49.
\(\Leftrightarrow m.sin2x+2\left(cos2x+1\right)=m+5\)
\(\Leftrightarrow m.sin2x+2cos2x=m+3\)
Pt có nghiệm khi:
\(m^2+2^2\ge\left(m+3\right)^2\)
\(\Leftrightarrow6m\le-5\Rightarrow m\le-\dfrac{5}{6}\)
\(\Rightarrow m=\left\{-3;-2;-1\right\}\)
50.
\(\Leftrightarrow m.2sin^2x+4sinx.cosx+3m.2cos^2x=2\)
\(\Leftrightarrow m\left(1-cos2x\right)+2sin2x+3m\left(1+cos2x\right)=2\)
\(\Leftrightarrow m.cos2x+sin2x=1-2m\)
Pt có nghiệm khi:
\(m^2+1\ge\left(1-2m\right)^2\Leftrightarrow3m^2-4m\le0\)
\(\Rightarrow m\in\left[0;\dfrac{4}{3}\right]\)
51.
ĐKXĐ: \(x\ne k\pi\)
\(\dfrac{5-4cosx}{sinx}=\dfrac{6tana}{1+tan^2a}\)
\(\Leftrightarrow\dfrac{5-4cosx}{sinx}=\dfrac{6sina}{cosa}.cos^2a=3sin2a\)
\(\Leftrightarrow5-4cosx=3sin2a.sinx\)
\(\Leftrightarrow3sin2a.sinx+4cosx=5\)
Pt có nghiệm khi:
\(\left(3sin2a\right)^2+4^2\ge5^2\)
\(\Leftrightarrow sin^22a\ge1\)
\(\Leftrightarrow sin^22a=1\Leftrightarrow cos2a=0\)
\(\Leftrightarrow2a=\dfrac{\pi}{2}+k\pi\Rightarrow a=\dfrac{\pi}{4}+\dfrac{k\pi}{2}\)
\(\Rightarrow a=\left\{\dfrac{\pi}{4};\dfrac{3\pi}{4};\dfrac{5\pi}{4};\dfrac{7\pi}{4}\right\}\)
Em tự cộng và chọn kết quả nhé
52.
\(\Leftrightarrow2sin\left(2x+\dfrac{\pi}{6}\right)+2sin\left(\dfrac{\pi}{2}\right)=m^2+\sqrt{3}sin2x-cos2x\)
\(\Leftrightarrow2sin2x.cos\left(\dfrac{\pi}{6}\right)+2cos2x.sin\left(\dfrac{\pi}{6}\right)+2=m^2+\sqrt{3}sin2x-cos2x\)
\(\Leftrightarrow\sqrt{3}sin2x+cos2x+2=m^2+\sqrt{3}sin2x-cos2x\)
\(\Leftrightarrow2cos2x=m^2-2\)
\(\Leftrightarrow cos2x=\dfrac{m^2-2}{2}\)
Do \(-1\le cos2x\le1\) nên pt có nghiệm khi:
\(-1\le\dfrac{m^2-2}{2}\le1\Leftrightarrow0\le m^2\le4\)
\(\Leftrightarrow m\in\left[-2;2\right]\)
\(\Rightarrow a+b=0\)
