17.
\(cosx.cos3x=-\dfrac{1}{2}\)
\(\Leftrightarrow\dfrac{1}{2}cos4x+\dfrac{1}{2}cos2x=-\dfrac{1}{2}\)
\(\Leftrightarrow2cos^22x-1+cos2x=-1\)
\(\Leftrightarrow2cos^22x-cos2x=0\)
\(\Rightarrow\left[{}\begin{matrix}cos2x=0\\cos2x=\dfrac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{4}+\dfrac{k\pi}{2}\\x=\pm\dfrac{\pi}{6}+k\pi\end{matrix}\right.\)
17: \(\Leftrightarrow\dfrac{1}{2}\cdot\left(cos\left(-2x\right)+cos4x\right)=\dfrac{-1}{2}\)
=>\(cos2x+cos4x=-1\)
=>\(2cos^22x-1+cos2x=-1\)
=>cos2x(2 cos 2x+1)=0
=>cos2x=0 hoặc cos2x=-1/2
\(\Leftrightarrow\left[{}\begin{matrix}2x=\dfrac{pi}{2}+kpi\\2x=\dfrac{2}{3}pi+k2pi\\2x=-\dfrac{2}{3}pi+k2pi\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{pi}{4}+\dfrac{kpi}{2}\\x=\dfrac{1}{3}pi+kpi\\x=-\dfrac{1}{3}pi+kpi\end{matrix}\right.\)
19: \(\Leftrightarrow4-4cos^2x+\left(2\sqrt{3}+2\right)cosx-4-\sqrt{3}=0\)
\(\Leftrightarrow4cos^2x-\left(2\sqrt{3}+2\right)cosx+\sqrt{3}=0\)
\(\text{Δ}=16-8\sqrt{3}\)
Pt sẽ có hai nghiệm là:
\(\left\{{}\begin{matrix}cosx=\dfrac{2\sqrt{3}+2-2\sqrt{3}+2}{8}=\dfrac{4}{8}=\dfrac{1}{2}\\cosx=\dfrac{2\sqrt{3}+2+2\sqrt{3}-2}{8}=\dfrac{4\sqrt{3}}{8}=\dfrac{\sqrt{3}}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\pm\dfrac{pi}{6}+k2pi\\x=\pm\dfrac{pi}{3}+k2pi\end{matrix}\right.\)
18.
\(3+sinx.sin3x=3cos2x\)
\(\Leftrightarrow3+sinx.sin3x=3\left(1-2sin^2x\right)\)
\(\Leftrightarrow6sin^2x+sinx.sin3x=0\)
\(\Leftrightarrow6.sin^2x+sinx\left(3sinx-4sin^3x\right)=0\)
\(\Leftrightarrow6sin^2x+sin^2x\left(3-4sin^2x\right)=0\)
\(\Leftrightarrow sin^2x\left(9-4sin^2x\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin^2x=0\\sin^2x=\dfrac{9}{4}\left(vn\right)\end{matrix}\right.\)
\(\Rightarrow x=k\pi\)
19.
\(4sin^2x+2\left(\sqrt{3}+1\right)cosx-\sqrt{3}=4\)
\(\Leftrightarrow4\left(1-cos^2x\right)+2\left(\sqrt{3}+1\right)cosx-\sqrt{3}=4\)
\(\Leftrightarrow-4cos^2x+2\left(\sqrt{3}+1\right)cosx-\sqrt{3}=0\)
\(\Rightarrow\left[{}\begin{matrix}cosx=\dfrac{1}{2}\\cosx=\dfrac{\sqrt{3}}{2}\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=\pm\dfrac{\pi}{3}+k2\pi\\x=\pm\dfrac{\pi}{6}+k2\pi\end{matrix}\right.\)
20.
ĐKXĐ: \(x\ne k\pi\)
\(\dfrac{3}{sin^2x}+\left(6+\sqrt{3}\right)cotx+2\sqrt{3}-3=0\)
\(\Leftrightarrow3\left(1+cot^2x\right)+\left(6+\sqrt{3}\right)cotx+2\sqrt{3}-3=0\)
\(\Leftrightarrow3cot^2x+\left(6+\sqrt{3}\right)cotx+2\sqrt{3}=0\)
\(\Rightarrow\left[{}\begin{matrix}cotx=-\dfrac{\sqrt{3}}{3}\\cotx=-2\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{3}+k\pi\\x=arccot\left(-2\right)+k\pi\end{matrix}\right.\)
