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初等数学公式集/初等関数の性質

三角関数

基本公式

• 三角関数相互の関係
• ${\displaystyle {\frac {\pi }{2}}+\theta }$
• ${\displaystyle \sin \left({\frac {\pi }{2}}+\theta \right)=\cos \theta }$
• ${\displaystyle \cos \left({\frac {\pi }{2}}+\theta \right)=-\sin \theta }$
• ${\displaystyle \tan \left({\frac {\pi }{2}}+\theta \right)=-{\frac {1}{\tan \theta }}}$
• ${\displaystyle \pi +\theta }$
• ${\displaystyle \sin(\pi +\theta )=-\sin \theta }$
• ${\displaystyle \cos(\pi +\theta )=-\cos \theta }$
• ${\displaystyle \tan(\pi +\theta )=\tan \theta }$
• 三角比の相互関係
• ${\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1}$(ピタゴラスの基本三角公式)
• ${\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}}$
• ${\displaystyle 1+\tan ^{2}\theta ={\frac {1}{\cos ^{2}\theta }}}$
• ${\displaystyle 1+{\frac {1}{\tan ^{2}\theta }}={\frac {1}{\sin ^{2}\theta }}}$

• 鋭角における三角比の相互関係（三角比のいずれかが有理数で表されている場合に有用）
• ${\displaystyle \sin \alpha ={\frac {a}{c}}}$であるとき。 ${\displaystyle \cos \alpha ={\frac {\sqrt {c^{2}-a^{2}}}{c}}}$, ${\displaystyle \tan \alpha ={\frac {a}{\sqrt {c^{2}-a^{2}}}}}$
• ${\displaystyle \cos \alpha ={\frac {b}{c}}}$であるとき。 ${\displaystyle \sin \alpha ={\frac {\sqrt {c^{2}-b^{2}}}{c}}}$, ${\displaystyle \tan \alpha ={\frac {\sqrt {c^{2}-b^{2}}}{b}}}$
• ${\displaystyle \tan \alpha ={\frac {a}{b}}}$であるとき。 ${\displaystyle \sin \alpha ={\frac {a}{\sqrt {a^{2}+b^{2}}}}}$, ${\displaystyle \cos \alpha ={\frac {b}{\sqrt {a^{2}+b^{2}}}}}$

負角の公式(還元公式)

• ${\displaystyle \sin(-\theta )=-\sin \theta }$
• ${\displaystyle \cos(-\theta )=\cos \theta }$
• ${\displaystyle \tan(-\theta )=-\tan \theta }$

補角の公式(還元公式)

• ${\displaystyle \sin(\pi -\theta )=\sin \theta }$
• ${\displaystyle \cos(\pi -\theta )=-\cos \theta }$
• ${\displaystyle \tan(\pi -\theta )=-\tan \theta }$
負角の公式との合成
• ${\displaystyle \sin(\theta -\pi )=-\sin \theta }$
• ${\displaystyle \cos(\theta -\pi )=-\cos \theta }$
• ${\displaystyle \tan(\theta -\pi )=\tan \theta }$

余角の公式(還元公式)

• ${\displaystyle \sin \left({\frac {\pi }{2}}-\theta \right)=\cos \theta }$
• ${\displaystyle \cos \left({\frac {\pi }{2}}-\theta \right)=\sin \theta }$
• ${\displaystyle \tan \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\tan \theta }}}$
負角の公式等との合成
• ${\displaystyle \sin \left(\theta -{\frac {\pi }{2}}\right)=-\sin \left({\frac {\pi }{2}}-\theta \right)=-\cos \theta }$
• ${\displaystyle \cos \left(\theta -{\frac {\pi }{2}}\right)=\cos \left({\frac {\pi }{2}}-\theta \right)=\sin \theta }$
• ${\displaystyle \tan \left(\theta -{\frac {\pi }{2}}\right)=-{\frac {1}{\tan \theta }}}$

加法定理

• ${\displaystyle \displaystyle \sin(\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta }$
• ${\displaystyle \cos(\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta }$
• ${\displaystyle \tan(\alpha \pm \beta )={\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }}}$

（すべて複号同順）

二倍角の公式

• ${\displaystyle \displaystyle \sin 2\theta =2\sin \theta \cos \theta }$
• ${\displaystyle \displaystyle \cos 2\theta =\cos ^{2}\theta -\sin ^{2}\theta =2\cos ^{2}\theta -1=1-2\sin ^{2}\theta }$
• ${\displaystyle \tan 2\theta ={\frac {2\tan \theta }{1-\tan ^{2}\theta }}}$

半角の公式

• ${\displaystyle \sin ^{2}\left({\frac {\theta }{2}}\right)={\frac {1-\cos \theta }{2}}}$ ← 倍角の公式より、${\displaystyle \cos \theta =1-2\sin ^{2}\left({\frac {\theta }{2}}\right)}$

• ${\displaystyle \cos ^{2}\left({\frac {\theta }{2}}\right)={\frac {1+\cos \theta }{2}}}$ ← 倍角の公式より、${\displaystyle \cos \theta =2\cos ^{2}\left({\frac {\theta }{2}}\right)-1}$

• ${\displaystyle \tan ^{2}\left({\frac {\theta }{2}}\right)={\frac {1-\cos \theta }{1+\cos \theta }}}$

• ${\displaystyle \tan \left({\frac {\theta }{2}}\right)={\frac {\sin \theta }{1+\cos \theta }}={\frac {1-\cos \theta }{\sin \theta }}}$

（拡張）${\displaystyle \tan \left({\frac {\theta }{2}}\right)=t}$ とするとき、
• ${\displaystyle \cos \theta ={\frac {1-t^{2}}{1+t^{2}}}}$ 　←　 ${\displaystyle \tan ^{2}\left({\frac {\theta }{2}}\right)=t^{2}={\frac {1-\cos \theta }{1+\cos \theta }}}$${\displaystyle \cos \theta }$について解く。
• ${\displaystyle \sin \theta ={\frac {2t}{1+t^{2}}}}$ 　← 　${\displaystyle \sin \theta =\tan \theta \cdot \cos \theta }$${\displaystyle \tan \theta ={\frac {2\tan \left({\frac {\theta }{2}}\right)}{1-\tan ^{2}\left({\frac {\theta }{2}}\right)}}={\frac {2t}{1-t^{2}}},\cos \theta ={\frac {1-t^{2}}{1+t^{2}}}}$を代入する。

三倍角の公式

• ${\displaystyle \displaystyle \sin 3\theta =3\sin \theta -4\sin ^{3}\theta }$
• ${\displaystyle \displaystyle \cos 3\theta =4\cos ^{3}\theta -3\cos \theta }$
• ${\displaystyle \tan 3\theta ={\frac {3\tan \theta -4\tan ^{3}\theta }{1-3\tan ^{2}\theta }}}$

和積の公式

• ${\displaystyle \sin \alpha +\sin \beta =2\sin {\frac {\alpha +\beta }{2}}\cos {\frac {\alpha -\beta }{2}}}$
• ${\displaystyle \sin \alpha -\sin \beta =2\cos {\frac {\alpha +\beta }{2}}\sin {\frac {\alpha -\beta }{2}}}$
• ${\displaystyle \cos \alpha +\cos \beta =2\cos {\frac {\alpha +\beta }{2}}\cos {\frac {\alpha -\beta }{2}}}$
• ${\displaystyle \cos \alpha -\cos \beta =-2\sin {\frac {\alpha +\beta }{2}}\sin {\frac {\alpha -\beta }{2}}}$

積和の公式

• ${\displaystyle \sin \alpha \cdot \cos \beta ={\frac {1}{2}}\left\{\sin(\alpha +\beta )+\sin(\alpha -\beta )\right\}}$
• ${\displaystyle \cos \alpha \cdot \sin \beta ={\frac {1}{2}}\left\{\sin(\alpha +\beta )-\sin(\alpha -\beta )\right\}}$
• ${\displaystyle \cos \alpha \cdot \cos \beta ={\frac {1}{2}}\left\{\cos(\alpha +\beta )+\cos(\alpha -\beta )\right\}}$
• ${\displaystyle \sin \alpha \cdot \sin \beta =-{\frac {1}{2}}\left\{\cos(\alpha +\beta )-\cos(\alpha -\beta )\right\}}$

三角関数の合成

• ${\displaystyle a\sin \theta +b\cos \theta ={\sqrt {a^{2}+b^{2}}}\sin(\theta +\alpha )}$ただし、${\displaystyle \sin \alpha ={\frac {b}{\sqrt {a^{2}+b^{2}}}},\cos \alpha ={\frac {a}{\sqrt {a^{2}+b^{2}}}}}$

${\displaystyle -\theta }$ の三角関数は、点 ${\displaystyle (\cos \theta ,\sin \theta )}$${\displaystyle x}$ 軸で線対称移動移動した点が ${\displaystyle (\cos(-\theta ),\sin(-\theta ))=(\cos \theta ,-\sin \theta )}$ であることから導出できます。

${\displaystyle \cos }$ の倍角の公式 ${\displaystyle \cos 2\theta =2\cos ^{2}\theta -1=1-2\sin ^{2}\theta }$${\displaystyle \pm 1\pm 2\mathrm {aaa} ^{2}\theta }$ という形を覚えて ${\displaystyle \sin }$ は符号が ${\displaystyle -}$、1 の符号はその逆と覚えます。

2乗の三角関数 ${\displaystyle \sin ^{2}\theta ={\frac {1-\cos 2\theta }{2}},\cos ^{2}\theta ={\frac {1+\cos 2\theta }{2}}}$ は、${\displaystyle {\frac {1\pm \cos 2\theta }{2}}}$ という形を覚えて、 ${\displaystyle \sin }$ は符号が${\displaystyle -}$ と考えます。

指数関数・対数関数

指数関数

• ${\displaystyle a^{0}=1,a^{1}=a}$
• ${\displaystyle \displaystyle a^{b}\times a^{c}=a^{b+c}}$
• ${\displaystyle a^{b}\div a^{c}=a^{b-c}}$
• ${\displaystyle a^{-b}={\frac {1}{a^{b}}}}$
• ${\displaystyle \displaystyle (a^{b})^{c}=a^{bc}}$
• ${\displaystyle \displaystyle (ab)^{c}=a^{c}b^{c}}$

対数関数

${\displaystyle a^{b}=c\Leftrightarrow b=\log _{a}c}$
• ${\displaystyle \log _{a}{1}=0}$, ${\displaystyle \log _{a}{a}=1}$
• ${\displaystyle \displaystyle \log _{a}(bc)=\log _{a}b+\log _{a}c}$
• ${\displaystyle \log _{a}\left({\frac {b}{c}}\right)=\log _{a}b-\log _{a}c}$
• ${\displaystyle \displaystyle \log _{a}b^{c}=c\log _{a}b}$
• ${\displaystyle \log _{a}b={\frac {\log _{c}b}{\log _{c}a}}}$
• 特に${\displaystyle \log _{a}b={\frac {1}{\log _{b}a}}}$,　${\displaystyle \log _{a}b\cdot \log _{b}a=1}$
• ${\displaystyle a=b^{\log _{b}a}}$