## 2016年5月26日木曜日

### 数学 – 円の中にひそむ関数 - 三角関数 – (三角関数の諸公式(和、差、積))

• 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. $\mathrm{sin}3\theta \mathrm{cos}\theta =\frac{\mathrm{sin}4\theta +\mathrm{sin}2\theta }{2}$

2. $\mathrm{cos}\theta \mathrm{cos}7\theta =\frac{\mathrm{cos}6\theta +\mathrm{cos}8\theta }{2}$

3. $\mathrm{sin}3\theta \mathrm{sin}2\theta =\frac{\mathrm{cos}\theta -\mathrm{cos}5\theta }{2}$

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

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

6. $\begin{array}{l}\mathrm{cos}9\theta +\mathrm{cos}\theta \\ =\mathrm{cos}\left(5\theta +4\theta \right)+\mathrm{cos}\left(5\theta -4\theta \right)\\ =2\mathrm{cos}5\theta \mathrm{cos}4\theta \end{array}$

7. $\begin{array}{l}\mathrm{cos}2\alpha -\mathrm{cos}2\beta \\ =\mathrm{cos}\left(\alpha +\beta +\alpha -\beta \right)-\mathrm{cos}\left(\alpha +\beta -\left(\alpha -\beta \right)\right)\\ =-2\mathrm{sin}\left(\alpha +\beta \right)\mathrm{sin}\left(\alpha -\beta \right)\end{array}$