## 2016年5月10日火曜日

### 数学 – 円の中にひそむ関数 - 三角関数 – (三角関数の合成、不等式)

• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Edge/Internet Explorer, Google Chrome...)用JavaScript Library: MathJax

1. $\begin{array}{l}\mathrm{sin}\theta -\mathrm{cos}\theta \\ =\sqrt{2}\mathrm{sin}\theta \mathrm{cos}\frac{\pi }{4}-\sqrt{2}\mathrm{cos}\theta \frac{\pi }{4}\\ =\sqrt{2}\mathrm{sin}\left(\theta -\frac{\pi }{4}\right)>0\\ 2n\pi <\theta -\frac{\pi }{4}<\pi +2n\pi \\ \frac{\pi }{4}+2n\pi <\theta <\frac{5}{4}\pi +2n\pi \\ \frac{\pi }{4}<\theta <\frac{5}{4}\pi \end{array}$

2. $\begin{array}{l}2\mathrm{sin}\frac{\pi }{3}\mathrm{cos}\theta -2\mathrm{cos}\frac{\pi }{3}\mathrm{sin}\theta \le 1\\ \mathrm{sin}\left(\frac{\pi }{3}-\theta \right)\le \frac{1}{2}\\ \mathrm{sin}\left(\theta -\frac{\pi }{3}\right)\ge -\frac{1}{2}\\ 2n\pi \le \theta -\frac{\pi }{3}\le \frac{7}{6}\pi +2n\pi ,\frac{11}{6}\pi +2n\pi \le \theta -\frac{\pi }{3}<2\pi +2n\pi \\ \frac{\pi }{3}+2n\pi \le \theta \le \frac{3}{2}\pi +2n\pi ,\frac{13}{6}\pi +2n\pi \le \theta <\frac{7}{3}\pi +2n\pi \\ \frac{\pi }{6}\le \theta \le \frac{3}{2}\pi \\ \end{array}$