## 2016年5月6日金曜日

### 数学 – 円の中にひそむ関数 - 三角関数 – (三角関数の合成2)

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

2. $\begin{array}{l}\sqrt{2}\mathrm{sin}\theta \mathrm{cos}\frac{\pi }{4}+\sqrt{2}\mathrm{cos}\theta \mathrm{sin}\frac{\pi }{4}=-1\\ \sqrt{2}\mathrm{sin}\left(\theta +\frac{\pi }{4}\right)=-1\\ \mathrm{sin}\left(\theta +\frac{\pi }{4}\right)=-\frac{1}{\sqrt{2}}\\ \theta +\frac{\pi }{4}=\frac{5}{4}\pi +2n\pi ,\frac{7}{4}\pi +2n\pi \\ \theta =\pi +2n\pi ,\frac{3}{2}\pi +2n\pi \\ \theta =\pi ,\frac{3}{2}\pi \end{array}$

3. $\begin{array}{l}2\mathrm{sin}\theta \mathrm{cos}\frac{\pi }{3}-2\mathrm{cos}\theta \mathrm{sin}\frac{\pi }{3}=0\\ \mathrm{sin}\theta \mathrm{cos}\frac{\pi }{3}-\mathrm{cos}\theta \mathrm{sin}\frac{\pi }{3}=0\\ \mathrm{sin}\left(\theta -\frac{\pi }{3}\right)=0\\ \theta -\frac{\pi }{3}=n\pi \\ \theta =\frac{\pi }{3}+n\pi \\ \theta =\frac{\pi }{3},\frac{4}{3}\pi \\ \end{array}$

4. $\begin{array}{l}2\mathrm{cos}\theta \mathrm{cos}\frac{\pi }{3}-2\mathrm{sin}\theta \mathrm{sin}\frac{\pi }{3}=1\\ \mathrm{cos}\theta \mathrm{cos}\frac{\pi }{3}-\mathrm{sin}\theta \mathrm{sin}\frac{\pi }{3}=\frac{1}{2}\\ \mathrm{cos}\left(\theta +\frac{\pi }{3}\right)=\frac{1}{2}\\ \theta +\frac{\pi }{3}=\frac{\pi }{3}+2n\pi ,\frac{5}{3}\pi +2n\pi \\ \theta =2n\pi ,\frac{4}{3}\pi +2n\pi \\ \theta =0,\frac{4}{3}\pi \end{array}$