4-2-4 和と積の公式(要点)

【積和の公式】

\(\displaystyle \sin\alpha\cos\beta=\frac{1}{2}\{\sin(\alpha+\beta)+\sin(\alpha-\beta)\}\)
\(\displaystyle \cos\alpha\sin\beta=\frac{1}{2}\{\sin(\alpha+\beta)-\sin(\alpha-\beta)\}\)
\(\displaystyle \cos\alpha\cos\beta=\frac{1}{2}\{\cos(\alpha+\beta)+\cos(\alpha-\beta)\}\)
\(\displaystyle \sin\alpha\sin\beta=-\frac{1}{2}\{\cos(\alpha+\beta)-\cos(\alpha-\beta)\}\)

【例題】次の式を和の形で表しなさい。

(1)\(\sin6\theta\cos4\theta\)

\(\displaystyle =\frac{1}{2}\{\sin(6\theta+4\theta)+\sin(6\theta-4\theta)\}\)
\(\displaystyle =\frac{1}{2}(\sin10\theta+\sin2\theta)\)

(2)\(\cos5\theta\cos3\theta\)

\(\displaystyle =\frac{1}{2}\{\cos(5\theta+3\theta)+\cos(5\theta-3\theta)\}\)
\(\displaystyle =\frac{1}{2}(\cos8\theta+\cos2\theta)\)

【和積の公式】

\(\displaystyle \sin A+\sin B=2\sin\frac{A+B}{2}\cos\frac{A-B}{2}\)
\(\displaystyle \sin A-\sin B=2\cos\frac{A+B}{2}\sin\frac{A-B}{2}\)
\(\displaystyle \cos A+\cos B=2\cos\frac{A+B}{2}\cos\frac{A-B}{2}\)
\(\displaystyle \cos A-\cos B=-2\sin\frac{A+B}{2}\sin\frac{A-B}{2}\)

【例題】次の式を積の形で表しなさい。

(1)\(\cos3\theta+\cos5\theta\)

\(=\cos(4\theta-\theta)+\cos(4\theta+\theta)\)
\(=2\cos4\theta\cos\theta\)

(2)\(\sin4\theta-\sin3\theta\)

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