\(sin^2x+\sqrt{3}sinx.cosx=1\)
\(\Leftrightarrow2sin^2x+2\sqrt{3}sinx.cosx=2\)
\(\Leftrightarrow\sqrt{3}sin2x-cos2x=1\)
\(\Leftrightarrow\dfrac{\sqrt{3}}{2}sin2x-\dfrac{1}{2}cos2x=\dfrac{1}{2}\)
\(\Leftrightarrow sin\left(2x-\dfrac{\pi}{6}\right)=\dfrac{1}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}2x-\dfrac{\pi}{6}=\dfrac{\pi}{6}+k2\pi\\2x-\dfrac{\pi}{6}=\pi-\dfrac{\pi}{6}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{6}+k\pi\\x=\dfrac{\pi}{2}+k\pi\end{matrix}\right.\)
Nghiệm âm lớn nhất, nghiệm dương nhỏ nhất lần lượt là \(x=-\dfrac{\pi}{2},x=\dfrac{\pi}{6}\)
\(\Rightarrow\) Tích \(P=-\dfrac{\pi}{2}.\dfrac{\pi}{6}=-\dfrac{\pi^2}{12}\)
