q) \(3sin3x-\sqrt{3}cos9x=1+4sin^33x\)
\(\Leftrightarrow3sin3x-\sqrt{3}cos9x=1+3sin3x-sin9x\)
\(\Leftrightarrow sin9x-\sqrt{3}cos9x=1\)
\(\Leftrightarrow sin9x.cos\dfrac{\pi}{3}-cos9x.sin\dfrac{\pi}{3}=\dfrac{1}{2}\)
\(\Leftrightarrow sin\left(9x-\dfrac{\pi}{3}\right)=\dfrac{1}{2}\) \(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{18}+\dfrac{k2\pi}{9}\\x=\dfrac{7\pi}{54}+\dfrac{k2\pi}{9}\end{matrix}\right.\) (\(k\in Z\))
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x) \(8sinx=\dfrac{\sqrt{3}}{cosx}+\dfrac{1}{sinx}\)
(đk: \(cosx\ne0;sinx\ne0\) \(\Rightarrow sin2x\ne0\) \(\Leftrightarrow x\ne\dfrac{k\pi}{2}\);\(k\in Z\))
\(\Leftrightarrow8sinx=\dfrac{\sqrt{3}sinx+cosx}{cosx.sinx}\)
\(\Leftrightarrow\)\(8sinx.cosx.sinx=\sqrt{3}sinx+cosx\)
\(\Leftrightarrow4sinx.sin2x=\sqrt{3}sinx+cosx\)
\(\Leftrightarrow2\left(cosx-cos3x\right)=\sqrt{3}sinx+cosx\)
\(\Leftrightarrow cosx-\sqrt{3}sinx=2cos3x\)
\(\Leftrightarrow cosx.cos\left(\dfrac{\pi}{3}\right)-sinx.sin\left(\dfrac{\pi}{3}\right)=cos3x\)
\(\Leftrightarrow cos\left(x+\dfrac{\pi}{3}\right)=cos3x\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{6}-k\pi\\x=-\dfrac{\pi}{12}+\dfrac{k\pi}{2}\end{matrix}\right.\)(\(k\in Z\)) (thỏa mãn)
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