21.
a) `2sin(x-30^@)-1=0`
`<=>sin(x-30^@)=1/2`
`<=> sin(x-30^@)=sin30^@`
`<=>[(x-30^@=30^@+k360^@),(x-30^@=180^@-30^@+k360^@):}`
`<=> [(x=60^@+k360^@),(x=180^@+k360^@):}`
b) `5sin^2x+3cosx+3=0`
`<=>5(1-cos^2x)+3cosx+3=0`
`<=>-5cos^2x+3cosx+8=0`
`<=>(cosx+1)(cosx=8/5)=0`
`<=>[(cosx=-1),(cosx=8/5\ (VN)):}`
`<=>x=180^@+k360^@`
22.
`-1<=sin2x<=1`
`<=>2<=3+sin2x<=4`
`=> y_(min)=2 ; y_(max)=4`

\(y=2\left(\dfrac{1}{2}sin2x+\dfrac{\sqrt{3}}{2}cos2x\right)=2sin\left(2x+\dfrac{\pi}{3}\right)\)

\(-1\le sin\left(2x+\dfrac{\pi}{3}\right)\le1\Rightarrow-2\le y\le2\)

\(y_{min}=-2\) khi \(sin\left(2x+\dfrac{\pi}{3}\right)=-1\Rightarrow x=-\dfrac{5\pi}{12}+k\pi\)

\(y_{max}=2\) khi \(sin\left(2x+\dfrac{\pi}{3}\right)=1\Rightarrow x=\dfrac{\pi}{12}+k\pi\)

\(y=sin2x+\sqrt{3}\left(\frac{1+cos2x}{2}\right)+1\)

\(=sin2x+\frac{\sqrt{3}}{2}cos2x+1+\frac{\sqrt{3}}{2}\)

\(=\frac{\sqrt{7}}{2}\left(sin2x.\frac{2\sqrt{7}}{7}+\frac{\sqrt{21}}{7}cos2x\right)+1+\frac{\sqrt{3}}{2}\)

\(=\frac{\sqrt{7}}{2}.sin\left(2x+a\right)+1+\frac{\sqrt{3}}{2}\)

(Với \(cosa=\frac{2\sqrt{7}}{7};sina=\frac{\sqrt{21}}{7}\))

\(\Rightarrow-\frac{\sqrt{7}}{2}+1+\frac{\sqrt{3}}{2}\le y\le\frac{\sqrt{7}}{2}+1+\frac{\sqrt{3}}{2}\)

\(y=1-cos^2x-6cosx+1=-cos^2x-6cosx+2\)

\(y=-cos^2x-6cosx-5+7\)

\(y=7-\left(cosx+1\right)\left(cosx+5\right)\)

Do \(cosx\ge-1\Rightarrow\left\{{}\begin{matrix}cosx+1\ge0\\cosx+5>0\end{matrix}\right.\)

\(\Rightarrow\left(cosx+1\right)\left(cosx+5\right)\ge0\)

\(\Rightarrow7-\left(cosx+1\right)\left(cosx+5\right)\le7-0=7\)

\(\Rightarrow y_{max}=7\) khi \(cosx=-1\Leftrightarrow x=\pi+k2\pi\)

\(y=2sin^2x-sin2x+7=1-cos2x-sin2x+7\)

\(y=8-\left(sin2x+cos2x\right)=8-\sqrt{2}sin\left(2x+\frac{\pi}{4}\right)\)

Do \(-1\le sin\left(2x+\frac{\pi}{4}\right)\le1\)

\(\Rightarrow8+\sqrt{2}\le y\le8-\sqrt{2}\)

\(y_{min}=8-\sqrt{2}\) khi \(sin\left(2x+\frac{\pi}{4}\right)=1\)

\(y_{max}=8+\sqrt{2}\) khi \(sin\left(2x+\frac{\pi}{4}\right)=-1\)

a) Ta có:

\(y=2\left(\dfrac{1}{2}cosx-\dfrac{\sqrt{3}}{2}sinx\right)=2sin\left(\dfrac{\pi}{6}-x\right)\)

\(\Rightarrow-2\le y\le2\) (Do \(-1\le sin\alpha\le1\))

Vậy min y = -2 , max y = 2