【高校数学Ⅰ】3-2-2 三角方程式・三角不等式｜問題集
1.次の方程式を満たす\(\theta\)を求めなさい。ただし、\(0^{\circ}\leqq \theta \leqq180^{\circ}\)とする。
(1)\(\displaystyle \sin\theta=\frac{\sqrt{3}}{2}\)
\(\theta=60^{\circ},120^{\circ}\)
(2)\(\displaystyle \sin\theta=\frac{1}{2}\)
\(\theta=30^{\circ},150^{\circ}\)
(3)\(\displaystyle \cos\theta=\frac{1}{2}\)
\(\theta=60^{\circ}\)
(4)\(\displaystyle \cos\theta=-\frac{1}{\sqrt{2}}\)
\(\theta=135^{\circ}\)
(5)\(\tan\theta=1\)
\(\theta=45^{\circ}\)
(6)\(\displaystyle \tan\theta=-\frac{1}{\sqrt{3}}\)
\(\theta=150^{\circ}\)
(7)\(\displaystyle \sin\theta=\frac{1}{\sqrt{2}}\)
\(\theta=45^{\circ},135^{\circ}\)
(8)\(\displaystyle \cos\theta=-\frac{1}{2}\)
\(\theta=120^{\circ}\)
(9)\(\tan\theta=-\sqrt{3}\)
\(\theta=120^{\circ}\)
(10)\(\sin\theta=1\)
\(\theta=90^{\circ}\)
(11)\(\cos\theta=1\)
\(\theta=0^{\circ}\)
2.次の不等式を満たす\(\theta\)を求めなさい。ただし、\(0^{\circ}\leqq \theta \leqq180^{\circ}\)とする。
(1)\(\displaystyle \sin\theta>\frac{1}{\sqrt{2}}\)
\(45^{\circ}<\theta<135^{\circ}\)
(2)\(\displaystyle \cos\theta\leqq-\frac{\sqrt{3}}{2}\)
\(150^{\circ}\leqq\theta\leqq180^{\circ}\)
(3)\(\tan\theta>\sqrt{3}\)
\(60^{\circ}<\theta<90^{\circ}\)
(4)\(\displaystyle \sin\theta<\frac{1}{2}\)
\(0^{\circ}\leqq\theta<30^{\circ},150^{\circ}<\theta\leqq180^{\circ}\)
(5)\(\displaystyle \cos\theta>-\frac{1}{\sqrt{2}}\)
\(0^{\circ}\leqq\theta<135^{\circ}\)
(6)\(\tan\theta\leqq1\)
\(0^{\circ}\leqq\theta\leqq45^{\circ},90^{\circ}<\theta\leqq180^{\circ}\)
3.次の問いに答えなさい。
(1)\(\displaystyle \sin\theta=\frac{1}{3}\)のとき、\(\cos\theta\),\(\tan\theta\)を求めなさい。ただし、\(0^{\circ}\leqq\theta\leqq180^{\circ}\)とする。
\(\cos^2\theta=1-\sin^2\theta\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =1-\left(\frac{1}{3}\right)^2\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =\frac{8}{9}\)
\(\cos\theta>0\)のとき、
\(\displaystyle \cos\theta=\frac{2\sqrt{2}}{3}\)
\(\displaystyle \tan\theta=\frac{1}{3}\div\frac{2\sqrt{2}}{3}\)
\(\displaystyle \ \ \ \ \ \ \ \ =\frac{\sqrt{2}}{4}\)
\(\cos\theta<0\)のとき、
\(\displaystyle \cos\theta=-\frac{2\sqrt{2}}{3}\)
\(\displaystyle \tan\theta=\frac{1}{3}\div\left(-\frac{2\sqrt{2}}{3}\right)\)
\(\displaystyle \ \ \ \ \ \ \ \ =-\frac{\sqrt{2}}{4}\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =1-\left(\frac{1}{3}\right)^2\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =\frac{8}{9}\)
\(\cos\theta>0\)のとき、
\(\displaystyle \cos\theta=\frac{2\sqrt{2}}{3}\)
\(\displaystyle \tan\theta=\frac{1}{3}\div\frac{2\sqrt{2}}{3}\)
\(\displaystyle \ \ \ \ \ \ \ \ =\frac{\sqrt{2}}{4}\)
\(\cos\theta<0\)のとき、
\(\displaystyle \cos\theta=-\frac{2\sqrt{2}}{3}\)
\(\displaystyle \tan\theta=\frac{1}{3}\div\left(-\frac{2\sqrt{2}}{3}\right)\)
\(\displaystyle \ \ \ \ \ \ \ \ =-\frac{\sqrt{2}}{4}\)
(2)\(\tan\theta=-2\)のとき、\(\sin\theta\),\(\cos\theta\)を求めなさい。ただし、\(0^{\circ}\leqq\theta\leqq180^{\circ}\)とする。
\(\displaystyle \cos^2\theta=\frac{1}{1+\tan^2\theta}\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =\frac{1}{1+(-2)^2}\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =\frac{1}{5}\)
\(\cos\theta<0\)より、
\(\displaystyle \cos\theta=-\frac{\sqrt{5}}{5}\)
\(\displaystyle \sin\theta=\cos\theta\tan\theta\)
\(\displaystyle \ \ \ \ \ \ \ \ =-\frac{\sqrt{5}}{5}\times(-2)\)
\(\displaystyle \ \ \ \ \ \ \ \ =\frac{2\sqrt{5}}{5}\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =\frac{1}{1+(-2)^2}\)
\(\displaystyle \ \ \ \ \ \ \ \ \ =\frac{1}{5}\)
\(\cos\theta<0\)より、
\(\displaystyle \cos\theta=-\frac{\sqrt{5}}{5}\)
\(\displaystyle \sin\theta=\cos\theta\tan\theta\)
\(\displaystyle \ \ \ \ \ \ \ \ =-\frac{\sqrt{5}}{5}\times(-2)\)
\(\displaystyle \ \ \ \ \ \ \ \ =\frac{2\sqrt{5}}{5}\)
4.次の直線と\(x\)軸の正の向きとなす角\(\theta\)を求めなさい。
(1)\(y=x\)
\(\tan\theta=1\)なので、
\(\theta=45^{\circ}\)
\(\theta=45^{\circ}\)
(2)\(\displaystyle y=\frac{1}{\sqrt{3}}x\)
\(\displaystyle \tan\theta=\frac{1}{\sqrt{3}}\)なので、
\(\theta=30^{\circ}\)
\(\theta=30^{\circ}\)
(3)\(y=-\sqrt{3}x\)
\(\tan\theta=-\sqrt{3}\)なので、
\(\theta=120^{\circ}\)
\(\theta=120^{\circ}\)
5.\(\displaystyle \sin\theta+\cos\theta=\frac{1}{2}\)のとき、次の式の値を求めなさい。ただし、\(0^{\circ}\leqq \theta \leqq180^{\circ}\)とする。
(1)\(\sin\theta\cos\theta\)
\(\displaystyle (\sin\theta+\cos\theta)^2=\left(\frac{1}{2}\right)^2\)
\(\displaystyle \sin^2\theta+2\sin\theta\cos\theta+\cos^2\theta=\frac{1}{4}\)
\(\displaystyle 2\sin\theta\cos\theta+1=\frac{1}{4}\)
\(\displaystyle 2\sin\theta\cos\theta=-\frac{3}{4}\)
\(\displaystyle \sin\theta\cos\theta=-\frac{3}{8}\)
\(\displaystyle \sin^2\theta+2\sin\theta\cos\theta+\cos^2\theta=\frac{1}{4}\)
\(\displaystyle 2\sin\theta\cos\theta+1=\frac{1}{4}\)
\(\displaystyle 2\sin\theta\cos\theta=-\frac{3}{4}\)
\(\displaystyle \sin\theta\cos\theta=-\frac{3}{8}\)
(2)\(\sin\theta-\cos\theta\)
\((\sin\theta-\cos\theta)^2\)
\(=\sin^2\theta-2\sin\theta\cos\theta+\cos^2\theta\)
\(\displaystyle =1-2\times\left(-\frac{3}{8}\right)\)
\(\displaystyle =\frac{7}{4}\)
\(\sin\theta>0,\cos\theta<0\)より、
\(\displaystyle \sin\theta-\cos\theta=\frac{\sqrt{7}}{2}\)
\(=\sin^2\theta-2\sin\theta\cos\theta+\cos^2\theta\)
\(\displaystyle =1-2\times\left(-\frac{3}{8}\right)\)
\(\displaystyle =\frac{7}{4}\)
\(\sin\theta>0,\cos\theta<0\)より、
\(\displaystyle \sin\theta-\cos\theta=\frac{\sqrt{7}}{2}\)
6.次の方程式を解きなさい。ただし、\(0^{\circ}\leqq \theta \leqq180^{\circ}\)とする。
(1)\(\sin^2\theta+\sin\theta-2=0\)
\((\sin\theta+2)(\sin\theta-1)=0\)
\(\sin\theta=-2,1\)
よって、
\(\theta=90^{\circ}\)
\(\sin\theta=-2,1\)
よって、
\(\theta=90^{\circ}\)
(2)\(\sin^2\theta+\cos\theta=1\)
\((1-\cos^2\theta)+\cos\theta=1\)
\(\cos\theta(\cos\theta-1)=0\)
\(\cos\theta=0,1\)
よって、
\(\theta=0^{\circ},90^{\circ}\)
\(\cos\theta(\cos\theta-1)=0\)
\(\cos\theta=0,1\)
よって、
\(\theta=0^{\circ},90^{\circ}\)
(3)\(2\sin\theta\cos\theta+2\sin\theta-\cos\theta-1=0\)
\(2\sin\theta(\cos\theta+1)-(\cos\theta+1)=0\)
\((2\sin\theta-1)(\cos\theta+1)=0\)
\(\displaystyle \sin\theta=\frac{1}{2},\cos\theta=-1\)
よって、
\(\theta=30^{\circ},150^{\circ},180^{\circ}\)
\((2\sin\theta-1)(\cos\theta+1)=0\)
\(\displaystyle \sin\theta=\frac{1}{2},\cos\theta=-1\)
よって、
\(\theta=30^{\circ},150^{\circ},180^{\circ}\)
7.次の不等式を解きなさい。ただし、\(0^{\circ}\leqq \theta \leqq180^{\circ}\)とする。
(1)\(2\cos^2\theta>1+\cos\theta\)
\(2\cos^2\theta-\cos\theta-1>0\)
\((2\cos\theta+1)(\cos\theta-1)>0\)
\(\displaystyle \cos\theta<-\frac{1}{2},1<\cos\theta\)
よって、
\(120^{\circ}<\theta\leqq180^{\circ}\)
\((2\cos\theta+1)(\cos\theta-1)>0\)
\(\displaystyle \cos\theta<-\frac{1}{2},1<\cos\theta\)
よって、
\(120^{\circ}<\theta\leqq180^{\circ}\)
(2)\(\cos^2\theta+\cos\theta-\sin^2\theta\leqq0\)
\(\cos^2\theta+\cos\theta-(1-\cos^2\theta)\leqq0\)
\(2\cos^2\theta+\cos\theta-1\leqq0\)
\((2\cos\theta-1)(\cos\theta+1)\leqq0\)
\(\displaystyle -1\leqq\cos\theta\leqq\frac{1}{2}\)
よって、
\(60^{\circ}\leqq\theta\leqq180^{\circ}\)
\(2\cos^2\theta+\cos\theta-1\leqq0\)
\((2\cos\theta-1)(\cos\theta+1)\leqq0\)
\(\displaystyle -1\leqq\cos\theta\leqq\frac{1}{2}\)
よって、
\(60^{\circ}\leqq\theta\leqq180^{\circ}\)
次の学習に進もう！