phương trình tương đương:

sin4x.sin7x-cos3x.cos6x=0

<=> \(\frac{-1}{2}\)cos11x+\(\frac{1}{2}\)cos3x-\(\frac{1}{2}\)cos9x-\(\frac{1}{2}\)cos3x=0

<=> -\(\frac{1}{2}\)( cos11x+cos9x)=0

<=> cos 11x+cos9x=0

<=> 2cos10x.cosx=0

<=>\(\left[\begin{array}{nghiempt}cos10x=0\\cosx=0\end{array}\right.\)

<=>\(\left[\begin{array}{nghiempt}x=\frac{\pi}{2}+k\pi\\x=\frac{\pi}{20}+\frac{k\pi}{10}\end{array}\right.\) với k \(\in\)Z

vậy có 2 nghiệm trên đó

ĐKXĐ: ...

Đặt \(tanx=t\Rightarrow cos2x=\frac{1-t^2}{1+t^2}\)

Pt trở thành: \(\frac{2-2t^2}{1+t^2}+t=\frac{4}{5}\Leftrightarrow5t^3-14t^2+5t+6=0\)

\(\Leftrightarrow\left(t-2\right)\left(5t^2-4t-3\right)=0\)

\(\Rightarrow\left[{}\begin{matrix}t=2\\t=\frac{2+\sqrt{19}}{5}\\t=\frac{2-\sqrt{19}}{5}\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=arctan\left(2\right)+k\pi\\x=arctan\left(\frac{2+\sqrt{19}}{5}\right)+k\pi\\x=arctan\left(\frac{2-\sqrt{19}}{5}\right)+k\pi\end{matrix}\right.\)

đọc o hiểu

Lời giải:

PT $\Leftrightarrow 2\sin 2x\cos 2x+2\cos 2x+4(\sin x+\cos x)=1+\cos ^22x-\sin ^22x=2\cos ^22x$

$\Leftrightarrow \sin 2x\cos 2x+\cos 2x+2(\sin x+\cos x)=\cos ^22x$

$\Leftrightarrow \cos 2x(\sin 2x+1-\cos 2x)+2(\sin x+\cos x)=0$

$\Leftrightarrow \cos 2x(2\sin x\cos x+2\sin ^2x)+2(\sin x+\cos x)=0$

$\Leftrightarrow \cos 2x\sin x(\cos x+\sin x)+(\sin x+\cos x)=0$

$\Leftrightarrow (\sin x+\cos x)(\cos 2x\sin x+1)=0$

Nếu $\sin x+\cos x=0$. Kết hợp $\sin ^2x+\cos ^2x=1$ suy ra $(\sin x, \cos x)=(\frac{1}{\sqrt{2}}; \frac{-1}{\sqrt{2}})$ và hoán vị

$\Rightarrow x=k\pi -\frac{\pi}{4}$ với $k$ nguyên.

Nếu $\cos 2x\sin x+1=0$

$\Leftrightarrow (1-2\sin ^2x)\sin x+1=0$

$\Leftrightarrow (1-\sin x)(2\sin ^2x+2\sin x+1)=0$

$\Rightarrow \sin x=1$

$\Rightarrow x=2k\pi +\frac{\pi}{2}$ với $k$ nguyên.

9.

\(\Leftrightarrow1-cos2x+m.sin2x=2m\)

\(\Leftrightarrow m.sin2x-cos2x=2m-1\)

Pt đã cho vô nghiệm khi và chỉ khi:

\(m^2+1< \left(2m-1\right)^2\)

\(\Leftrightarrow3m^2-4m>0\Rightarrow\left[{}\begin{matrix}m< 0\\m>\frac{4}{3}\end{matrix}\right.\)

10.

\(sin\left(x+\frac{\pi}{4}\right)+sin\left(x-\frac{\pi}{4}\right)=0\)

\(\Leftrightarrow2sinx.cos\frac{\pi}{4}=0\)

\(\Leftrightarrow sinx=0\)

\(\Leftrightarrow x=k\pi\)

\(\Rightarrow x=\left\{\pi;2\pi;3\pi\right\}\Rightarrow\sum x=6\pi\)

7.

\(sin3x-cosx=0\)

\(\Leftrightarrow sin3x=cosx\)

\(\Leftrightarrow sin3x=sin\left(\frac{\pi}{2}-x\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}3x=\frac{\pi}{2}-x+k2\pi\\3x=\frac{\pi}{2}+x+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{8}+\frac{k\pi}{2}\\x=\frac{\pi}{4}+k\pi\end{matrix}\right.\)

8.

\(\Leftrightarrow2sinx.cosx-\left(1-2sin^2x\right)-3sinx-cosx+2=0\)

\(\Leftrightarrow cosx\left(2sinx-1\right)+2sin^2x-3sinx+1=0\)

\(\Leftrightarrow cosx\left(2sinx-1\right)+\left(2sinx-1\right)\left(sinx-1\right)=0\)

\(\Leftrightarrow\left(2sinx-1\right)\left(cosx+sinx-1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=\frac{1}{2}\\sin\left(x+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{6}+k2\pi\\x=\frac{5\pi}{6}+k2\pi\\x=k2\pi\\x=\frac{\pi}{2}+k2\pi\end{matrix}\right.\) pt có đúng 1 nghiệm trên khoảng đã cho

\(\Leftrightarrow\left(sin^2x+cos^2x\right)^2-2sin^2x.cos^2x=2cos2x\)

\(\Leftrightarrow1-\frac{1}{2}sin^22x=2cos2x\)

\(\Leftrightarrow2-\left(1-cos^22x\right)=4cos2x\)

\(\Leftrightarrow cos^22x-4cos2x+1=0\)

\(\Leftrightarrow\left[{}\begin{matrix}cos2x=2+\sqrt{3}>1\left(l\right)\\cos2x=2-\sqrt{3}\end{matrix}\right.\)

\(\Leftrightarrow x=\pm\frac{1}{2}arccos\left(2-\sqrt{3}\right)+k\pi\)

ĐKXĐ: ...

\(\Leftrightarrow\left(tan^2x-1\right)\left(tan^2x-3\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}tan^2x=1\\tan^2x=3\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}sin^2x=cos^2x\\sin^2x=3cos^2x\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}cos^2x-sin^2x=0\\\frac{1}{2}-\frac{1}{2}cos2x=\frac{3}{2}+\frac{3}{2}cos2x\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}cos2x=0\\cos2x=-\frac{1}{2}\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{4}+\frac{k\pi}{2}\\x=\pm\frac{\pi}{3}+k\pi\end{matrix}\right.\)

\(sinx=-\dfrac{4}{3}< -1\)

\(\Rightarrow\)Phương trình vô nghiệm

\(\Leftrightarrow3.\left(1-cos6x\right)+2.cos^26x-1=4\)

\(\Leftrightarrow2.cos^26x-3.cos6x-2=0\)

\(\Leftrightarrow\left(cos6x-2\right)\left(2.cos6x+1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}cos6x=2\left(vn\right)\\cos6x=-\dfrac{1}{2}\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}6x=\dfrac{2\pi}{3}+k2\pi\\6x=-\dfrac{2\pi}{3}+k2\pi\end{matrix}\right.\) (\(k\in Z\)) \(\Leftrightarrow\)\(\left[{}\begin{matrix}x\ne\dfrac{\pi}{9}+\dfrac{k\pi}{3}\\x\ne\dfrac{-\pi}{9}+\dfrac{k\pi}{3}\end{matrix}\right.\) (\(k\in Z\))

Vậy...