## 2016年4月28日木曜日

### 数学 – 円の中にひそむ関数 - 三角関数 – (正弦・余弦の加法定理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(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax

1. $\begin{array}{l}\mathrm{sin}\frac{\pi }{12}\\ =\mathrm{sin}\left(\frac{\pi }{3}-\frac{\pi }{4}\right)\\ =\mathrm{sin}\frac{\pi }{3}\mathrm{cos}\frac{\pi }{4}-\mathrm{cos}\frac{\pi }{3}\mathrm{sin}\frac{\pi }{4}\\ =\frac{\sqrt{3}}{2}·\frac{1}{\sqrt{2}}-\frac{1}{2}\frac{1}{\sqrt{2}}\\ =\frac{\sqrt{3}-1}{2\sqrt{2}}\\ =\frac{\sqrt{6}-\sqrt{2}}{4}\end{array}$

2. $\begin{array}{l}\mathrm{cos}\frac{\pi }{12}\\ =\mathrm{cos}\left(\frac{\pi }{3}-\frac{\pi }{4}\right)\\ =\mathrm{cos}\frac{\pi }{3}\mathrm{cos}\frac{\pi }{4}+\mathrm{sin}\frac{\pi }{3}\mathrm{sin}\frac{\pi }{4}\\ =\frac{1}{2}·\frac{1}{\sqrt{2}}+\frac{\sqrt{3}}{2}·\frac{1}{\sqrt{2}}\\ =\frac{\sqrt{2}+\sqrt{6}}{4}\end{array}$

3. $\begin{array}{l}\mathrm{sin}\left(\frac{\pi }{3}+\frac{\pi }{4}\right)\\ =\mathrm{sin}\frac{\pi }{3}\mathrm{cos}\frac{\pi }{4}+\mathrm{cos}\frac{\pi }{3}\mathrm{sin}\frac{\pi }{4}\\ =\frac{\sqrt{3}}{2}·\frac{1}{\sqrt{2}}+\frac{1}{2}·\frac{1}{\sqrt{2}}\\ =\frac{\sqrt{6}+\sqrt{2}}{4}\end{array}$

4. $\begin{array}{l}\mathrm{cos}\left(\frac{\pi }{3}+\frac{\pi }{4}\right)\\ =\mathrm{cos}\frac{\pi }{3}\mathrm{cos}\frac{\pi }{4}-\mathrm{sin}\frac{\pi }{3}\mathrm{sin}\frac{\pi }{4}\\ =\frac{1}{2}·\frac{1}{\sqrt{2}}-\frac{\sqrt{3}}{2}·\frac{1}{\sqrt{2}}\\ =\frac{\sqrt{2}-\sqrt{6}}{4}\end{array}$

5. $\mathrm{sin}\frac{2\pi }{3}=\frac{\sqrt{3}}{2}$

6. $\mathrm{cos}\frac{2\pi }{3}=-\frac{1}{2}$

7. $\mathrm{sin}{15}^{°}=\frac{\sqrt{6}-\sqrt{2}}{4}$

8. $-\mathrm{cos}{15}^{°}=-\frac{\sqrt{6}+\sqrt{2}}{4}$