1.次の三角比を\(45^{\circ}\)以下の角の三角比で表しなさい。
(1)\(\sin 70^{\circ}\)
\(\cos 20^{\circ}\)
(2)\(\cos 75^{\circ}\)
\(\sin 15^{\circ}\)
(3)\(\tan 58^{\circ}\)
\(\displaystyle \frac{1}{\tan 32^{\circ}}\)
(4)\(\sin 80^{\circ}\)
\(\cos 10^{\circ}\)
(5)\(\cos 50^{\circ}\)
\(\sin 40^{\circ}\)
(6)\(\tan 64^{\circ}\)
\(\displaystyle \frac{1}{\tan 26^{\circ}}\)
2.次の値を求めなさい。ただし、\(\theta\)は鋭角とする。
(1)\(\displaystyle \cos\theta=\frac{2}{7}\)のとき、\(\sin\theta\)
\(\sin^2\theta=1-\cos^2\theta\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =1-\left(\frac{2}{7}\right)^2\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =\frac{45}{49}\)
\(\sin\theta>0\)より、
\(\displaystyle \sin\theta=\frac{3\sqrt{5}}{7}\)
(2)\(\displaystyle \cos\theta=\frac{2}{7}\)のとき、\(\tan\theta\)
\(\displaystyle \tan\theta=\frac{\sin\theta}{\cos\theta}\)
\(\displaystyle \ \ \ \ \ \ \ \ =\frac{3\sqrt{5}}{7}\div\frac{2}{7}\)
\(\displaystyle \ \ \ \ \ \ \ \ =\frac{3\sqrt{5}}{2}\)
(3)\(\displaystyle \tan\theta=\frac{1}{2}\)のとき、\(\cos\theta\)
\(\displaystyle \cos^2\theta=\frac{1}{1+\tan^2\theta}\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =\frac{1}{1+(\frac{1}{2})^2}\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =\frac{4}{5}\)
\(\cos\theta>0\)より、
\(\displaystyle \cos\theta=\frac{2\sqrt{5}}{5}\)
(4)\(\displaystyle \tan\theta=\frac{1}{2}\)のとき、\(\sin\theta\)
\(\displaystyle \sin\theta=\cos\theta\tan\theta\)
\(\displaystyle \ \ \ \ \ \ \ \ =\frac{2\sqrt{5}}{5}\times\frac{1}{2}\)
\(\displaystyle \ \ \ \ \ \ \ \ =\frac{\sqrt{5}}{5}\)
(5)\(\displaystyle \cos\theta=\frac{1}{3}\)のとき、\(\sin\theta\)
\(\sin^2\theta=1-\cos^2\theta\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =1-\left(\frac{1}{3}\right)^2\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =\frac{8}{9}\)
\(\sin\theta>0\)より、
\(\displaystyle \sin\theta=\frac{2\sqrt{2}}{3}\)
(6)\(\displaystyle \cos\theta=\frac{1}{3}\)のとき、\(\tan\theta\)
\(\displaystyle \tan\theta=\frac{\sin\theta}{\cos\theta}\)
\(\displaystyle \ \ \ \ \ \ \ \ =\frac{2\sqrt{2}}{3}\div\frac{1}{3}\)
\(\ \ \ \ \ \ \ \ =2\sqrt{2}\)
(7)\(\displaystyle \tan\theta=\frac{1}{7}\)のとき、\(\cos\theta\)
\(\displaystyle \cos^2\theta=\frac{1}{1+\tan^2\theta}\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =\frac{1}{1+(\frac{1}{7})^2}\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =\frac{49}{50}\)
\(\cos\theta>0\)より、
\(\displaystyle \cos\theta=\frac{7\sqrt{2}}{10}\)
(8)\(\displaystyle \tan\theta=\frac{1}{7}\)のとき、\(\sin\theta\)
\(\displaystyle \sin\theta=\cos\theta\tan\theta\)
\(\displaystyle \ \ \ \ \ \ \ \ =\frac{7\sqrt{2}}{10}\times\frac{1}{7}\)
\(\displaystyle \ \ \ \ \ \ \ \ =\frac{\sqrt{2}}{10}\)