加法定理の問題

■問題

$3$\displaystyle{ {0}^{^\circ }< \alpha < { 180 }^{^\circ } }$ , $3$\displaystyle{ \cos \alpha =\frac{\hspace{2} 12 \hspace{2}}{\hspace{2} 13 \hspace{2}} }$ のとき　 $3$\displaystyle{ 2\alpha }$ ， $3$\displaystyle{ \frac{\hspace{2}\alpha \hspace{2}}{\hspace{2}2\hspace{2}} }$ の正弦，余弦，正接の値を求めよ．

■答

$3$\displaystyle{ \sin 2{\alpha}=\frac{\hspace{2} 120 \hspace{2}}{\hspace{2} 169 \hspace{2}} }$ ， $3$\displaystyle{ \cos 2{\alpha}=\frac{\hspace{2} 119 \hspace{2}}{\hspace{2} 169 \hspace{2}} }$ ， $3$\displaystyle{ \tan 2{\alpha}=\frac{\hspace{2} 120 \hspace{2}}{\hspace{2} 119 \hspace{2}} }$

$3$\displaystyle{ \sin \frac{\hspace{2}{\alpha}\hspace{2}}{\hspace{2}2\hspace{2}}=\frac{\hspace{2} \sqrt{ 26 } \hspace{2}}{\hspace{2} 26 \hspace{2}} }$ ， $3$\displaystyle{ \cos \frac{\hspace{2}{\alpha}\hspace{2}}{\hspace{2}2\hspace{2}}=\frac{\hspace{2} 5\sqrt{ 26 } \hspace{2}}{\hspace{2} 26 \hspace{2}} }$ ， $3$\displaystyle{ \tan \frac{\hspace{2}{\alpha}\hspace{2}}{\hspace{2}2\hspace{2}}=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 5 \hspace{2}} }$

■ヒント

$3$\displaystyle{ 2{\alpha} }$ と $3$\displaystyle{ \frac{\hspace{2}{\alpha}\hspace{2}}{\hspace{2}2\hspace{2}} }$ の値の正負の範囲に注意して２倍角の公式と半角の公式に代入する．

■解説

$3$\displaystyle{ {0}^{^\circ }< {\alpha}< { 180 }^{^\circ } }$ より $3$\displaystyle{ \sin {\alpha}> 0 }$ である．よって

$3$\displaystyle{ \sin \alpha =\sqrt{ 1- { \cos }^{2}\alpha } }$ $3$\displaystyle{ =\sqrt{ 1- { $ \frac{\hspace{2} 12 \hspace{2}}{\hspace{2} 13 \hspace{2}} $ }^{2} } }$ $3$\displaystyle{ =\sqrt{ \frac{\hspace{2} 25 \hspace{2}}{\hspace{2} 169 \hspace{2}} } }$ $3$\displaystyle{ =\frac{\hspace{2}5\hspace{2}}{\hspace{2} 13 \hspace{2}} }$

$3$\displaystyle{ 2\alpha }$ の場合

$3$\displaystyle{ \sin 2\alpha =2\sin \alpha \cos \alpha =2\times \frac{\hspace{2}5\hspace{2}}{\hspace{2} 13 \hspace{2}}\times \frac{\hspace{2} 12 \hspace{2}}{\hspace{2} 13 \hspace{2}} }$ $3$\displaystyle{ =\frac{\hspace{2} 120 \hspace{2}}{\hspace{2} 169 \hspace{2}} }$

$3$\displaystyle{ \cos 2\alpha =2{ \cos }^{2}\alpha - 1=2\times { $ \frac{\hspace{2} 12 \hspace{2}}{\hspace{2} 13 \hspace{2}} $ }^{2}- 1 }$ $3$\displaystyle{ =\frac{\hspace{2} 288 \hspace{2}}{\hspace{2} 169 \hspace{2}}- 1 }$ $3$\displaystyle{ =\frac{\hspace{2} 119 \hspace{2}}{\hspace{2} 169 \hspace{2}} }$

$3$\displaystyle{ \tan 2{\alpha}=\frac{\hspace{2} \sin 2{\alpha} \hspace{2}}{\hspace{2} \cos 2{\alpha} \hspace{2}}=\frac{\hspace{2} \displaystyle{\frac{\hspace{2} 120 \hspace{2}}{\hspace{2} 169 \hspace{2}}} \hspace{2}}{\hspace{2} \displaystyle{\frac{\hspace{2} 119 \hspace{2}}{\hspace{2} 169 \hspace{2}}} \hspace{2}}=\frac{\hspace{2} 120 \hspace{2}}{\hspace{2} 119 \hspace{2}} }$

$3$\displaystyle{ \frac{\hspace{2}{\alpha}\hspace{2}}{\hspace{2}2\hspace{2}} }$ の場合

$3$\displaystyle{ {0}^{^\circ }< \frac{\hspace{2}{\alpha}\hspace{2}}{\hspace{2}2\hspace{2}}< { 90 }^{^\circ } }$ より $3$\displaystyle{ \sin \frac{\hspace{2}{\alpha}\hspace{2}}{\hspace{2}2\hspace{2}}> 0 }$ ， $3$\displaystyle{ \cos \frac{\hspace{2}\alpha \hspace{2}}{\hspace{2}2\hspace{2}}> 0 }$ である．よって

$3$\displaystyle{ \sin \frac{\hspace{2}\alpha \hspace{2}}{\hspace{2}2\hspace{2}}=\sqrt{ \frac{\hspace{2} 1- \cos \alpha \hspace{2}}{\hspace{2}2\hspace{2}} }=\sqrt{ \frac{\hspace{2} 1- \frac{\hspace{2} 12 \hspace{2}}{\hspace{2} 13 \hspace{2}} \hspace{2}}{\hspace{2}2\hspace{2}} } }$ $3$\displaystyle{ =\sqrt{ \frac{\hspace{2} \frac{\hspace{2}1\hspace{2}}{\hspace{2} 13 \hspace{2}} \hspace{2}}{\hspace{2}2\hspace{2}} } }$ $3$\displaystyle{ =\sqrt{ \frac{\hspace{2}1\hspace{2}}{\hspace{2} 26 \hspace{2}} } }$ $3$\displaystyle{ =\frac{\hspace{2} \sqrt{ 26 } \hspace{2}}{\hspace{2} 26 \hspace{2}} }$

$3$\displaystyle{ \cos \frac{\hspace{2}\alpha \hspace{2}}{\hspace{2}2\hspace{2}}=\sqrt{ \frac{\hspace{2} 1+\cos \alpha \hspace{2}}{\hspace{2}2\hspace{2}} }=\sqrt{ \frac{\hspace{2} 1+\frac{\hspace{2} 12 \hspace{2}}{\hspace{2} 13 \hspace{2}} \hspace{2}}{\hspace{2}2\hspace{2}} } }$ $3$\displaystyle{ =\sqrt{ \frac{\hspace{2} \frac{\hspace{2} 25 \hspace{2}}{\hspace{2} 13 \hspace{2}} \hspace{2}}{\hspace{2}2\hspace{2}} } }$ $3$\displaystyle{ =\sqrt{ \frac{\hspace{2} 25 \hspace{2}}{\hspace{2} 26 \hspace{2}} } }$ $3$\displaystyle{ =\frac{\hspace{2} 5\sqrt{ 26 } \hspace{2}}{\hspace{2} 26 \hspace{2}} }$

$3$\displaystyle{ \tan \frac{\hspace{2}{\alpha}\hspace{2}}{\hspace{2}2\hspace{2}}=\frac{\hspace{2} \sin \frac{\hspace{2}{\alpha}\hspace{2}}{\hspace{2}2\hspace{2}} \hspace{2}}{\hspace{2} \cos \frac{\hspace{2}{\alpha}\hspace{2}}{\hspace{2}2\hspace{2}} \hspace{2}}=\frac{\hspace{2} \displaystyle{\frac{\hspace{2} \sqrt{ 26 } \hspace{2}}{\hspace{2} 26 \hspace{2}}} \hspace{2}}{\hspace{2} \displaystyle{\frac{\hspace{2} 5\sqrt{ 26 } \hspace{2}}{\hspace{2} 26 \hspace{2}}} \hspace{2}}=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 5 \hspace{2}} }$

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最終更新日： 2023年3月15日