Nghiệm dương nhỏ nhất của phương trình \(\sin x-\cos x+\sin 2x=2{{\cos }^{2}}x\) là:
\(x=\dfrac{\pi }{6}\)
\(x=\dfrac{2\pi }{3}\)
\(x=\dfrac{\pi }{4}\)
\(x=\dfrac{\pi }{3}\)
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\(\begin{array}{l}\sin x - \cos x + \sin 2x = 2{\cos ^2}x\\ \Leftrightarrow \sin x - \cos x + 2\sin x\cos x - 2{\cos ^2}x = 0\\ \Leftrightarrow \sin x\left( {1 + 2\cos x} \right) - \cos x\left( {1 + 2\cos x} \right) = 0\\ \Leftrightarrow \left( {1 + 2\cos x} \right)\left( {\sin x - \cos x} \right) = 0\\ \Leftrightarrow \left[ \begin{array}{l}1 + 2\cos x = 0\\\sin x = \cos x\end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l}\cos x = - \frac{1}{2}\\\tan x = 1\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = \frac{{2\pi }}{3} + k2\pi \,\,\,\,\,\,\,\,\left( 1 \right)\\x = - \frac{{2\pi }}{3} + k2\pi \,\,\,\,\left( 2 \right)\\x = \frac{\pi }{4} + k\pi \,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 3 \right)\end{array} \right.\,\left( {k \in Z} \right)\end{array}\)
Nghiệm dương nhỏ nhất của (1) là \(x=\dfrac{2\pi }{3}\)
Nghiệm dương nhỏ nhất của (2) là \(x=\dfrac{4\pi }{3}.\)
Nghiệm dương nhỏ nhất của (3) là \(x=\dfrac{\pi }{4}.\)
Vậy nghiệm dương nhỏ nhất của phương trình ban đầu là \(x=\dfrac{\pi }{4}.\)
Chọn C.
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