Giải phương trình: sin( π/6+x)=cos 2x

Đáp án:

 $sin(\dfrac{\pi}{6}+x)=cos2x$

$⇒sin(\dfrac{\pi}{6}+x)=sin(\dfrac{\pi}{2}-2x)$

$⇔\left[\begin{matrix} \dfrac{\pi}{6}+x=\dfrac{\pi}{2}-2x+k2\pi\\\dfrac{\pi}{6}+x=\pi-\dfrac{\pi}{2}+2x+k2\pi\end{matrix}\right.(k \in \mathbb{Z}).$

$⇔\left[\begin{matrix} \dfrac{\pi}{6}+3x=\dfrac{\pi}{2}+k2\pi\\\dfrac{\pi}{6}-x=\pi-\dfrac{\pi}{2}+k2\pi\end{matrix}\right.(k \in \mathbb{Z}).$

$⇔\left[\begin{matrix} 3x=\dfrac{\pi}{3}+k2\pi\\-x=\dfrac{\pi}{3}+k2\pi\end{matrix}\right.(k \in \mathbb{Z}).$

$⇔\left[\begin{matrix} x=\dfrac{\pi}{9}+k\dfrac{2\pi}{3}\\x=-\dfrac{\pi}{3}-k2\pi\end{matrix}\right.(k \in \mathbb{Z}).$

$\text{Vậy pt có các nghiệm} x=-\dfrac{\pi}{3}-k2\pi,x=\dfrac{\pi}{9}+k\dfrac{2\pi}{3}(k \in \mathbb{Z}).$

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