1596

已知 ${\displaystyle \sin (\alpha +\beta )=\sin \alpha -\cos \beta }$, 则 ${\displaystyle \sin (\alpha +\beta )}$ 的最大值为_____.

采用主元的思想: 令 ${\displaystyle \gamma =\alpha +\beta }$, 则$$\begin{array}{rl}\sin \gamma & =\sin \alpha -\cos (\gamma -\alpha )\\ & =\sin \alpha (1-\sin \gamma )-\cos \alpha \cos \gamma \\ & \le \sqrt{(1-\sin \gamma {)}^2+{\cos }^2\gamma }\\ & =\sqrt{(1-\sin \gamma {)}^2+1-{\sin }^2\gamma },\end{array}$$ 解得 ${\displaystyle \sin \gamma \le \sqrt{3}-1}$.