Question

c. (sin2x+1)(\(\sqrt{3}\)cosx-sinx-1)=0

Answer 1

\(\left(sin2x+1\right)\left(\sqrt{3}cosx-sinx-1\right)=0\)

\(\Leftrightarrow\left(sin2x+1\right)\left(\dfrac{\sqrt{3}}{2}cosx-\dfrac{1}{2}sinx-\dfrac{1}{2}\right)=0\)

\(\Leftrightarrow\left(sin2x+1\right)\left(cos\left(x+\dfrac{\pi}{6}\right)-\dfrac{1}{2}\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sin2x=-1\\cos\left(x+\dfrac{\pi}{6}\right)=\dfrac{1}{2}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}2x=-\dfrac{\pi}{2}+k2\pi\\x+\dfrac{\pi}{6}=\dfrac{\pi}{3}+k2\pi\\x+\dfrac{\pi}{6}=-\dfrac{\pi}{3}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{4}+k\pi\\x=\dfrac{\pi}{6}+k2\pi\\x=-\dfrac{\pi}{2}+k2\pi\end{matrix}\right.\)