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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
6.
\(\Leftrightarrow\frac{1}{2}cos6x+\frac{1}{2}cos4x=\frac{1}{2}cos6x+\frac{1}{2}cos2x+\frac{3}{2}+\frac{3}{2}cos2x+1\)
\(\Leftrightarrow cos4x=4cos2x+5\)
\(\Leftrightarrow2cos^22x-1=4cos2x+5\)
\(\Leftrightarrow cos^22x-2cos2x-3=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos2x=-1\\cos2x=3>1\left(ktm\right)\end{matrix}\right.\)
\(\Leftrightarrow...\)
7.
Thay lần lượt 4 đáp án ta thấy chỉ có đáp án C thỏa mãn
8.
\(\Leftrightarrow\left[{}\begin{matrix}sinx=1\\sinx=\frac{1}{2}\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{2}+k2\pi\\x=\frac{\pi}{6}+k2\pi\\x=\frac{5\pi}{6}+k2\pi\end{matrix}\right.\)
\(\Rightarrow x=\left\{\frac{\pi}{6};\frac{\pi}{2}\right\}\)
9.
Đặt \(sinx+cosx=t\Rightarrow\left\{{}\begin{matrix}-1\le t\le1\\sinx.cosx=\frac{t^2-1}{2}\end{matrix}\right.\)
\(\Rightarrow mt+\frac{t^2-1}{2}+1=0\)
\(\Leftrightarrow t^2+2mt+1=0\)
Pt đã cho có đúng 1 nghiệm thuộc \(\left[-1;1\right]\) khi và chỉ khi: \(\left[{}\begin{matrix}m\ge1\\m\le-1\end{matrix}\right.\)
10.
\(\frac{\sqrt{3}}{2}cos5x-\frac{1}{2}sin5x=cos3x\)
\(\Leftrightarrow cos\left(5x-\frac{\pi}{6}\right)=cos3x\)
\(\Leftrightarrow\left[{}\begin{matrix}5x-\frac{\pi}{6}=3x+k2\pi\\5x-\frac{\pi}{6}=-3x+k2\pi\end{matrix}\right.\)
4.
\(\Leftrightarrow2sinx.cosx-\left(1-2sin^2x\right)+3sinx-cosx-1=0\)
\(\Leftrightarrow cosx\left(2sinx-1\right)+2sin^2x+3sinx-2=0\)
\(\Leftrightarrow cosx\left(2sinx-1\right)+\left(2sinx-1\right)\left(sinx+2\right)=0\)
\(\Leftrightarrow\left(2sinx-1\right)\left(sinx+cosx+2\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}2sinx-1=0\\sinx+cosx=-2\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx=\frac{1}{2}\\sin\left(x+\frac{\pi}{4}\right)=-\sqrt{2}< -1\left(l\right)\end{matrix}\right.\)
\(\Leftrightarrow...\)
2.
ĐKXĐ: ...
\(\Leftrightarrow cot\left(\frac{\pi}{4}-x\right)=-\frac{1}{\sqrt{3}}\)
\(\Leftrightarrow\frac{\pi}{4}-x=-\frac{\pi}{3}+k\pi\)
\(\Leftrightarrow x=\frac{7\pi}{12}+k\pi\)
3.
\(\Leftrightarrow cos\frac{x}{4}sinx+sin\frac{x}{4}.cosx-3\left(sin^2x+cos^2x\right)+cosx=0\)
\(\Leftrightarrow sin\left(x+\frac{x}{4}\right)=-cosx\)
\(\Leftrightarrow sin\frac{5x}{4}=sin\left(x-\frac{\pi}{2}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}\frac{5x}{4}=x-\frac{\pi}{2}+k2\pi\\\frac{5x}{4}=\frac{3\pi}{2}-x+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow...\)
\(2\left(1-sin^2x\right)+3sinx+3=0\)
\(\Leftrightarrow-2sin^2x+3sinx+5=0\Rightarrow\left[{}\begin{matrix}sinx=-1\\sinx=\frac{5}{2}\left(l\right)\end{matrix}\right.\)
\(\Leftrightarrow x=-\frac{\pi}{2}+k2\pi\)
\(0\le-\frac{\pi}{2}+k2\pi\le200\pi\Rightarrow1\le k\le100\) (có 100 nghiệm)
Tổng các nghiệm:
\(\sum x=-\frac{\pi}{2}.100+\sum\limits^{100}_{k=1}2k\pi=10050\pi\)
2.
\(\Leftrightarrow2cos^2x-1+3\left|cosx\right|-1=0\)
\(\Leftrightarrow2\left|cosx\right|^2+3\left|cosx\right|-2=0\Rightarrow\left[{}\begin{matrix}\left|cosx\right|=\frac{1}{2}\\\left|cosx\right|=-2\left(l\right)\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}cosx=\frac{1}{2}\\cosx=-\frac{1}{2}\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x=\pm\frac{\pi}{3}+k2\pi\\x=\pm\frac{2\pi}{3}+k2\pi\end{matrix}\right.\)
Pt có 2 nghiệm trên đoạn đã cho \(x=\pm\frac{\pi}{3}\)
a. ĐKXĐ: ...
\(cot\left(2\pi-\frac{\pi}{3}-3x\right)=tan\left(2x+\frac{\pi}{3}\right)\)
\(\Leftrightarrow cot\left(-3x-\frac{\pi}{3}\right)=tan\left(2x+\frac{\pi}{3}\right)\)
\(\Leftrightarrow tan\left(3x+\frac{5\pi}{6}\right)=tan\left(2x+\frac{\pi}{3}\right)\)
\(\Leftrightarrow3x+\frac{5\pi}{6}=2x+\frac{\pi}{3}+k\pi\)
\(\Leftrightarrow...\)
b. ĐKXĐ: \(x\ne\frac{k\pi}{2}\)
\(\frac{cosx.cos2x}{sinx.sin2x}=-1\)
\(\Leftrightarrow cosx.cos2x=-sinx.sin2x\)
\(\Leftrightarrow cosx.cos2x+sinx.sin2x=0\)
\(\Leftrightarrow cosx=0\)
\(\Leftrightarrow x=\frac{\pi}{2}+k\pi\) (ktm)
Vậy pt vô nghiệm
c. ĐKXĐ: ...
\(tanx=\frac{3}{tanx}\)
\(\Leftrightarrow tan^2x=3\)
\(\Rightarrow tanx=\pm\sqrt{3}\)
\(\Rightarrow x=\pm\frac{\pi}{3}+k\pi\)
d.
\(2sin^2x+1-2sin^2x=2\)
\(\Leftrightarrow1=2\) (vô lý)
Vậy pt vô nghiệm