202601202

题目

$已知 3 \sin α + \sin β = \sqrt{10}, 且 α + β = \frac{3 π}{2}, 求 \sin α$

解析

$\sin β = \sin (\frac{3 π}{2} - α) = - \cos α$
${\begin{cases} 3 \sin α - \cos α = \sqrt{10} \dots ① \\ \sin^{2} α + \cos^{2} α = 1 \dots ② \end{cases}$
$由 ① \cos α = 3 \sin α - \sqrt{10} 代入 ②$
$整理得 (\sqrt{10} \sin α - 3)^{2} = 0$
$\sin α = \frac{3 \sqrt{10}}{10}$