Question

Trong khoảng \(\left(0;\frac{\pi}{2}\right)\) thì pt: \(\sin^24x+3\sin4x.\cos4x-4\cos^24x=0\) có bn nghiệm?

Answer 1

\(\Leftrightarrow\left(sin4x-cos4x\right)\left(sin4x+4cos4x\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sin4x-cos4x=0\\sin4x=-4cos4x\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}sin\left(4x-\frac{\pi}{4}\right)=0\\tan4x=-4\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}4x-\frac{\pi}{4}=k\pi\\4x=arctan\left(-4\right)+k\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{16}+\frac{k\pi}{4}\\x=\frac{1}{4}arctan\left(-4\right)+\frac{k\pi}{4}\end{matrix}\right.\)

Pt có 4 nghiệm trên khoảng đã cho