ĐKXĐ: \(sinx\ne\pm1\)

\(\dfrac{3cos2x-2sinx+5}{2\left(1-sin^2x\right)}=0\)

\(\Leftrightarrow3\left(1-2sin^2x\right)-2sinx+5=0\)

\(\Leftrightarrow-6sin^2x-2sinx+8=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=1\left(loại\right)\\sinx=-\dfrac{4}{3}< -1\left(loại\right)\end{matrix}\right.\)

Vậy pt vô nghiệm

a

\(\Leftrightarrow\left(3sinx-sin3x\right)cos3x+\left(3cosx+cos3x\right)sin3x+3\sqrt{3}cos4x=3\)

\(\Leftrightarrow\left(sinx.cos3x+sin3x.cosx\right)+\sqrt{3}cos4x=1\)

\(\Leftrightarrow sin4x+\sqrt{3}cos4x=1\)

Tới đây thôi, mình lười ghi rồi =))

b

\(\Leftrightarrow\left(1-cos2x\right)\left(2sin^2x-1\right)\left(2sin^2+1\right)=cos2x\left(7cos^22x+3cos2x-4\right)\)

\(\Leftrightarrow\left(1-cos2x\right)\left(-cos2x\right)\left(2-cos2x\right)=cos2x\left(7cos^22x+3cos2x+4\right)\)

\(\Leftrightarrow-cos^22x+3cos2x-2=7cos^22x+3cos2x+4\)

\(\Leftrightarrow4cos^22x+3=0\)

=> pt vô nghiệm

ĐKXĐ: \(\left\{{}\begin{matrix}2x+5>=0\\4-2x>=0\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}2x>=-5\\2x< =4\end{matrix}\right.\Leftrightarrow-\dfrac{5}{2}< =x< =2\)

\(x^2+\sqrt{2x+5}+\sqrt{4-2x}=4x-1\)

=>\(x^2-4+\sqrt{2x+5}-3+\sqrt{4-2x}=4x-1-7\)

=>\(\left(x-2\right)\left(x+2\right)+\dfrac{2x+5-9}{\sqrt{2x+5}+3}+\sqrt{4-2x}=4x-8\)

=>\(\left(x-2\right)\left[\left(x+2\right)+\dfrac{2}{\sqrt{2x+5}+3}-4\right]+\sqrt{4-2x}=0\)

=>\(-\left(2-x\right)\left[\left(x-2\right)+\dfrac{2}{\sqrt{2x+5}+3}\right]+\sqrt{2\left(2-x\right)}=0\)

=>\(\sqrt{2-x}\left[-\sqrt{2-x}\left(x-2+\dfrac{2}{\sqrt{2x+5}+3}\right)+\sqrt{2}\right]=0\)

=>\(\sqrt{2-x}=0\)

=>x=2(nhận)

a) ta có : \(2sin^2x+3cos2x=0\Leftrightarrow2sin^2x+3\left(1-2sin^2x\right)=0\)

\(\Leftrightarrow3-4sin^2x=0\Leftrightarrow sin^2x=\dfrac{3}{4}\Leftrightarrow sinx=\pm\dfrac{\sqrt{3}}{2}\)

th1 : \(sinx=\dfrac{\sqrt{3}}{2}\Leftrightarrow sinx=sin\dfrac{\pi}{3}\) \(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{3}+k2\pi\\x=\pi-\dfrac{\pi}{3}+k2\pi\end{matrix}\right.\)

th2 : \(sinx=\dfrac{-\sqrt{3}}{2}\Leftrightarrow sinx=sin\left(\dfrac{-\pi}{3}\right)\) \(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{-\pi}{3}+k2\pi\\x=\pi+\dfrac{\pi}{3}+k2\pi\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{-\pi}{3}+k2\pi\\x=\dfrac{4\pi}{3}+k2\pi\end{matrix}\right.\)

vậy phương trình có 4 hệ nghiệm : \(x=\dfrac{\pi}{3}+k2\pi;x=\dfrac{2\pi}{3}+k2\pi;x=\dfrac{-\pi}{3}+k2\pi;x=\dfrac{4\pi}{3}+k2\pi\)

a) ta có : \(2sin^2x+3cos2x=0\Leftrightarrow2sin^2x+3\left(1-2sin^2x\right)=0\)

\(\Leftrightarrow3-4sin^2x=0\Leftrightarrow sin^2x=\dfrac{3}{4}\Leftrightarrow sinx=\pm\dfrac{\sqrt{3}}{2}\)

th1 : \(sinx=\dfrac{\sqrt{3}}{2}\Leftrightarrow sinx=sin\dfrac{\pi}{3}\) \(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{3}+k2\pi\\x=\pi-\dfrac{\pi}{3}+k2\pi\end{matrix}\right.\)

th2 : \(sinx=\dfrac{-\sqrt{3}}{2}\Leftrightarrow sinx=sin\left(\dfrac{-\pi}{3}\right)\) \(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{-\pi}{3}+k2\pi\\x=\pi+\dfrac{\pi}{3}+k2\pi\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{-\pi}{3}+k2\pi\\x=\dfrac{4\pi}{3}+k2\pi\end{matrix}\right.\)

vậy phương trình có 4 hệ nghiệm : \(x=\dfrac{\pi}{3}+k2\pi;x=\dfrac{2\pi}{3}+k2\pi;x=\dfrac{-\pi}{3}+k2\pi;x=\dfrac{4\pi}{3}+k2\pi\)

câu b bn làm tương tự cho quen nha

Pt\(\Leftrightarrow3\left(cos^2x-sin^2x\right)-8.sinx.cosx=sin^2x+cos^2x\)

\(\Leftrightarrow2cos^2x-8sinx.cosx-4sin^2x=0\)

\(\Leftrightarrow\left[{}\begin{matrix}cosx=\left(2+\sqrt{6}\right)sinx\\cosx=\left(2-\sqrt{6}\right)sinx\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}tanx=\dfrac{1}{2+\sqrt{6}}\\tanx=\dfrac{1}{2-\sqrt{6}}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=arc.tan\left(\dfrac{1}{2+\sqrt{6}}\right)+k\pi\\x=arc.tan\left(\dfrac{1}{2-\sqrt{6}}\right)+k\pi\end{matrix}\right.\), k nguyên

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