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\(\left(sin\dfrac{x}{2}-cox\dfrac{x}{2}\right)^2+\sqrt{3}cosx=2sin5x+1\)
⇔\(sin^2\dfrac{x}{2}+cos^2\dfrac{x}{2}-2sin\dfrac{x}{2}cos\dfrac{x}{2}+\sqrt{3}cosx=2sin5x+1\)
⇔\(1-sinx+\sqrt{3}cosx=2sin5x+1\)
⇔\(sin\left(\dfrac{\Pi}{3}-x\right)=sin5x\)
\(2sinx\left(\sqrt{3}cosx+sinx+2sin3x\right)=1\)
⇔\(2\sqrt{3}sinxcosx+2sin^2x+4sinxsin3x=1\)
⇔\(\sqrt{3}sin2x+1-cos2x+cos2x-2cos4x=1\)
⇔\(\sqrt{3}sin2x+cos2x=2cos4x\)
⇔\(cos\left(2x-\dfrac{\Pi}{3}\right)=cos4x\)
a/
\(\left(m+1\right)^2+\left(m-1\right)^2\ge\left(2m+3\right)^2\)
\(\Leftrightarrow2m^2+12m+7\le0\)
\(\Leftrightarrow\frac{-6-\sqrt{22}}{2}\le m\le\frac{-6+\sqrt{22}}{2}\)
b/ \(\Leftrightarrow\left\{{}\begin{matrix}m\ge0\\\left(m-1\right)^2+4m\ge m^4\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}m\ge0\\m^4-\left(m+1\right)^2\le0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}m\ge0\\\left(m^2+m+1\right)\left(m^2-m-1\right)\le0\end{matrix}\right.\)
\(\Leftrightarrow0\le m\le\frac{1+\sqrt{5}}{2}\)
c/ \(\Leftrightarrow\frac{1}{2}sin2x-\frac{\sqrt{3}}{2}cos2x+\frac{1}{2}=m\)
\(\Leftrightarrow sin\left(2x-\frac{\pi}{3}\right)+\frac{1}{2}=m\)
Do \(-\frac{1}{2}\le sin\left(2x-\frac{\pi}{3}\right)\le\frac{3}{2}\Rightarrow-\frac{1}{2}\le m\le\frac{3}{2}\)
a/ ĐKXĐ:
\(1-sinx>0\Leftrightarrow sinx\ne1\)
\(\Rightarrow x\ne\frac{\pi}{2}+k2\pi\)
b/ ĐKXĐ:
\(\left\{{}\begin{matrix}sinx.cosx\ne0\\tanx+4cotx+2\ne0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}sin2x\ne0\\tan^2x+2tanx+4\ne0\left(luôn-đúng\right)\end{matrix}\right.\)
\(\Rightarrow2x\ne k\pi\Rightarrow x\ne\frac{k\pi}{2}\)
c/
\(\left\{{}\begin{matrix}\frac{1-cosx}{2+cosx}\ge0\\2+cosx\ne0\end{matrix}\right.\) \(\Rightarrow x\in R\)
\(m^2\left(sinx-cosx\right)-m\left(sin^2x-cos^2x\right)=cosx-sinx\)
\(\Leftrightarrow m\left(sinx-cosx\right)\left(sinx+cosx\right)-m^2\left(sinx-cosx\right)-\left(sinx-cosx\right)=0\)
\(\Leftrightarrow\left(sinx-cosx\right)\left[m\left(sinx+cosx\right)-m^2-1\right]=0\)
\(\Leftrightarrow sinx-cosx=0\)
\(\Leftrightarrow sin\left(x-\frac{\pi}{4}\right)=0\)
\(\Leftrightarrow x-\frac{\pi}{4}=k\pi\Rightarrow x=\frac{\pi}{4}+k\pi\)
\(\Rightarrow x=\frac{\pi}{4}\)
1.
a, Phương trình có nghiệm khi:
\(\left(m+2\right)^2+m^2\ge4\)
\(\Leftrightarrow m^2+4m+4+m^2\ge4\)
\(\Leftrightarrow2m^2+4m\ge0\)
\(\Leftrightarrow\left[{}\begin{matrix}m\ge0\\m\le-2\end{matrix}\right.\)
b, Phương trình có nghiệm khi:
\(m^2+\left(m-1\right)^2\ge\left(2m+1\right)^2\)
\(\Leftrightarrow2m^2+6m\le0\)
\(\Leftrightarrow-3\le m\le0\)
2.
a, Phương trình vô nghiệm khi:
\(\left(2m-1\right)^2+\left(m-1\right)^2< \left(m-3\right)^2\)
\(\Leftrightarrow4m^2-4m+1+m^2-2m+1< m^2-6m+9\)
\(\Leftrightarrow4m^2-7< 0\)
\(\Leftrightarrow-\dfrac{\sqrt{7}}{2}< m< \dfrac{\sqrt{7}}{2}\)
b, \(2sinx+cosx=m\left(sinx-2cosx+3\right)\)
\(\Leftrightarrow\left(m-2\right)sinx-\left(2m+1\right)cosx=-3m\)
Phương trình vô nghiệm khi:
\(\left(m-2\right)^2+\left(2m+1\right)^2< 9m^2\)
\(\Leftrightarrow m^2-4m+4+4m^2+4m+1< 9m^2\)
\(\Leftrightarrow m^2-1>0\)
\(\Leftrightarrow\left[{}\begin{matrix}m>1\\m< -1\end{matrix}\right.\)