【高校数学Ⅱ】4-1-4 三角方程式と三角不等式｜問題集

Q: 三角不等式の解き方のコツはありますか？

三角不等式では、まず不等式をsinθやcosθなど1種類の関数で表し、関数の増減やグラフの形を利用して範囲を求めるのがコツです。周期性を考慮して、一般解まで導く練習をしましょう。

Q: 二次式を含む三角方程式はどのように解きますか？

sinθやcosθを文字（たとえばx）に置き換えて二次方程式として解き、得られた解をもとにθを求めます。置き換えた後、必ずsinθやcosθの値の範囲（-1≦x≦1）を確認することが大切です。

Q: この問題集はどんな学習に役立ちますか？

定期テスト対策や大学入試の基礎固めに最適です。代表的な三角方程式・三角不等式の問題を通じて、考え方と計算手順を効率よく身につけることができます。

Question 1

(1)\(\displaystyle \sin\theta=\frac{\sqrt{3}}{2}\)

Answer

\(\displaystyle \theta=\frac{\pi}{3},\frac{2\pi}{3}\)

Question 2

(2)\(2\cos\theta+1=0\)

Answer

\(\displaystyle \theta=\frac{2\pi}{3},\frac{4\pi}{3}\)

Question 3

(3)\(\displaystyle \cos\theta=-\frac{\sqrt{2}}{2}\)

Answer

\(\displaystyle \theta=\frac{3\pi}{4},\frac{5\pi}{4}\)

Question 4

(4)\(\tan\theta=-\sqrt{3}\)

Answer

\(\displaystyle \theta=\frac{2\pi}{3},\frac{5\pi}{3}\)

Question 5

(5)\(\sin\theta+1=0\)

Answer

\(\displaystyle \theta=\frac{3\pi}{2}\)

Question 6

(6)\(\cos\theta=0\)

Answer

\(\displaystyle \theta=\frac{\pi}{2},\frac{3\pi}{2}\)

Question 7

(7)\(\displaystyle \sin\theta=-\frac{1}{\sqrt{2}}\)

Answer

\(\displaystyle \theta=\frac{5\pi}{4},\frac{7\pi}{4}\)

Question 8

(8)\(\displaystyle \sin\left(\theta+\frac{\pi}{3}\right)=\frac{1}{\sqrt{2}}\)

Answer

\(\displaystyle \theta+\frac{\pi}{3}=\frac{\pi}{4},\frac{3\pi}{4}\)
\(0\leqq \theta \leqq2\pi\)より、
\(\displaystyle \theta=\frac{5\pi}{12},\frac{23\pi}{12}\)

Question 9

(9)\(\displaystyle \sqrt{2}\cos\left(\theta+\frac{\pi}{3}\right)=1\)

Answer

\(\displaystyle \theta+\frac{\pi}{3}=\frac{\pi}{4},\frac{7\pi}{4}\)
\(0\leqq \theta \leqq2\pi\)より、
\(\displaystyle \theta=\frac{5\pi}{12},\frac{17\pi}{12}\)

Question 10

(10)\(\displaystyle 2\sin\left(\theta-\frac{\pi}{6}\right)=-\sqrt{3}\)

Answer

\(\displaystyle \theta-\frac{\pi}{6}=\frac{4\pi}{3},\frac{5\pi}{3}\)
\(0\leqq \theta \leqq2\pi\)より、
\(\displaystyle \theta=\frac{3\pi}{2},\frac{11\pi}{6}\)

Question 11

(1)\(\displaystyle \cos\theta\leqq-\frac{1}{2}\)

Answer

\(\displaystyle \frac{2\pi}{3}\leqq\theta\leqq\frac{4\pi}{3}\)

Question 12

(2)\(\displaystyle \sin\theta>\frac{1}{\sqrt{2}}\)

Answer

\(\displaystyle \frac{\pi}{4}<\theta<\frac{3\pi}{4}\)

Question 13

(3)\(\displaystyle \cos\theta>\frac{1}{2}\)

Answer

\(\displaystyle 0\leqq\theta<\frac{\pi}{3},\frac{5\pi}{3}<\theta<2\pi\)

Question 14

(4)\(\displaystyle \sin\theta\leqq\frac{\sqrt{3}}{2}\)

Answer

\(\displaystyle 0\leqq\theta\leqq\frac{\pi}{3},\frac{2\pi}{3}\leqq\theta<2\pi\)

Question 15

(5)\(\displaystyle \tan\theta\leqq-1\)

Answer

\(\displaystyle \frac{\pi}{2}<\theta\leqq\frac{3\pi}{4},\frac{3\pi}{2}<\theta\leqq\frac{7\pi}{4}\)

Question 16

(6)\(\displaystyle \tan\theta<\sqrt{3}\)

Answer

\(\displaystyle 0\leqq\theta\leqq\frac{\pi}{3},\frac{\pi}{2}<\theta<\frac{4\pi}{3},\frac{3\pi}{2}<\theta<2\pi\)

Question 17

(7)\(\displaystyle \sin\theta>-\frac{1}{2}\)

Answer

\(\displaystyle 0\leqq\theta<\frac{7\pi}{6},\frac{11\pi}{6}<\theta<2\pi\)

Question 18

(1)\(2\cos^2\theta+3\cos\theta-2=0\)

Answer

\((2\cos\theta-1)(\cos\theta+2)=0\)
\(2\cos\theta-1=0\)より、
\(\displaystyle \theta=\frac{\pi}{3},\frac{5\pi}{3}\)
\(\cos\theta+2=0\)より、解なし
よって、
\(\displaystyle \theta=\frac{\pi}{3},\frac{5\pi}{3}\)

Question 19

(2)\(2\cos^2\theta-\sin\theta-1=0\)

Answer

\(2(1-\sin^2\theta)-\sin\theta-1=0\)
\(2\sin^2\theta+\sin\theta-1=0\)
\((2\sin\theta-1)(\sin\theta+1)=0\)
\(2\sin\theta-1=0\)より、
\(\displaystyle \theta=\frac{\pi}{6},\frac{5\pi}{6}\)
\(\sin\theta+1=0\)より、
\(\displaystyle \theta=\frac{3\pi}{2}\)
よって、
\(\displaystyle \theta=\frac{\pi}{6},\frac{5\pi}{6},\frac{3\pi}{2}\)

Question 20

(3)\(2\sin^2\theta+3\cos\theta-3=0\)

Answer

\(2(1-\cos^2\theta)+3\cos\theta-3=0\)
\(2\cos^2\theta-3\cos\theta+1=0\)
\((2\cos\theta-1)(\cos\theta-1)=0\)
\(2\cos\theta-1=0\)より、
\(\displaystyle \theta=\frac{\pi}{3},\frac{5\pi}{3}\)
\(\cos\theta-1=0\)より、
\(\theta=0\)
よって、
\(\displaystyle \theta=0,\frac{\pi}{3},\frac{5\pi}{3}\)

Question 21

(1)\(2\cos^2\theta-\sin\theta-1\geqq0\)

Answer

\(2(1-\sin^2\theta)-\sin\theta-1\geqq0\)
\(2\sin^2\theta+\sin\theta-1\leqq0\)
\((2\sin\theta-1)(\sin\theta+1)\leqq0\)
\(\displaystyle -1\leqq\sin\theta\leqq\frac{1}{2}\)
よって、
\(\displaystyle 0\leqq\theta\leqq\frac{\pi}{6},\frac{5\pi}{6}\leqq\theta\leqq2\pi\)

Question 22

(2)\(2\sin^2\theta+3\cos\theta-3\geqq0\)

Answer

\(2(1-\cos^2\theta)+3\cos\theta-3\geqq0\)
\(2\cos^2\theta-3\cos\theta+1\leqq0\)
\((2\cos\theta-1)(\cos\theta-1)\leqq0\)
\(\displaystyle \frac{1}{2}\leqq\cos\theta\leqq1\)
よって、
\(\displaystyle 0\leqq\theta\leqq\frac{\pi}{3},\frac{5\pi}{3}\leqq\theta\leqq2\pi\)

Question 23

(1)\(y=-\sin^2\theta-\cos\theta+1\)

Answer

\(y=-(1-\cos^2\theta)-\cos\theta+1\)
\(\ \ =\cos^2\theta-\cos\theta\)
\(\displaystyle \ \ =\left(\cos\theta-\frac{1}{2}\right)^2-\frac{1}{4}\)
\(-1\leqq\cos\theta\leqq1\)より、
\(\displaystyle \cos\theta=-1\)のとき、最大値\(2\)
\(\theta=\pi\)
\(\displaystyle \cos\theta=\frac{1}{2}\)のとき、最小値\(\displaystyle -\frac{1}{4}\)
\(\displaystyle \theta=\frac{\pi}{3},\frac{5\pi}{3}\)
よって、
最大値は\(2\)(\(\theta=\pi\)のとき)
最小値は\(\displaystyle -\frac{1}{4}\)(\(\displaystyle \theta=\frac{\pi}{3},\frac{5\pi}{3}\)のとき)

Question 24

(2)\(y=-\cos^2\theta+\sin\theta\)

Answer

\(y=-(1-\sin^2\theta)+\sin\theta\)
\(\ \ =\sin^2\theta+\sin\theta-1\)
\(\displaystyle \ \ =\left(\sin\theta+\frac{1}{2}\right)^2-\frac{5}{4}\)
\(-1\leqq\sin\theta\leqq1\)より、
\(\displaystyle \sin\theta=1\)のとき、最大値\(1\)
\(\displaystyle \theta=\frac{\pi}{2}\)
\(\displaystyle \cos\theta=-\frac{1}{2}\)のとき、最小値\(\displaystyle -\frac{5}{4}\)
\(\displaystyle \theta=\frac{7\pi}{6},\frac{11\pi}{6}\)
よって、
最大値は\(1\)(\(\displaystyle \theta=\frac{\pi}{2}\)のとき)
最小値は\(\displaystyle -\frac{5}{4}\)(\(\displaystyle \theta=\frac{7\pi}{6},\frac{11\pi}{6}\)のとき)

Question 25

(3)\(y=\sin^2\theta+\cos\theta\)

Answer

\(y=(1-\cos^2\theta)+\cos\theta\)
\(\ \ =-\cos^2\theta+\cos\theta+1\)
\(\displaystyle \ \ =-\left(\cos\theta-\frac{1}{2}\right)^2+\frac{5}{4}\)
\(-1\leqq\cos\theta\leqq1\)より、
\(\displaystyle \cos\theta=\frac{1}{2}\)のとき、最大値\(\displaystyle \frac{5}{4}\)
\(\displaystyle \theta=\frac{\pi}{3},\frac{5\pi}{3}\)
\(\cos\theta=-1\)のとき、最小値\(-1\)
\(\displaystyle \theta=\pi\)
よって、
最大値は\(\displaystyle \frac{5}{4}\)(\(\displaystyle \theta=\frac{\pi}{3},\frac{5\pi}{3}\)のとき)
最小値は\(-1\)(\(\theta=\pi\)のとき)