Question

Tìm nghiệm của các phương trình sau

a) \(2\left(sin2x.cosx-sinx.cos2x\right)-1=0\) (k thuoc Z)

b) \(\left(sin2x.cosx+sinx.cos2x\right)+2sin\frac{3x}{2}.cos\frac{3x}{2}-\sqrt{2}=0\)

Answer 1

a/ \(\Leftrightarrow2sin\left(2x-x\right)-1=0\)

\(\Leftrightarrow2sinx-1=0\Rightarrow sinx=\frac{1}{2}=sin\left(\frac{\pi}{6}\right)\)

\(\Rightarrow\left[{}\begin{matrix}x=\frac{\pi}{6}+k2\pi\\x=\frac{5\pi}{6}+k2\pi\end{matrix}\right.\)

b/ \(\Leftrightarrow sin\left(2x+x\right)+sin3x=\sqrt{2}\)

\(\Leftrightarrow2sin3x=\sqrt{2}\)

\(\Leftrightarrow sin3x=\frac{\sqrt{2}}{2}=sin\left(\frac{\pi}{4}\right)\)

\(\Rightarrow\left[{}\begin{matrix}3x=\frac{\pi}{4}+k2\pi\\3x=\frac{3\pi}{4}+k2\pi\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=\frac{\pi}{12}+\frac{k2\pi}{3}\\x=\frac{\pi}{4}+\frac{k2\pi}{3}\end{matrix}\right.\)