三角函数之和差化积公式(贰)
perface
Invoke: 积化和差公式
从 积化和差 推衍得到 和差化积
First
\[
\begin{align}
已知积化和差公式: \\
\sin\alpha\cos\beta=
\frac{\sin\left(\alpha+\beta\right)+\sin\left(\alpha-\beta\right)}{2}
\\ \\
设\alpha\Rightarrow\frac{\alpha+\beta}{2}, \enspace
\beta\Rightarrow\frac{\alpha-\beta}{2}
\enspace,将假设代入积化和差公式:
\\ \\
2\sin(\frac{\alpha+\beta}{2})\cos(\frac{\alpha-\beta}{2})
=\sin(\frac{\alpha+\beta}{2}+\frac{\alpha-\beta}{2}) +
\sin(\frac{\alpha+\beta}{2}-\frac{\alpha-\beta}{2})
\\ \\ \\
\Rightarrow\sin\left(\frac{2\alpha+\beta-\beta}{2}\right)+\sin\left(\frac{\alpha-\alpha+\beta+\beta}{2}\right)
\\ \\
\Rightarrow\sin\left(\frac{2\alpha}{2}\right)+\sin\left(\frac{2\beta}{2}\right)=\sin\alpha+\sin\beta
\\ \\
\therefore
2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}
=\sin\alpha+\sin\beta
\\ \\
获得公式1: \enspace
\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}
\end{align}
\]
Second
\[\begin{align}
已知积化和差公式: \\
\cos\alpha\sin\beta=\frac{\sin\left(\alpha+\beta\right)-\sin\left(\alpha-\beta\right)}{2}
\\ \\
设\alpha\Rightarrow\frac{\alpha+\beta}{2}, \enspace
\beta\Rightarrow\frac{\alpha-\beta}{2}
\enspace,将假设代入积化和差公式:
\\ \\
2\cos(\frac{\alpha+\beta}{2})\sin(\frac{\alpha-\beta}{2})=\sin(\frac{\alpha+\beta}{2}+\frac{\alpha-\beta}{2})
-\sin(\frac{\alpha+\beta}{2}-\frac{\alpha-\beta}{2})
\\ \\
\Rightarrow\sin(\frac{2\alpha}{2})-\sin(\frac{2\beta}{2})=\sin\alpha-\sin\beta
\\ \\
\therefore
2\cos(\frac{\alpha+\beta}{2})\sin(\frac{\alpha-\beta}{2})=\sin\alpha-\sin\beta
\\ \\
获得公式2: \enspace
\sin\alpha-\sin\beta=2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}
\end{align}
\]
Third
\[\begin{align}
已知积化和差公式: \\
\cos\alpha\cos\beta=\frac{\cos(\alpha+\beta)+\cos(\alpha-\beta)}{2}
\\ \\
设\alpha\Rightarrow\frac{\alpha+\beta}{2}, \enspace
\beta\Rightarrow\frac{\alpha-\beta}{2}
\enspace,将假设代入积化和差公式:
\\ \\
2\cos(\frac{\alpha+\beta}{2})\cos(\frac{\alpha-\beta}{2})=\cos(\frac{\alpha+\beta}{2}+\frac{\alpha-\beta}{2})
+\cos(\frac{\alpha+\beta}{2}-\frac{\alpha-\beta}{2})
\\ \\
\Rightarrow\cos(\frac{2\alpha+\beta-\beta}{2})+\cos(\frac{\alpha-\alpha+\beta+\beta}{2})
\\
\Rightarrow\cos(\frac{2\alpha}{2})+\cos(\frac{2\beta}{2})=\cos\alpha+\cos\beta
\\ \\
\therefore2\cos(\frac{\alpha+\beta}{2})\cos(\frac{\alpha-\beta}{2})
=\cos\alpha+\cos\beta
\\ \\
获得公式3: \enspace
\cos\alpha+\cos\beta =
2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}
\end{align}
\]
Fourth
\[\begin{align}
已知积化和差公式: \\
\sin\alpha\sin\beta=\frac{\cos(\alpha+\beta)-\cos(\alpha-\beta)}{-2}
\\
设\alpha\Rightarrow\frac{\alpha+\beta}{2}, \enspace
\beta\Rightarrow\frac{\alpha-\beta}{2}
\enspace,将假设代入积化和差公式:
\\ \\
-2\sin(\frac{\alpha+\beta}{2})\sin(\frac{\alpha-\beta}{2})=\cos(\frac{\alpha+\beta}{2}+
\frac{\alpha-\beta}{2})-\cos(\frac{\alpha+\beta}{2}-\frac{\alpha-\beta}{2})
\\ \\
\Rightarrow\cos\left(\frac{2\alpha}{2}\right)-\cos\left(\frac{2\beta}{2}\right)
=\cos\alpha-\cos\beta
\\ \\
\therefore
-2\sin\left( \frac{\alpha+\beta}{2} \right)\sin\left(\frac{\alpha-\beta}{2}\right)
=\cos\alpha-\cos\beta
\\ \\
获得公式4: \enspace
\cos\alpha-\cos\beta =
-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}
\end{align}
\]
Summarize
\[\begin{align}
公式1: \enspace
\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}
\\ \\
公式2: \enspace
\sin\alpha-\sin\beta
=2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}
\\ \\
公式3: \enspace
\cos\alpha+\cos\beta =
2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}
\\ \\
公式4: \enspace
\cos\alpha-\cos\beta =
-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}
\end{align}
\]
