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\(y=\sqrt{5}\left(\frac{2}{\sqrt{5}}sinx+\frac{1}{\sqrt{5}}cosx\right)=\sqrt{5}.sin\left(x+a\right)\)
Do \(-1\le sina\le1\)
\(\Rightarrow-\sqrt{5}\le y\le\sqrt{5}\)
a/ \(y=sin2x+\left(\sqrt{3}+1\right)cos2x+sin^2x-cos^2x-1\)
\(=sin2x+\sqrt{3}cos2x-1=2sin\left(2x+\frac{\pi}{3}\right)-1\)
Do \(-1\le sin\left(2x+\frac{\pi}{3}\right)\le1\Rightarrow-3\le y\le1\)
b/ \(y=2sin^2x-2cos^2x-3sinx.cosx-1\)
\(=-2cos2x-\frac{3}{2}sin2x-1=-\frac{5}{2}\left(\frac{3}{5}sinx+\frac{4}{5}cosx\right)-1\)
\(=-\frac{5}{2}sin\left(x+a\right)-1\Rightarrow-\frac{7}{2}\le y\le\frac{3}{2}\)
c/ \(y=1-sin2x+2cos2x+\frac{3}{2}sin2x=\frac{1}{2}sin2x+2cos2x+1\)
\(=\frac{\sqrt{17}}{2}\left(\frac{1}{\sqrt{17}}sin2x+\frac{4}{\sqrt{17}}cos2x\right)+1=\frac{\sqrt{17}}{2}sin\left(2x+a\right)+1\)
\(\Rightarrow-\frac{\sqrt{17}}{2}+1\le y\le\frac{\sqrt{17}}{2}+1\)
\(y=5\left(\frac{3}{5}sin2x-\frac{4}{5}cos2x\right)=5.sin\left(2x-a\right)\)
Mà \(-1\le sin\left(2x-a\right)\le1\)
\(\Rightarrow-5\le y\le5\)
\(y=2\left(\frac{1}{2}sin2x+\frac{\sqrt{3}}{2}cos2x\right)+1=2sin\left(2x+\frac{\pi}{3}\right)+1\)
Do \(-1\le sin\left(2x+\frac{\pi}{3}\right)\le1\)
\(\Rightarrow-1\le y\le3\)