余角の公式
【余角の公式(\(90°-θ\)の三角比)】
\(\sin (90^{\circ}-\theta)=\cos \theta\)
\(\cos (90^{\circ}-\theta)=\sin \theta\)
\(\displaystyle \tan (90^{\circ}-\theta)=\frac{1}{\tan \theta}\)
【例題】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)\(\cos 52^{\circ}\)
\(\sin 38^{\circ}\)
(3)\(\tan 56^{\circ}\)
\(\displaystyle \frac{1}{\tan 34^{\circ}}\)
三角比の相互関係
【三角比の相互関係】
\(\sin^2\theta+\cos^2\theta=1\)
\(\displaystyle \tan\theta=\frac{\sin\theta}{\cos\theta}\)
\(\displaystyle 1+\tan^2\theta=\frac{1}{\cos^2\theta}\)
【例題】次の値を求めなさい。ただし、\(\theta\)は鋭角とする。
(1)\(\displaystyle \sin\theta=\frac{2}{3}\)のとき、\(\cos\theta\)
\(\cos^2\theta=1-\sin^2\theta\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =1-\left(\frac{2}{3}\right)^2\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =\frac{5}{9}\)
\(\cos\theta>0\)より、
\(\displaystyle \cos\theta=\frac{\sqrt{5}}{3}\)
(2)\(\displaystyle \sin\theta=\frac{2}{3}\)のとき、\(\tan\theta\)
\(\displaystyle \tan\theta=\frac{\sin\theta}{\cos\theta}\)
\(\displaystyle \ \ \ \ \ \ \ \ =\frac{2}{3}\div\frac{\sqrt{5}}{3}\)
\(\displaystyle \ \ \ \ \ \ \ \ =\frac{2}{\sqrt{5}}\)
\(\displaystyle \ \ \ \ \ \ \ \ =\frac{2\sqrt{5}}{5}\)
(3)\(\tan\theta=2\)のとき、\(\cos\theta\)
\(\displaystyle \cos^2\theta=\frac{1}{1+\tan^2\theta}\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =\frac{1}{1+2^2}\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =\frac{1}{5}\)
\(\cos\theta>0\)より、
\(\displaystyle \cos\theta=\frac{\sqrt{5}}{5}\)
(4)\(\tan\theta=2\)のとき、\(\sin\theta\)
\(\displaystyle \sin\theta=\cos\theta\tan\theta\)
\(\displaystyle \ \ \ \ \ \ \ \ =\frac{\sqrt{5}}{5}\times2\)
\(\displaystyle \ \ \ \ \ \ \ \ =\frac{2\sqrt{5}}{5}\)