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現在，One Moreの数学II・B版を作成中です！
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  2. 数学I（One More）
  3. 4章：図形と計量
  4. 1節：三角比の定義・性質
  5. 【数学I】4章：図形と計量（基本事項）

【数学I】4章：図形と計量（基本事項）

数学I（One More） 4章：図形と計量 1節：三角比の定義・性質 2節：正弦定理と余弦定理 3節：図形の計量
【数学I】基本事項4章，図形と計量
【数学I】基本事項4章，三角比の定義性質，三角比，三角比の拡張
【数学I】基本事項4章，三角比の値と符号，三角比の相互関係，有名な三角比の値，余角補角の三角比
【数学I】基本事項4章，正弦定理と余弦定理，正弦定理，余弦定理，角と辺の大小関係
【数学I】基本事項4章，図形の計量，三角形の面積，三角形の面積と内接円
検索用コード（LaTeX）
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% 基本事項I4.1.1：三角比（One More）
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(1) $\angle\mathrm{C}$が直角の直角三角形$\mathrm{ABC}$において，鋭角$\mathrm{A}$に対する辺の長さの比の値を$\angle\mathrm{A}$の正弦，余弦，正接といい，これらをまとめて三角比という．

\begin{align*}
\text{正弦(sine)}&:\sin A=\frac{\text{対辺}}{\text{斜辺}}=\frac{\mathrm{BC}}{\mathrm{AB}}=\frac{a}{c}\\
\text{余弦(cosine)}&:\cos A=\frac{\text{底辺}}{\text{斜辺}}=\frac{\mathrm{AC}}{\mathrm{AB}}=\frac{b}{c}\\
\text{正接(tangent)}&:\tan A=\frac{\text{対辺}}{\text{底辺}}=\frac{\mathrm{BC}}{\mathrm{AC}}=\frac{a}{b}
\end{align*}

(2) 有名な角（$30^{\circ},45^{\circ},60^{\circ}$）の三角比

\begin{tabular}{|c||c|c|c|}\hline
$A$&$30^\circ$&$45^\circ$&$60^\circ$\\\hline
$\sin{A}$&$\frac{1}{2}$&$\frac{1}{\sqrt{2}}$&$\frac{\sqrt{3}}{2}$\\\hline
$\cos{A}$&$\frac{\sqrt{3}}{2}$&$\frac{1}{\sqrt{2}}$&$\frac{1}{2}$\\\hline
$\tan{A}$&$\frac{1}{\sqrt{3}}$&$1$&$\sqrt{3}$\\\hline
\end{tabular}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 基本事項I4.1.2：三角比の拡張（One More）
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$0^{\circ}\leqq\theta\leqq 180^{\circ}$のとき，下の図において

\[
\sin\theta=\frac{y}{r},\cos\theta=\frac{x}{r},\tan\theta=\frac{y}{x}
\]

ただし，$\tan 90^{\circ}$の値は定義されない．とくに，原点を中心とする半径が1の半円上において，

\[
\sin\theta=y,\cos\theta=x
\]

ここで，$-1\leqq x\leqq 1,0\leqq y\leqq 1$であるから$-1\leqq\cos\theta\leqq 1,0\leqq\sin\theta\leqq 1$である．

また，直線$x=1$と，直線OPの交点を$\mathrm{T}(1,m)$とすると，$\tan\theta=\frac{y}{x}=\frac{m}{1}$

よって，$\tan\theta=m(\theta\neq 90^{\circ})$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 基本事項I4.1.3：三角比の値と符号（One More）
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{tabular}{|c||c|c|c|c|c|}
\hline$\theta$&$0^{\circ}$&鋭角&$90^{\circ}$&鈍角&$180^{\circ}$\\

\hline$\sin\theta$&$0$&$+$&$1$&$+$&$0$\\

\hline$\cos\theta$&$1$&$+$&$0$&$-$&$-1$\\

\hline$\tan\theta$&$0$&$+$&------&$-$&$0$\\

\hline
\end{tabular}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 基本事項I4.1.4：三角比の相互関係（One More）
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\[
\tan\theta=\frac{\sin\theta}{\cos\theta},\sin^2\theta+\cos^2\theta=1,1+\tan^2\theta=\frac{1}{\cos^2\theta}
\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 基本事項I4.1.5：有名な三角比の値（One More）
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|}\hline
$\theta$&$0^\circ$&$30^\circ$&$45^\circ$&$60^\circ$&$90^\circ$&$120^\circ$&$135^\circ$&$150^\circ$&$180^\circ$\\\hline
$\sin{\theta}$&$0$&$\frac{1}{2}$&$\frac{1}{\sqrt{2}}$&$\frac{\sqrt{3}}{2}$&$1$&$\frac{\sqrt{3}}{2}$&$\frac{1}{\sqrt{2}}$&$\frac{1}{2}$&$0$\\\hline
$\cos{\theta}$&$1$&$\frac{\sqrt{3}}{2}$&$\frac{1}{\sqrt{2}}$&$\frac{1}{2}$&$0$&$-\frac{1}{2}$&$-\frac{1}{\sqrt{2}}$&$-\frac{\sqrt{3}}{2}$&$-1$\\\hline
$\tan{\theta}$&$0$&$\frac{1}{\sqrt{3}}$&$1$&$\sqrt{3}$&------&$-\sqrt{3}$&$-1$&$-\frac{1}{\sqrt{3}}$&$0$\\\hline
\end{tabular}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 基本事項I4.1.6：余角・補角の三角比（One More）
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(1) $90^{\circ}-\theta$（余角）の三角比$(0^{\circ}\leqq\theta\leqq 90^{\circ})$

\begin{align*}
&\sin(90^{\circ}-\theta)=\cos\theta,\\
&\cos(90^{\circ}-\theta)=\sin\theta,\\
&\tan(90^{\circ}-\theta)=\frac{1}{\tan\theta}(\theta\neq 0^{\circ},\theta\neq 90^{\circ})
\end{align*}

(2) $180^{\circ}-\theta$（補角）の三角比$(0^{\circ}\leqq\theta\leqq 180^{\circ})$

\begin{align*}
&\sin(180^{\circ}-\theta)=\sin\theta,\\
&\cos(180^{\circ}-\theta)=-\cos\theta,\\
&\tan(180^{\circ}-\theta)=-\tan\theta(\theta\neq 90^{\circ})
\end{align*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 基本事項I4.2.1：正弦定理（One More）
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

三角形の外接円の半径を$R$とすると，

\[
\frac{a}{\sin{A}}=\frac{b}{\sin{B}}=\frac{c}{\sin{C}}=2R
\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 基本事項I4.2.2：余弦定理（One More）
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{align*}
&a^2=b^2+c^2-2bc\cos{A},\\
&b^2=c^2+a^2-2ca\cos{B},\\
&c^2=a^2+b^2-2ab\cos{C}
\end{align*}

式を整理すると，次の等式が得られる．

\begin{align*}
&\cos{A}=\frac{b^2+c^2-a^2}{2bc},\\
&\cos{B}=\frac{c^2+a^2-b^2}{2ca},\\
&\cos{C}=\frac{a^2+b^2-c^2}{2ab}
\end{align*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 基本事項I4.2.3：角と辺の大小関係（One More）
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$\triangle\mathrm{ABC}$において，$\mathrm{AB}=c,\mathrm{BC}=a,\mathrm{CA}=b$とする．ここで，$\angle\mathrm{A}$が鋭角，直角，鈍角であるとき，それぞれ，$\cos A>0$，$\cos A=0$，$\cos A<0$であることから，次の関係がいえる．

\[
A\text{が鋭角}\Longleftrightarrow b^2+c^2>a^2,
\]

\[
A\text{が直角}\Longleftrightarrow b^2+c^2=a^2,
\]

\[
A\text{が鈍角}\Longleftrightarrow b^2+c^2<a^2
\]

また，角の大小と辺の大小は一致する．すなわち，

\[
A>B\Longleftrightarrow a>b
\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 基本事項I4.3.1：三角形の面積（One More）
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$\triangle\mathrm{ABC}$の面積を$S$とする．

(1) 2辺とその間の角が与えられているとき，
\[
S=\frac{1}{2}bc\sin A=\frac{1}{2}ca\sin B=\frac{1}{2}ab\sin C
\]

(2) ヘロンの公式
\[
S=\sqrt{s(s-a)(s-b)(s-c)}(s=\frac{a+b+c}{2})
\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 基本事項I4.3.2：三角形の面積と内接円（One More）
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

三角形の3辺に接する円を，その三角形の内接円という．

$\triangle\mathrm{ABC}$の面積を$S$，内接円の半径を$r$とすると，

\[
S=\frac{1}{2}(a+b+c)r
\]
数学I（One More） 4章：図形と計量 1節：三角比の定義・性質 2節：正弦定理と余弦定理 3節：図形の計量
Onemath OneMore 三角比の定義・性質 図形と計量 図形の計量 基本事項I 数学I 正弦定理と余弦定理
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