Question

Giải phương trình lượng giác: \(sin^23x.cos2x+sin^2x=0\)

Answer 1

\(sin^23x.cos2x+sin^2x=0\)

\(\left(3sinx-4sin^3x\right)^2.cos2x+sin^2x=0\)

\(sin^2x\left[\left(3-4sin^2x\right)^2.cos2x+1\right]=0\)

\(sin^2x\left[\left(1+2cos2x\right)^2.cos2x+1\right]=0\)

\(sin^2x\left(4cos^22x+1\right)\left(cos2x+1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=0\\cos2x=1\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=k\text{π}\\2x=k2\text{π}\end{matrix}\right.\)\(\Leftrightarrow x=k\text{π}\)

Answer 2

\(sin^23xcos2x+sin^2x=0\rightarrow\dfrac{1-cos6x}{2}.cos2x+\dfrac{1-cos2x}{2}=0\\ \rightarrow cos6xcos2x=1\rightarrow cos8x+cos4x=2\\ \rightarrow cos8x=cos4x=1\rightarrow x=\dfrac{k\pi}{2}\left(k\in Z\right)\)