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\(\Leftrightarrow\frac{4sin2x+cos2x+17}{3cos2x+sin2x+m+1}-2\ge0\) (tất nhiên là với mọi x)
\(\Leftrightarrow\frac{2sin2x-5cos2x-2m+15}{3cos2x+sin2x+m+1}\ge0\)
TH1: \(\left\{{}\begin{matrix}2sin2x-5cos2x-2m+15\ge0\\3cos2x+sin2x+m+1>0\end{matrix}\right.\) ;\(\forall x\)
\(\Leftrightarrow\left\{{}\begin{matrix}\frac{2}{\sqrt{29}}sin2x-\frac{5}{\sqrt{29}}cos2x\ge\frac{2m-15}{\sqrt{29}}\\\frac{1}{\sqrt{10}}sin2x+\frac{3}{\sqrt{10}}cos2x>\frac{-m-1}{\sqrt{10}}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}sin\left(2x-a\right)\ge\frac{2m-15}{\sqrt{29}}\\sin\left(2x+b\right)>\frac{-m-1}{\sqrt{10}}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\frac{2m-15}{\sqrt{29}}\le-1\\\frac{-m-1}{\sqrt{10}}< -1\end{matrix}\right.\) tới đây chắc bạn tự giải được
TH2: tương tự:
\(\left\{{}\begin{matrix}2sin2x-5cos2x-2m+15\le0\\3cos2x+sin2x+m+1< 0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\frac{2m-15}{\sqrt{29}}\ge1\\\frac{-m-1}{\sqrt{10}}>1\end{matrix}\right.\) \(\Leftrightarrow...\)
e/
\(\Leftrightarrow1+cos2x+1+cos4x+1+cos6x=3+3cosx.cos4x\)
\(\Leftrightarrow cos2x+cos6x+cos4x-3cosx.cos4x=0\)
\(\Leftrightarrow2cos4x.cos2x+cos4x-3cosx.cos4x=0\)
\(\Leftrightarrow cos4x\left(2cos2x+1-3cosx\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos4x=0\Rightarrow x=\frac{\pi}{8}+\frac{k\pi}{4}\\2cos2x-3cosx+1=0\left(1\right)\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow2\left(2cos^2x-1\right)-3cosx+1=0\)
\(\Leftrightarrow4cos^2x-3cosx-1=0\)
\(\Rightarrow\left[{}\begin{matrix}cosx=1\\cosx=-\frac{1}{4}\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=k2\pi\\x=\pm arccos\left(-\frac{1}{4}\right)+k2\pi\end{matrix}\right.\)
d/
\(\Leftrightarrow5\left(1+cosx\right)=2+\left(sin^2x-cos^2x\right)\left(sin^2x+cos^2x\right)\)
\(\Leftrightarrow5\left(1+cosx\right)=2+sin^2x-cos^2x\)
\(\Leftrightarrow5+5cosx=2+1-cos^2x-cos^2x\)
\(\Leftrightarrow2cos^2x+5cosx+2=0\)
\(\Rightarrow\left[{}\begin{matrix}cosx=-\frac{1}{2}\\cosx=-2\left(l\right)\end{matrix}\right.\)
\(\Rightarrow x=\pm\frac{2\pi}{3}+k2\pi\)
1.
\(0\le cos^2\left(\frac{x}{2}-\frac{\pi}{9}\right)\le1\)
\(\Rightarrow-5\le y\le-1\)
\(y_{min}=-5\) khi \(cos\left(\frac{x}{2}-\frac{\pi}{9}\right)=0\)
\(y_{max}=-1\) khi \(cos^2\left(\frac{x}{2}-\frac{\pi}{9}\right)=1\)
2.
Hàm \(y=3-sin7x\) có chu kì \(T=\frac{2\pi}{7}\)
Hàm \(y=\frac{sin2x.cos2x}{25}=\frac{1}{50}sin4x\) có chu kì \(T=\frac{2\pi}{4}=\frac{\pi}{2}\)
23.
\(tan^2x\ge0\Rightarrow y\le2\)
\(y_{max}=2\) khi \(tanx=0\)
\(y_{min}\) không tồn tại
24.
\(-1\le cosx\le1\Rightarrow0< 1+cosx\le2\)
\(\Rightarrow y\ge\frac{1}{2}\)
\(y_{min}=\frac{1}{2}\) khi \(cosx=1\)
\(y_{max}\) ko tồn tại
19.
\(y=\sqrt{5-\frac{1}{2}\left(2sinxcosx\right)^2}=\sqrt{5-\frac{1}{2}sin^22x}\)
\(0\le sin^22x\le1\Rightarrow\frac{3\sqrt{2}}{2}\le y\le\sqrt{5}\)
\(y_{min}=\frac{3\sqrt{2}}{2}\) khi \(sin^22x=1\)
\(y_{max}=\sqrt{5}\) khi \(sin^22x=0\)
21.
\(y=2sin^2x-\left(1-2sin^2x\right)=4sin^2x-1\)
\(0\le sin^2x\le1\Rightarrow-1\le y\le3\)
\(y_{min}=-1\) khi \(sin^2x=0\)
\(y_{max}=3\) khi \(sin^2x=1\)
ĐKXĐ: \(sin2x\ne0\)
\(\Leftrightarrow\frac{cos^2x-sin^2x}{sinx.cosx}+4sin2x=\frac{2}{sin2x}\)
\(\Leftrightarrow\frac{2cos2x}{sin2x}+4sin2x=\frac{2}{sin2x}\)
\(\Leftrightarrow cos2x+2sin^22x=1\)
\(\Leftrightarrow cos2x-2\left(1-cos^22x\right)-1=0\)
\(\Leftrightarrow2cos^22x+cos2x-3=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos2x=1\left(ktm\right)\\cos2x=-\frac{3}{2}>1\left(l\right)\end{matrix}\right.\)
Phương trình vô nghiệm
