三角関数のグラフに関する問題

■問題

$3$\displaystyle{y=\sin \left({ {\theta}- \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}4\hspace{2}} }\right)}$ ， $3$\displaystyle{y=\sin \left({ {\theta}+\frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}4\hspace{2}} }\right)}$ のグラフを描け

■解説動画

◆三角関数の動画一覧のページへ

■答

■ヒント

単位円を描き，グラフの平行移動に注意してグラフを描く．

■解説

● $3$\displaystyle{y=\sin \left({ {\theta}- \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}4\hspace{2}} }\right)}$ のグラフ

グラフの平行移動の式より， $3$\displaystyle{ y=\sin {\theta} }$ のグラフを $3$\displaystyle{{\theta}}$ 軸方向に $3$\displaystyle{a}$ 平行移動したグラフを表す式は， $3$\displaystyle{ y=\sin ${\theta}- a$ }$ となる．このことより

$3$\displaystyle{ y=\sin $ {\theta}- \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}4\hspace{2}} $ }$ のグラフは

$3$\displaystyle{{\theta}}$ 軸方向に $3$\displaystyle{ y=\sin {\theta} }$ のグラフを $3$\displaystyle{\frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}4\hspace{2}}}$ 平行移動したもの

になる．

$3${\theta}$	$3$0$	$3$\frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$	$3$\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{\pi}$	$3$\frac{\hspace{2}3\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$	$3${\pi}$	$3$\frac{\hspace{2}5\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$	$3$\frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}}{\pi}$	$3$\frac{\hspace{2}7\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$	$3$2{\pi}$
$3${\theta}+\frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$	$3$\frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$	$3$\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{\pi}$	$3$\frac{\hspace{2}3\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$	$3${\pi}$	$3$\frac{\hspace{2}5\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$	$3$\frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}}{\pi}$	$3$\frac{\hspace{2}7\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$	$3$2{\pi}$	$3$\frac{\hspace{2}9\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$
$3$\sin {\theta}$	$3$0$	$3$\frac{\hspace{2} \sqrt{2} \hspace{2}}{\hspace{2}2\hspace{2}}$	$3$1$	$3$\frac{\hspace{2} \sqrt{2} \hspace{2}}{\hspace{2}2\hspace{2}}$	$3$0$	$3$- \frac{\hspace{2} \sqrt{2} \hspace{2}}{\hspace{2}2\hspace{2}}$	$3$- 1$	$3$- \frac{\hspace{2} \sqrt{2} \hspace{2}}{\hspace{2}2\hspace{2}}$	$3$0$
$3$\sin \left({ {\theta}+\frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}{\pi} }\right)$	$3$\frac{\hspace{2} \sqrt{2} \hspace{2}}{\hspace{2}2\hspace{2}}$	$3$1$	$3$\frac{\hspace{2} \sqrt{2} \hspace{2}}{\hspace{2}2\hspace{2}}$	$3$0$	$3$- \frac{\hspace{2} \sqrt{2} \hspace{2}}{\hspace{2}2\hspace{2}}$	$3$- 1$	$3$- \frac{\hspace{2} \sqrt{2} \hspace{2}}{\hspace{2}2\hspace{2}}$	$3$0$	$3$\frac{\hspace{2} \sqrt{2} \hspace{2}}{\hspace{2}2\hspace{2}}$

● $3$\displaystyle{y=\sin \left({ {\theta}+\frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}4\hspace{2}} }\right)}$ のグラフ

$3$\displaystyle{ y=\sin $ {\theta}+\frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}4\hspace{2}} $ }$ を書き換えると $3$\displaystyle{ y=\sin $ {\theta}- \( - \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}4\hspace{2}} $ \) }$ となる．よって

$3$\displaystyle{ y=\sin $ {\theta}+\frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}4\hspace{2}} $ }$ のグラフは

$3$\displaystyle{ y=\sin {\theta} }$ のグラフをを $3$\displaystyle{{\theta}}$ 軸方向に $3$\displaystyle{- \frac{\hspace{2}{\pi}\hspace{2}}{\hspace{2}4\hspace{2}}}$ 平行移動したもの

になる．

$3${\theta}$	$3$0$	$3$\frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$	$3$\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{\pi}$	$3$\frac{\hspace{2}3\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$	$3${\pi}$	$3$\frac{\hspace{2}5\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$	$3$\frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}}{\pi}$	$3$\frac{\hspace{2}7\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$	$3$2{\pi}$
$3${\theta}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$	$3$- \frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$	$3$0$	$3$\frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$	$3$\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{\pi}$	$3$\frac{\hspace{2}3\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$	$3${\pi}$	$3$\frac{\hspace{2}5\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$	$3$\frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}}{\pi}$	$3$\frac{\hspace{2}7\hspace{2}}{\hspace{2}4\hspace{2}}{\pi}$
$3$\sin {\theta}$	$3$0$	$3$\frac{\hspace{2} \sqrt{2} \hspace{2}}{\hspace{2}2\hspace{2}}$	$3$1$	$3$\frac{\hspace{2} \sqrt{2} \hspace{2}}{\hspace{2}2\hspace{2}}$	$3$0$	$3$- \frac{\hspace{2} \sqrt{2} \hspace{2}}{\hspace{2}2\hspace{2}}$	$3$- 1$	$3$- \frac{\hspace{2} \sqrt{2} \hspace{2}}{\hspace{2}2\hspace{2}}$	$3$0$
$3$\sin \left({ {\theta}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}{\pi} }\right)$	$3$- \frac{\hspace{2} \sqrt{2} \hspace{2}}{\hspace{2}2\hspace{2}}$	$3$0$	$3$\frac{\hspace{2} \sqrt{2} \hspace{2}}{\hspace{2}2\hspace{2}}$	$3$1$	$3$\frac{\hspace{2} \sqrt{2} \hspace{2}}{\hspace{2}2\hspace{2}}$	$3$0$	$3$- \frac{\hspace{2} \sqrt{2} \hspace{2}}{\hspace{2}2\hspace{2}}$	$3$- 1$	$3$- \frac{\hspace{2} \sqrt{2} \hspace{2}}{\hspace{2}2\hspace{2}}$