} \times \displaystyle \frac{\sqrt{3}}{2}=3$

$AB=\sqrt{6},\, AC=3$

}̎Op`ABCɂāC\\
$\angle$A $=45^\circ$C$\angle$C $=30^\circ$CBC$=6 $cm\\
łB̂ƂAB̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic229.tex}
\end{center}

process
藝C\\
$\displaystyle \frac{c}{\sin 30^\circ}=\displaystyle \frac{6}{\sin 45^\circ}$\\
$c=6 \times \displaystyle \frac{\sin 30^\circ}{\sin 45^\circ}
=6 \times \displaystyle \frac{\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{2}}{2}}
=6 \times \displaystyle \frac{1}{\sqrt{2}}=6 \times \displaystyle \frac{\sqrt{2}}{2}\\
=3\sqrt{2}$

$3\sqrt{2}$

$\triangle $ABCɂ$a=4,\,A=45^\circ,\,B=60^\circ$łƂCCA̒$b$߂B

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic101.tex}
\end{center}
$\displaystyle \frac{b}{\sin 60^\circ}=\displaystyle \frac{4}{\sin 45^\circ}$\\
$b=4 \times \displaystyle \frac{\sin 60^\circ}{\sin 45^\circ}=4 \times \displaystyle \frac{\displaystyle \frac{\sqrt{3}}{2}}{\displaystyle \frac{\sqrt{2}}{2}}$

$2\sqrt{6}$

$\triangle$ABCɂāC̊֌WƂC$a$߂B\\
$A=30^\circ,\,B=45^\circ,\,b=7\,cm$

process
藝C\\
$\displaystyle \frac{a}{\sin 30^\circ}=\displaystyle \frac{7}{\sin 45^\circ}$\\
$a=7 \times \displaystyle \frac{\sin 30^\circ}{\sin 45^\circ}
=7 \times \displaystyle \frac{\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{2}}{2}}
=7 \times \displaystyle \frac{\sqrt{2}}{2}$

$\displaystyle \frac{7 \sqrt{2}}{2}$

}̎Op`ɂāCAC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic237.tex}
\end{center}

process
$\angle$A$=45^\circ$ł邩C\\
$\displaystyle\frac{x}{\sin 60^\circ}=\displaystyle\frac{10}{\sin 45^\circ}$\\
$x=10\times \displaystyle\frac{\sin 60^\circ}{\sin 45^\circ}
=\displaystyle\frac{\displaystyle\frac{\sqrt{3}}{2}}{\displaystyle\frac{\sqrt{2}}{2}}=5\sqrt{6}$

$5\sqrt{6}$

}̎Op`ABCɂāC\\
$\angle \rm{A}=45^\circ,\,\angle\rm{B}=60^\circ,\,\rm{AC}=3\sqrt{6}\,\rm{cm}$\\
łB̂ƂCBC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic243.tex}
\end{center}

process
$\displaystyle\frac{a}{\sin 45^\circ}=\displaystyle\frac{3\sqrt{6}}{\sin 60^\circ}
=\displaystyle\frac{a}{\displaystyle\frac{\sqrt{2}}{2}}=\displaystyle\frac{3\sqrt{6}}{\displaystyle\frac{\sqrt{3}}{2}}$\\
$a=6$

6\,cm

$\triangle$ABCɂāCBC$=4$C$\cos A=\displaystyle\frac{1}{3}$łB\\
iIj\,$\sin A$̒l߂ȂB\\
iI\hspace{-.1em}Ij\,$\triangle$ABC̊Oډ~̔aR̒߂ȂB

process
iIj\,$\sin^2 A+\cos^2 A=1$\\
$\sin^2 A+\left( \displaystyle\frac{1}{3} \right)^2=1$\\
$\sin^2 A=\displaystyle\frac{8}{9}$\\
ŁC$A<180$$\sin A>0$C$\sin A=\displaystyle\frac{2\sqrt{2}}{3}$\\
iI\hspace{-.1em}Ij\,
$\displaystyle\frac{4}{\sin A}=2R$\\
$\displaystyle\frac{4}{\displaystyle\frac{2\sqrt{2}}{3}}=2R$\\
$R=\displaystyle\frac{3}{\sqrt{2}}=\displaystyle\frac{3\sqrt{2}}{2}$

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

$\triangle$ABCɂāCAC$=5$C$\angle$ABC$=30^\circ$C$\angle$ACB$=45^\circ$łB
AB̒߂ȂB

process
$\displaystyle\frac{c}{\sin 45 ^\circ}=\displaystyle\frac{5}{\sin 30^\circ}$\\
$\displaystyle\frac{c}{\displaystyle\frac{\sqrt{2}}{2}}=\displaystyle\frac{5}{\displaystyle\frac{1}{2}}$\\
$\displaystyle\frac{c}{\sqrt{2}}=\displaystyle\frac{5}{1}$C$c=5\sqrt{2}$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic307.tex}
\end{center} 

$c=5\sqrt{2}$

}̎Op`ABCɂāC\\
${\rm AB}=6 {\rm cm}$C$\sin B=\displaystyle\frac{2}{7}$C$\sin C=\displaystyle\frac{3}{7}$łB\\
\hspace{8pt} ̂ƂAC̒\fbox{\,I\,}cmłB\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/h2701m0503.tex}
\end{center} 

process
$\displaystyle\frac{\rm AC}{\sin B}=\displaystyle\frac{6}{\sin C}$\\
${\rm AC}=6\times \displaystyle\frac{\sin B}{\sin C}=6\times\displaystyle\frac{2/7}{3/7}=6\times\displaystyle\frac{2}{3}=4$

4

}̎Op`ABCɂāC\\
$\angle {\rm A}=30^\circ$C$\angle {\rm B}=105^\circ$C${\rm AB=4 cm}$
łB\\
\hspace{8pt} ̂ƂCBC̒\fbox{\,I\,}$\sqrt{\fbox{\,J\,}}$ cmłB\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/h2602m0502.tex}
\end{center} 

process
$\displaystyle\frac{B}{\sin 30^\circ}=\displaystyle\frac{4}{\sin 45^\circ}$\\
$\therefore {\rm BC}=4 \times \displaystyle\frac{\sin 30^\circ}{\sin 45^\circ}
=4 \times\displaystyle\frac{\displaystyle\frac{1}{2}}{\displaystyle\frac{\sqrt{2}}{2}}
=4\times \displaystyle\frac{1}{\sqrt{2}}=4\times \displaystyle\frac{\sqrt{2}}{2} \\=2\sqrt{2}$

$2\sqrt{2}$

}̎Op`ABCɂāC\\
AB$=6 cm$C$\sin A=\displaystyle\frac{1}{4}$C$\sin B=\displaystyle\frac{3}{4}$\\
łB̂ƂCBC͉̒$\fbox{\,J\,}$ cmłB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/h2801m0503.tex}
\end{center}

process
$\displaystyle\frac{a}{\sin A}=\displaystyle\frac{6}{\sin B}$\\
$a=6\times \displaystyle\frac{\sin A}{\sin B}=6\times\displaystyle\frac{1/4}{3/4}=2$

$2$

\hspace{8pt}
}̎Op`ABCɂāC\\
$\angle A=30^\circ$C$\angle C=45^\circ$C${\rm BC=3cm}$łB\\
\hspace{8pt} ̂ƂC
AB̒$\fbox{\, L \,}\sqrt{\fbox{\, N \,}}{\rm cm^2}$łB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/h2802m0503.tex}
\end{center}

process
$\displaystyle\frac{c}{\sin 45^\circ}=\displaystyle\frac{3}{\sin 30^\circ}$\\
$\therefore c=\displaystyle\frac{\sin 45^\circ}{\sin 30^\circ}=\displaystyle\frac{\sqrt{2}/2}{1/2}=3\sqrt{2}$

$3\sqrt{2}$

}$\triangle {\rm ABC}$ŁC$\angle {A}=30^\circ$C${\rm AC}=2$C${BC}=\sqrt{2}$̂ƂC$\angle {\rm B}$̑傫߂ȂBCB͉spƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic504.tex}
\end{center}

process
$\displaystyle\frac{\sqrt{2}}{\sin {30}^\circ}=\displaystyle\frac{2}{\sin B}$\\
$\sin B=\sin {30}^\circ\times \displaystyle\frac{2}{\sqrt{2}}
=\displaystyle\frac{1}{2}\times \displaystyle\frac{2}{\sqrt{2}}=\displaystyle\frac{1}{\sqrt{2}}$

${45}^\circ$

}̎Op`ABCɂāC
${\rm AB=10cm}$C$\sin B=\displaystyle\frac{1}{6}$C
$\sin C=\displaystyle\frac{5}{6}$
łB̂ƂCAC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/h3001m505.tex}
\end{center}

process
$\displaystyle\frac{b}{\sin B}=\displaystyle\frac{10}{\sin C}$\\
$b=\displaystyle\frac{\sin B}{\sin C}=10 \times \displaystyle\frac{1}{5}=2$

$2$

AnBnʂRC܂ŃhCu邱ƂɂȂB
AnBnƎRފp$30^\circ$łCAn$20km$ꂽBnɒĂC
AnƎRCފp𑪂$135^\circ$łB
BnRC܂ł̋߂BC$\sin 15^\circ=0.25$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic103.tex}
\end{center}

process
$\displaystyle \frac{a}{\sin 30^\circ}=\displaystyle \frac{20}{\sin 15^\circ}$\\
$a=20 \times \displaystyle \frac{\sin 30^\circ}{\sin 15^\circ}
=20 \times\displaystyle \frac{\displaystyle \frac{1}{2}}{0.25}
=20\times \displaystyle \frac{0.5}{0.25}=40$

$40km$

}̎Op`ABCɂāC
${\rm AB=9cm}$C$\sin B=\displaystyle\frac{2}{5}$C
$\sin C=\displaystyle\frac{3}{5}$
łB̂ƂAC̒\fbox{\bf\, G \,}cmłB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic508.tex}
\end{center}

process
$\displaystyle\frac{b}{2/5}=\displaystyle\frac{9}{3/5}$\\
$\therefore b=\displaystyle\frac{2}{3}=6$

$6$





[Level4]
藝ipx͓݊pCpj
̐}ɂāCABԂ̋߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic102.tex}
\end{center}

process
$B=30^\circ$C\\
$\displaystyle \frac{c}{\sin 120^\circ}=\displaystyle \frac{1000}{\sin 30^\circ}$\\
$c=1000 \times \displaystyle \frac{\sin 120^\circ}{\sin 30^\circ}
=\displaystyle \frac{\displaystyle \frac{\sqrt{3}}{2}}{\displaystyle \frac{1}{2}}=1000 \sqrt{3}$

$1000 \sqrt{3} \, m$

$\triangle$ABCŁC$A=120^\circ,\,a=2\sqrt{3},\,b=2$̂ƂC$B$$c$߂B

process
$\displaystyle \frac{2\sqrt{3}}{\sin 120^\circ}=\displaystyle \frac{2}{\sin B}$\\
$\sin B=\displaystyle \frac{2 \sin 120^\circ}{2\sqrt{3}}
=\displaystyle \frac{2 \times \displaystyle \frac{\sqrt{3}}{2}}{2\sqrt{3}}
=\displaystyle \frac{1}{2}$\\
$B=30^\circ$CāC$C=180^\circ-(120^\circ+30^\circ)=30^\circ$C
$\triangle$ABC͓񓙕ӎOp`ł邩C$c=2$

$B=30^\circCc=2$

}̎Op`ABCɂāC\\
$\angle \rm{A}=30^\circ,\,\angle\rm{B}=135^\circ,\,\rm{BC}=5\,\rm{cm}$\\
łB̂ƂCAC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic239.tex}
\end{center}

process
$\displaystyle\frac{\rm{AC}}{\sin 135^\circ}=\displaystyle\frac{5}{\sin 30^\circ}$\\
$\rm{AC}=5\times\displaystyle\frac{\sin 135^\circ}{\sin 30^\circ}
=5 \times \displaystyle\frac{\displaystyle\frac{\sqrt{2}}{2}}{\displaystyle\frac{1}{2}}
=5\sqrt{2}$

$5\sqrt{2}$

$\triangle {\rm ABC}$ŁC$\angle {A}=30^\circ$C${\rm AC}=2$C${BC}=\sqrt{2}$̂ƂC$\angle {\rm B}$̑傫߂ȂB$90^\circ <{\rm B}<180^\circ$ƂB

process
$\displaystyle\frac{\sqrt{2}}{\sin {30}^\circ}=\displaystyle\frac{2}{\sin B}$\\
$\sin B=\sin {30}^\circ\times \displaystyle\frac{2}{\sqrt{2}}
=\displaystyle\frac{1}{2}\times \displaystyle\frac{2}{\sqrt{2}}=\displaystyle\frac{1}{\sqrt{2}}$

${135}^\circ$







[Level5]
藝iOډ~̔aj
$\triangle$ABCɂāCAC$=5$C$\angle$ABC$=30^\circ$C$\angle$ACB$=45^\circ$łB
$\triangle$ABC̊Oډ~̔aR߂ȂB

process
$\displaystyle\frac{5}{\sin 30^\circ}=2R$\\
$\displaystyle\frac{5}{\displaystyle\frac{1}{2}}=2R$\\
$\therefore R=5$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic307.tex}
\end{center} 

$R=5$

$\triangle $ABCɂāC$a=6CA=45^\circ$̂ƂC$\triangle $ABC̊Oډ~̔a$R$߂B

process
$\displaystyle \frac{6}{\sin 45^\circ}=2R$\\
$R=\displaystyle \frac{6}{2\times \displaystyle \frac{\sqrt{2}}{2}}
=3\sqrt{2}$

$3\sqrt{2}$

̂悤$\triangle $ABCɂāCOډ~̔a$R$߂B\\
$a=3,\,A=60^\circ$

process
$\displaystyle \frac{3}{\sin 60^\circ}=2R$\\
$R=\displaystyle \frac{3}{2\times \displaystyle \frac{\sqrt{3}}{2}}
=\sqrt{3}$

$\sqrt{3}$

̂悤$\triangle $ABCɂāCOډ~̔a$R$߂B\\
$b=5,\,B=30^\circ$

process
$\displaystyle \frac{5}{\sin 30^\circ}=2R$\\
$R=\displaystyle \frac{5}{2\times \displaystyle \frac{1}{2}}
=5$

$5$

̂悤$\triangle $ABCɂāCOډ~̔a$R$߂B\\
$c=\sqrt{2},\,C=135^\circ$

process
$\displaystyle \frac{\sqrt{2}}{\sin 135^\circ}=2R$\\
$R=\displaystyle \frac{\sqrt{2}}{2\times \displaystyle \frac{\sqrt{2}}{2}}
=1$

$1$

̂悤$\triangle $ABCɂāCOډ~̔a$R$߂B\\
$a=4,\,B=45^\circ,\,C=15^\circ$

process
$A=120^\circ$C\\
$\displaystyle \frac{4}{\sin 120^\circ}=2R$\\
$R=\displaystyle \frac{4}{2\times \displaystyle \frac{\sqrt{3}}{2}}
=\displaystyle \frac{4\sqrt{3}}{3}$

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

$a=8$ł$\triangle $ABCɂāCOډ~̔a$R=4$̂ƂCp$A$߂B

process
$\displaystyle \frac{8}{\sin A}=2 \times 4$\\
$\sin A=1$C$A=90^\circ$

$90^\circ$

$\triangle {\rm ABC}$ɂāC
${\rm BC}=6$C$\cos A=\displaystyle\frac{3}{4}$łB
$\triangle {\rm ABC}$̊Oډ~̔a$R$̒߂ȂB

process
$\sin A=\sqrt{1-\cos^2 A}=\sqrt{1-\left( \displaystyle\frac{3}{4} \right)^2}
=\displaystyle\frac{\sqrt{7}}{4}$\\
$\displaystyle\frac{a}{\sin A}=2R$C\\
$R=\displaystyle\frac{a}{2\sin A}=\displaystyle\frac{6}{2\times \displaystyle\frac{\sqrt{7}}{4}}=\displaystyle\frac{12}{\sqrt{7}}=
\displaystyle\frac{12\sqrt{7}}{7}$

$\displaystyle\frac{12\sqrt{7}}{7}$








[Level6]
]藝ispj
}̎Op`ABCɂāCAB$=3cm$,\, AC$=1cm$,\, $\cos A=-\displaystyle \frac{1}{3}$łB
̂ƂCBC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic204.tex}
\end{center}

process
$a^2=b^2+c^2-2bc \cos A \\
=3^2+1^2-2 \times 3 \times 1 \times \left( -\displaystyle \frac{1}{3}\right)=12$\\
$a=\sqrt{12}=\sqrt{4 \times 3}=\sqrt{2^2 \times 3}=2 \sqrt{3}$

$2 \sqrt{3}$

}̎Op`ABCɂāC
AB $= 4cm$,AC $=3 cm$, $\cos A=\displaystyle \frac{1}{8}$łB
̂ƂCBC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic202.tex}
\end{center}

process
$a^2=b^2+c^2-2bc \cos A \\
=3^2+4^2-2 \times 3 \times 4 \times  \displaystyle \frac{1}{8}
=22$\\
$a=\sqrt{22}$

$a=\sqrt{22}$

}̎Op`ABCɂāC\\
AB$=3 cm$CAC$=7 cm$C$\cos A =\displaystyle \frac{11}{14}$\\
łB̂ƂCBC͉̒$cm$B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic215.tex}
\end{center}

process
$a^2=b^2+c^2-2bc \cos A \\
=3^2+7^2-2 \times 3 \times 7 \times  \displaystyle \frac{11}{14}
=25$\\
$a=5$

$5 cm$

}̎Op`ABCɂāC\\
AB $=4 cm$CAC $=3 \sqrt{3}$cmC$\angle $A $=30^\circ$\\
łB̂ƂCBC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic222.tex}
\end{center}

process
$a^2=b^2+c^2-2bc \cos A \\
=(3\sqrt{3})^2+4^2-2 \times 3\sqrt{3} \times 4 \times  \displaystyle \frac{\sqrt{3}}{2}\\
=9 \times 3+16-2 \times 3\sqrt{3} \times 4 \times \displaystyle \frac{\sqrt{3}}{2}=7$\\
$a=\sqrt{7}$

$\sqrt{7}$

}̎Op`ABCɂāC\\
AB$=5$cmCAC$=6$cmC$\angle$A$=60^\circ$\\
łB̂ƂCBC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic227.tex}
\end{center}

process
$a^2=b^2+c^2-2bc \cos A \\
=5^2+6^2-2 \times 5 \times 6 \times  \displaystyle \frac{1}{2}
=25+36-30=31$\\
$a=\sqrt{31}$

$\sqrt{31}$

}̎Op`ABCɂāC\\
AB$=3$\,cmCAC$=$5 cmC$\cos A=\displaystyle \frac{4}{5}$\\
łB̂ƂCBC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic232.tex}
\end{center}

process
$a^2=b^2+c^2-2bc \cos A \\
=5^2+3^2-2 \times 5 \times 3 \times \displaystyle \frac{4}{5}\\
=25+9-24=10$\\
$a=\sqrt{10}$

$\sqrt{10}$\,cm

}̎Op`ABCɂāCAB$=\sqrt{3} cm$,\, AC$=5 cm$,\, $\angle $ A $=30^\circ$łB
̂ƂCBC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic236.tex}
\end{center}

process
$a^2=b^2+c^2-2bc \cos A \\
=5^2+(\sqrt{3})^2-2 \times 5 \times \sqrt{3} \times \displaystyle \frac{\sqrt{3}}{2}=13$\\
$a=\sqrt{13}$

$\sqrt{13}$

}̂悤ɁCR͂2̒n_ACBB
2n_ACB̒߂邽߂ɁCn_ACBƂɌnn_CɈړCʂƂC\\
AC$=3km$CBC$=4km$C$\angle $ACB$=60^\circ$\\
łB̂ƂCACBԂ̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic217.tex}
\end{center}

process
$c^2=4^2+3^2-2 \cdot 4 \cdot 3 \cos 60^\circ\\
=16+9-2 \cdot 4 \cdot 3\cdot \displaystyle \frac{1}{2}\\
=13$\\
$c=\sqrt{13}$

$\sqrt{13}$\,km

$\triangle$ABCɂāC$b=3,\,c=8,\,A=60^\circ$łƂCBC̒$a$߂B

process
]藝ɂC\\
$a^2=b^2+c^2-2bc \cos A\\
=3^2+8^2-2 \cdot 3 \cdot 8 \cos 60^\circ\\
=9+64-2 \cdot 3 \cdot 8 \cdot \displaystyle \frac{1}{2}=49$\\
$a>0$ł邩C$a=7$

$7$

̂悤$\triangle$ABCɂāCw肳ꂽ̂߂B\\
$b=4,\,c=3,\,A=60^\circ $̂ƂCBC̒$a$

process
]藝ɂC\\
$a^2=b^2+c^2-2bc \cos A\\
=4^2+3^2-2 \cdot 4 \cdot 3 \cos 60^\circ\\
=16+9-2 \cdot 4 \cdot 3 \cdot \displaystyle \frac{1}{2}=13$\\
$a>0$ł邩C$a=\sqrt{13}$

$\sqrt{13}$

̂悤$\triangle$ABCɂāCw肳ꂽ̂߂B\\
$a=1+\sqrt{3},\,c=\sqrt{2},\,B=45^\circ $̂ƂCCA̒$b$

process
]藝ɂC\\
$b^2=c^2+a^2-2ca \cos B\\
=(\sqrt{2})^2+(1+\sqrt{3})^2-2 \cdot \sqrt{2} \cdot (1+\sqrt{3}) \cos 45^\circ\\
=(\sqrt{2})^2+(1+\sqrt{3})^2-2 \cdot \sqrt{2} \cdot (1+\sqrt{3}) \cdot \displaystyle \frac{1}{\sqrt{2}}\\
=2+1+3+2\sqrt{3}-2(1+\sqrt{3})=4$\\
$b>0$ł邩C$b=2$

$2$

$\triangle$ABCɂāC$a=8,\,b=7,\,c=5$̂ƂC$\cos B$̒lƊp$B$߂B

process
$\cos B=\displaystyle \frac{c^2+a^2-b^2}{2ca}\\
=\displaystyle \frac{5^2+8^2-7^2}{2\cdot 5 \cdot 8}\\
=\displaystyle \frac{1}{2}$\\
$B=60^\circ$

$\cos B=\displaystyle \frac{1}{2},\,B=60^\circ$

̂悤$\triangle$ABCɂāCw肳ꂽ̂߂B\\
$a=\sqrt{37},\,b=4,\,c=7$̂ƂC$\cos A$̒lƊp$A$

process
$\cos A=\displaystyle \frac{b^2+c^2-a^2}{2bc}\\
=\displaystyle \frac{4^2+7^2-(\sqrt{37})^2}{2\cdot 4 \cdot 7}=\displaystyle \frac{1}{2}$\\
$A=60^\circ$

$\cos A=\displaystyle \frac{1}{2},\,A=60^\circ$

̂悤$\triangle$ABCɂāCw肳ꂽ̂߂B\\
$a=7,\,b=5,\,c=4\sqrt{2}$̂ƂC$\cos B$̒lƊp$B$

process
$\cos B=\displaystyle \frac{c^2+a^2-b^2}{2ca}\\
=\displaystyle \frac{(4\sqrt{2})^2+7^2-5^2}{2\cdot 4\sqrt{2} \cdot 7}=\displaystyle \frac{1}{\sqrt{2}}$\\
$B=45^\circ$

$\cos B=\displaystyle \frac{1}{\sqrt{2}},\,B=45^\circ$

}̎Op`ABCɂāC\\
$\rm{AB}=6\,cm,\,\rm{BC}=4\,cm,\,\rm{CA}=5\,cm$\\
łB̂ƂC$\cos A$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic238.tex}
\end{center}

process
$\cos A=\displaystyle\frac{b^2+c^2-a^2}{2bc}
=\displaystyle\frac{5^2+6^2-4^2}{2 \cdot 5 \cdot 6}=\displaystyle\frac{3}{4}$

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

}̎Op`ABCɂāC\\
$\rm{AB}=2\,cm,\,\rm{AC}=3\,cm,\,\angle \rm{A}=60^\circ$\\
łB̂ƂCBC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic240.tex}
\end{center}

process
$a^2=b^2+c^2-2bc\cos A
=3^2+2^2-2\times 3 \times 2 \times \displaystyle\frac{1}{2}=7$\\
$a=\sqrt{7}$

$\sqrt{7}$

}̎Op`ABCɂāC\\
$\rm{AB}=5\,cm,\,\rm{AC}=8\,cm,\,\cos A=\displaystyle\frac{9}{10}$\\
łB̂ƂCBC̒߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic242.tex}
\end{center}

process
$a^2=b^2+c^2-2bc\cos A\\
=8^2+5^2-2\times 8\times 5 \times \displaystyle\frac{9}{10}
=17$\\
$a=\sqrt{17}$

$\sqrt{17}\rm{cm}$

$\triangle$ ABCɂāC\\
AB$=5$cmCAC$=2$cmC$\angle${\rm BAC}$=60^\circ$łB\\
BC̒߂ȂB

process
$x^2=2^2+5^2-2\cdot 2 \cdot 5 \cos 60^\circ$\\
$x^2=4+25-2\cdot 2 \cdot 5 \cdot \displaystyle\frac{1}{2}=19$\\
$x=\sqrt{19}$

$\sqrt{19}$

}̎Op`ABCɂāC\\${\rm AB=4 cm}$C${\rm AC=3 cm}$C$\cos A=-\displaystyle\frac{1}{3}$̂ƂC
BC̒$\sqrt{ \fbox{\,G\,} }$łB
\begin{center}\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/h2702m0502.tex}\end{center}

process
]藝\\
 $x^2=3^2+4^2-2\cdot 3 \cdot 4 \cos A\\
=9+16-2\cdot 3 \cdot 4 \cdot\left( -\displaystyle\frac{1}{3} \right)
=9+16+8=33$\\
$\therefore x=\sqrt{33}$

33

}̎Op`ABCɂāC\\
${\rm AB=5 cm}$C${\rm AC=6 cm}$C$\cos A=\displaystyle\frac{1}{5}$
łB̂ƂCBC̒\fbox{\,G\,}cmłB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/h2701m0502.tex}
\end{center}

process
${\rm BC}^2=6^2+5^2-2\times 6\times 5 \times \displaystyle\frac{1}{5}\\
=36+25-12=49$\\
${\rm BC}=7$

7

}̎Op`ABCɂāC
AB$=5\sqrt{3}$cmCAC$=5$cmC$\angle A=30^\circ$
łB\\
̂ƂCBC̒́C\fbox{\,G\,}łB\\
\begin{center}
 \input{D:/texlive/2018/bin/win32/mathtex/406_menseki/h2602m0501.tex}
 \end{center}

process
${\rm BC}^2=5^2+\left(5\sqrt{3} \right)^2-2\cdot 5 \cdot 5\sqrt{3} \cos 30^\circ\\
=25+75-2\cdot 5 \cdot 5\sqrt{3} \cdot \displaystyle\frac{\sqrt{3}}{2}\\
=25+75-75=25$\\
$\therefore {\rm BC}=5$

5

}̎Op`ABCɂāC\\
${\rm AB=\sqrt{2} cm}$C${\rm AC=4 cm}$C$\angle A = 45^\circ$łB
̂ƂCBC̒$\sqrt{\fbox{IJ}}$ cmłB
\begin{center}
 \input{D:/texlive/2018/bin/win32/mathtex/406_menseki/h2601m0502.tex}
 \end{center}

process
${\rm BC}^2=4^2+\left( \sqrt{2} \right)^2-2\cdot 4 \cdot \sqrt{2} \cdot \cos 45^\circ\\
=16+2-2\cdot 4 \cdot \sqrt{2} \cdot \displaystyle\frac{1}{\sqrt{2}} =10$\\
$\therefore {\rm BC}=\sqrt{10}$

10

}̎Op`ABCɂāC\\
${\rm AB=7 cm}$C${\rm AC=5 cm}$C$\cos A=\displaystyle\frac{1}{7}$
łB̂ƂCBC̒\fbox{\,G\,}cmłB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic403.tex}
\end{center}

process
${\rm BC}^2=5^2+7^2-2\times 5\times 7 \times \displaystyle\frac{1}{7}\\
=25+49-10=64$\\
${\rm BC}=8$

8

}̎lp`ABCDɂāC\\
AB$=4 cm$CBC$=5 cm$C$\angle B=60^\circ$\\
łB̂ƂCΊpAC͉̒$\sqrt{\,\fbox{\,GI\,}\,}$łB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/h2801502.tex}
\end{center}

process
$b^2=c^2+a^2-2ca \cos B \\
=4^2+5^2-2 \times 4 \times 5 \times  \displaystyle \frac{1}{2}
=21$\\
$b=\sqrt{21}$

$21$

}̎Op`ABCɂāC
${\rm AB=6cm}$C${\rm AC=7cm}$C$\cos A=\displaystyle\frac{5}{7}$
łB̂ƂCBC̒\fbox{\hspace{4pt} {\bf G} \hspace{4pt}}cmłB\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/405_sincos/h29m020502.tex}
\end{center}

process
$x^2=7^2+6^2-2\times 7\times 6\times \displaystyle\frac{5}{7}=25$\\
$\therefore x=5$

$5$

}$\triangle {\rm ABC}$ŁCAC̒߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic501.tex}
\end{center}

process
$b^2=c^2+a^2-2ca\cos {60}^\circ\\
=4^2+5^2-2\times 4 \times 5 \times \displaystyle\frac{1}{2}=21$

$\sqrt{21}$

}$\triangle {\rm ABC}$ŁCAB̒߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic502.tex}
\end{center}

process
$c^2=a^2+b^2-2ab\cos {45}^\circ\\
={\sqrt{2}}^2+3^2-2\times \sqrt{2} \times 3 \times \displaystyle\frac{1}{\sqrt{2}}=5$

$\sqrt{5}$

}̎Op`ABCɂāC
${\rm AB=5cm}$C${\rm AC=8cm}$C$\angle A={60}^\circ$
łB̂ƂCBC̒\fbox{\bf\, G \,}cmłB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic507.tex}
\end{center}

process
$a^2=8^2+5^2-2\times 8 \times 5 \times \displaystyle\frac{1}{2}
=64+25-40=49$\\
$\therefore a=7$

$7$

$\triangle {\rm ABC}$ɂāC
${\rm AB}=7$C${\rm BC}=8$C${\rm CA}=9$ƂƂC
$\cos B$߂ȂB

process
$\cos B=\displaystyle\frac{c^2+a^2-b^2}{2ca}
=\displaystyle\frac{7^2+8^2-9^2}{2 \times 7 \times 8}\\
=\displaystyle\frac{32}{2 \times 7 \times 8}
=\displaystyle\frac{2}{7}$

$\displaystyle\frac{2}{7}$

$\triangle {\rm ABC}$ɂāC${\rm AB}=7$C
${\rm BC}=8$C${\rm CA}=9$C$\cos B=\displaystyle\frac{2}{7}$ƂƂC
BC${\rm AB=AD}$ƂȂC_BƂ͈قȂ_D߂B
BD̒߂ȂB

process
${\rm BD}=x$ƂB
$7^2=7^2+x^2-2\times 7 \times x \times \displaystyle\frac{2}{7}\\
0=x^2-4x=x(x-4)$\\
$\therefore x=4 $
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/h29401.tex}
\end{center}

BD$=4$




[Level7]
]藝i݊pj
}$\triangle {\rm ABC}$ŁCBC̒߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic503.tex}
\end{center}

process
$a^2=b^2+c^2-2bc\cos A \\
=1^2+(3\sqrt{2})^2-2\cdot 1\cdot 3\sqrt{2} \left( -\displaystyle\frac{1}{\sqrt{2}} \right)=25$

$5$

$\triangle$ABCɂāC$a=3,\,b=5,\,C=120^\circ$łƂCAB̒$c$߂B

process
]藝ɂC\\
$c^2=a^2+a^2-2ab \cos C\\
=3^2+5^2-2 \cdot 3 \cdot 5 \cos 120^\circ\\
=9+25-2 \cdot 3 \cdot 5 \cdot \left(-\displaystyle \frac{1}{2}\right)=49$\\
$c>0$ł邩C$c=7$

$7$

$\triangle$ ABCɂāC\\
AB$=7$cmCAC$=8$cmC$\angle${\rm BAC}$=120^\circ$łB\\
BC̒߂ȂB

process
$x^2=8^2+7^2-2\cdot 8 \cdot 7 \cos 120^\circ$\\
$x^2=64+49-2\cdot 8 \cdot 7 \cdot \left(-\displaystyle\frac{1}{2} \right)=169$\\
$x=13$

$13$

̂悤$\triangle$ABCɂāCw肳ꂽ̂߂B\\
$a=3,\,b=5,\,c=7$̂ƂC$\cos C$̒lƊp$C$

process
$\cos C=\displaystyle \frac{a^2+b^2-c^2}{2ab}\\
=\displaystyle \frac{3^2+5^2-7^2}{2\cdot 3 \cdot 5}=-\displaystyle \frac{1}{2}$\\
$C=120^\circ$

$\cos C=\displaystyle \frac{1}{2},\,C=120^\circ$

$\triangle$ABCŁCAB$=3$CBC$=7$CCA$=5$̂ƂC$\angle $Ȃ傫߂ȂB

process
$\cos A=\displaystyle \frac{5^2+3^2-7^2}{2 \times 5 \times 3}
=\displaystyle \frac{25+9-49}{2 \times 5 \times 3}=-\displaystyle \frac{1}{2}$\\
$A=120^\circ$

$A=120^\circ$

$\triangle$ABCɂāC̊֌WƂC$a$߂B\\
$A=120^\circ,\,b=13\,cm,\,c=9\, cm$

process
]藝ɂC\\
$a^2=b^2+c^2-2bc \cos A\\
=13^2+9^2-2 \cdot 13 \cdot 9 \cos 120^\circ\\
=169+81-2 \cdot 13 \cdot 9 \cdot \left(-\displaystyle \frac{1}{2}\right)=367$\\
$a>0$ł邩C$a=\sqrt{367}$

$\sqrt{367}\, cm$

}$\triangle {\rm ABC}$ŁC$\angle {\rm C}$̑傫߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/pic505.tex}
\end{center} 

process
$c^2=a^2+b^2-2bc\cos C$C\\
$\cos C=\displaystyle\frac{a^2+b^2-c^2}{2ab}=\displaystyle\frac{8^2+7^2-13^2}{2\times 8 \times 7}\\ =\displaystyle\frac{64+49-169}{2\times 8 \times 7}=-\displaystyle\frac{1}{2}$\\
$0^\circ < C< 180^\circ$${120}^\circ$

${120}^\circ$

}̎Op`ABCɂāC
${\rm AB}=4{\rm cm}$C${\rm AC=5cm}$C$\cos A=-\displaystyle\frac{1}{5}$
łB̂ƂCBC̒\fbox{\bf\, G \,}cmłB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/picA04.tex}
\end{center}

process
$a^2=b^2+c^2-2bc\cos A=\\ 5^2+4^2-2\times 5 \times 4 \left( -\displaystyle\frac{1}{5} \right)=25+16+8=49$\\
$\therefore a=7$

$7$

$\triangle {\rm ABC}$ɂāC
${\rm AB}=3$C
${\rm AC}=2$C
$\angle {\rm BAC}={120}^\circ$łB
BC̒߂ȂB

process
$a^2=b^2+c^2-2bc \cos A
\\
=2^2+3^2-2\times 2 \times 3 \times \cos {120}^\circ\\
=4+9-2\times 6 \times \left( -\displaystyle\frac{1}{2} \right)
=19$\\
$\therefore a=\sqrt{19}$

$\sqrt{19}$




[Level8]
藝C]藝̑
$\triangle {\rm ABC}$ɂāC$BC=4$C$CA=5$C$AB=6$ƂƂC
̊e₢ɂēȂB\\
iij\,$\cos A$߂ȂB\\
ii\hspace{-.1em}ij\,$\sin A$߂ȂB\\
ii\hspace{-.1em}i\hspace{-.1em}ij\,
$\triangle {\rm ABC}$̖ʐ$S$߂ȂB\\
ii\hspace{-.1em}vj\,
$\triangle {\rm ABC}$̓ډ~̔a$r$߂ȂB

process
iij\,$\cos A=\displaystyle\frac{b^2+c^2-a^2}{2bc}=\displaystyle\frac{5^2+6^2-4^2}{2 \times 5\times 6}=\displaystyle\frac{3}{4}$\\
ii\hspace{-.1em}ij\,$0<\cos A <1$$0<A<\displaystyle\frac{\pi}{2}$
ł邩C$\sin A>0$B\\
$\sin A=\sqrt{1-\left( \displaystyle\frac{3}{4} \right)^2}=\displaystyle\frac{\sqrt{7}}{4}$\\
ii\hspace{-.1em}i\hspace{-.1em}ij\,
$S=\displaystyle\frac{1}{2} bc \sin A=\displaystyle\frac{1}{2} \times 5 \times 6 \times \displaystyle\frac{\sqrt{7}}{4}=\displaystyle\frac{15\sqrt{7}}{4}$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/406_menseki/fig_naisetsu.tex}
\end{center}
ii\hspace{-.1em}vj\,}$S=\displaystyle\frac{r}{2}(a+b+c)$\\
$\therefore r=\displaystyle\frac{2S}{a+b+c}=\displaystyle\frac{2}{15} \times \displaystyle\frac{15\sqrt{7}}{4}=\displaystyle\frac{\sqrt{7}}{2}$

iij\,$\displaystyle\frac{3}{4}$C
ii\hspace{-.1em}ij\,$\displaystyle\frac{\sqrt{7}}{4}$C
ii\hspace{-.1em}i\hspace{-.1em}ij\,$\displaystyle\frac{15\sqrt{7}}{4}$C
ii\hspace{-.1em}vj\,$\displaystyle\frac{\sqrt{7}}{2}$





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