已知 0<α<π, cos α 2 = 5 5，则 sin…——2025 高考数学第 8 题答案解析

【解答】
$0<\alpha<\pi, \quad \cos \frac{\alpha}{2}=\frac{\sqrt{5}}{5}, \quad \sin \left(\alpha-\frac{\pi}{4}\right)=$
A．$\frac{\sqrt{2}}{10}$
B．$\frac{\sqrt{2}}{5}$
C．$\frac{3 \sqrt{2}}{10}$
D．$\frac{7 \sqrt{2}}{10}$

【答案】D
【解答】

$$ \begin{aligned} & \because \cos \frac{\alpha}{2}=\frac{\sqrt{5}}{5}, \\ & \therefore \cos \alpha=2 \cos ^{2} \frac{\alpha}{2}-1=-\frac{3}{5}, \end{aligned} $$

又 $\because 0<\alpha<\pi$ ，

$$ \therefore \sin \alpha=\frac{4}{5}, $$

则 $\sin \left(\alpha-\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}(\sin \alpha-\cos \alpha)=\frac{7 \sqrt{2}}{10}$ ．