2.Biểu thức luôn xác định

\(y=\dfrac{4}{\sqrt{5-2cos^2sin^2x}}=\dfrac{4}{\sqrt{5-\dfrac{1}{2}sin^22x}}\)

Có: \(1\ge sin^22x\ge0\)

\(\Leftrightarrow-\dfrac{1}{2}\le-\dfrac{1}{2}sin^22x\le0\)

\(\Leftrightarrow\dfrac{3\sqrt{2}}{2}\le\sqrt{5-\dfrac{1}{2}sin^22x}\le\sqrt{5}\)

\(\Rightarrow\dfrac{4\sqrt{2}}{3}\ge y\ge\dfrac{4\sqrt{5}}{5}\)

miny=\(\dfrac{4\sqrt{5}}{5}\) \(\Leftrightarrow sin2x=0\)\(\Leftrightarrow x=\dfrac{k\pi}{2}\left(k\in Z\right)\)

maxy=\(\dfrac{4\sqrt{2}}{3}\Leftrightarrow\left[{}\begin{matrix}sin2x=1\\sin2x=-1\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{4}+k\pi\\x=\dfrac{-\pi}{4}+k\pi\end{matrix}\right.\)\(\left(k\in Z\right)\)

1.Biểu thức luôn xác định

Xét \(sin2x=0\) \(\Leftrightarrow x=\dfrac{k\pi}{2}\left(k\in Z\right)\) khi đó \(y=-6\)

Xét \(sin2x\ne0\) 

=> \(1\ge sin^52x\ge-1\)

\(\Leftrightarrow4-1\le4-sin^52x\le4+1\)

\(\Leftrightarrow\sqrt{3}\le\sqrt{4-sin^52x}\le\sqrt{5}\)

\(\Leftrightarrow\sqrt{3}-8\le y\le\sqrt{5}-8\)

\(y=\sqrt{3}-8< -6\) , \(y=\sqrt{5}-8>-6\)

=>min= \(\sqrt{3}-8\) \(\Leftrightarrow sin2x=1\left(tm\right)\) \(\Leftrightarrow x=\dfrac{\pi}{4}+k\pi\left(k\in Z\right)\)

maxy=\(\sqrt{5}-8\)\(\Leftrightarrow sin2x=-1\left(tm\right)\) \(\Leftrightarrow x=-\dfrac{\pi}{4}+k\pi\left(k\in Z\right)\)

(câu này e ko chắc)

a: -1<=sin x<=1

=>-1+3<=sin x+3<=1+3

=>2<=sinx+3<=4

=>\(\dfrac{1}{2}>=\dfrac{1}{sinx+3}>=\dfrac{1}{4}\)

=>\(2>=\dfrac{4}{sinx+3}>=1\)

=>\(-2< =-\dfrac{4}{sinx+3}< =-1\)

=>-2+3<=y<=-1+3

=>1<=y<=2

y=1 khi \(\dfrac{-4}{sinx+3}+3=1\)

=>\(\dfrac{-4}{sinx+3}=-2\)

=>sinx+3=2

=>sin x=-1

=>x=-pi/2+k2pi

y=3 khi sin x=1

=>x=pi/2+k2pi

b: -1<=cosx<=1

=>4>=-4cosx>=-4

=>9>=-4cosx+5>=1

=>2/9<=2/5-4cosx<=2

=>2/9<=y<=2

\(y_{min}=\dfrac{2}{9}\) khi \(\dfrac{2}{5-4cosx}=\dfrac{2}{9}\)

=>\(5-4\cdot cosx=9\)

=>4*cosx=4

=>cosx=1

=>x=k2pi

y max khi cosx=-1

=>x=pi+k2pi

c: \(0< =cos^2x< =1\)

=>\(0< =2\cdot cos^2x< =2\)

=>\(-1< =y< =2\)

y min=-1 khi cos^2x=0

=>x=pi/2+kpi

y max=2 khi cos^2x=1

=>sin^2x=0

=>x=kpi

a) làm tương tự 2 bài mk đã giải nha.

b) \(y=2\cos^2x-2\sqrt{3}\sin x\cos x+1\)

\(=1-\left(\cos2x+\sqrt{3}\sin2x\right)\)

Lại có \(-2\le\cos2x+\sqrt{3}\sin2x\le2\) \(\Rightarrow-1\le y\le3\)

c) Vì \(\left\{{}\begin{matrix}0\le\sqrt[4]{\sin x}\le1\\0\le\sqrt{\cos x}\le1\end{matrix}\right.\)

Do đó \(-1\le y\le1\)

1: ĐKXĐ: 3-cosx>0

=>cosx<3(luôn đúng)

2: ĐKXĐ: 1-sin 3x>=0

=>sin 3x<=1(luôn đúng)

3: ĐKXĐ: sin x<>0 và 2x<>pi/2+kpi

=>x<>kpi và x<>pi/4+kpi/2

4: ĐKXĐ: 2x-1>=0

=>x>=1/2

a.

\(-1\le sinx\le1\Rightarrow-7\le y\le-3\)

\(y_{min}=-7\) khi \(sinx=-1\)

\(y_{max}=-3\) khi \(sinx=1\)

b.

\(-1\le cos\left(x+\frac{\pi}{3}\right)\le1\Rightarrow1\le y\le5\)

\(y_{min}=1\) khi \(cos\left(x+\frac{\pi}{3}\right)=-1\)

\(y_{max}=5\) khi \(cos\left(x+\frac{\pi}{3}\right)=1\)

c.

\(0\le1-cosx\le2\Rightarrow-5\le y\le3\sqrt{2}-5\)

\(y_{min}=-5\) khi \(cosx=1\)

\(y_{max}=3\sqrt{2}-5\) khi \(cosx=-1\)

d.

ĐKXĐ: \(0\le sinx\Rightarrow0\le sinx\le1\Rightarrow1\le y\le3\)

\(y_{min}=1\) khi \(sinx=0\)

\(y_{max}=3\) khi \(sinx=1\)

a.

\(0\le cos^2x\le1\Rightarrow2\le y\le1+\sqrt{3}\)

\(y_{min}=2\) khi \(cosx=0\)

\(y_{max}=1+\sqrt{3}\) khi \(cos^2x=1\)

b.

\(-1\le sin\left(2x-\frac{\pi}{4}\right)\le1\Rightarrow-2\le y\le4\)

\(y_{min}=-2\) khi \(sin\left(2x-\frac{\pi}{4}\right)=-1\)

\(y_{max}=4\) khi \(sin\left(2x-\frac{\pi}{4}\right)=1\)

c.

\(0\le cos^23x\le1\Rightarrow1\le y\le3\)

\(y_{min}=1\) khi \(cos^23x=1\)

\(y_{max}=3\) khi \(cos3x=0\)

d.

\(-1\le sin2x\le1\Rightarrow2\le y\le1+\sqrt{3}\)

\(y_{min}=2\) khi \(sin2x=-1\)

\(y_{max}=1+\sqrt{3}\) khi \(sin2x=1\)

e.

\(0\le sin^2x\le1\Rightarrow\frac{4}{3}\le y\le2\)

\(y_{min}=\frac{4}{3}\) khi \(sin^2x=1\)

\(y_{max}=2\) khi \(sinx=0\)

1. \(sin\left(\dfrac{\pi}{3}-x\right)\ne0\Leftrightarrow\dfrac{\pi}{3}-x\ne k\pi\Leftrightarrow x\ne\dfrac{\pi}{3}-k\pi\)

2. \(cos2x\ne0\Leftrightarrow2x\ne\dfrac{\pi}{2}+k\pi\Leftrightarrow x\ne\dfrac{\pi}{4}+\dfrac{k\pi}{2}\)

3. \(\sqrt{1+sinx}-\sqrt{2}\ge0\Leftrightarrow1+sinx\ge2\Leftrightarrow sinx\ge1\Leftrightarrow sinx=1\Leftrightarrow x=\dfrac{\pi}{2}+k2\pi\)

4. \(\sqrt{2-2cosx}-2\ne0\Leftrightarrow2-2cosx\ne4\Leftrightarrow cosx\ne-1\Leftrightarrow x\ne\pi+k2\pi\)

5. \(1-\sqrt{1+sin3x}\ne0\Leftrightarrow sin3x\ne0\Leftrightarrow3x\ne k\pi\Leftrightarrow x\ne\dfrac{k\pi}{3}\)