1.

\(3sin^22x-2sin2x.cos2x-4cos^22x=2\)

\(\Leftrightarrow-\dfrac{3}{2}\left(1-2sin^22x\right)-2sin2x.cos2x-2\left(2cos^22x-1\right)=\dfrac{5}{2}\)

\(\Leftrightarrow sin4x+\dfrac{7}{2}cos4x=-\dfrac{5}{2}\)

\(\Leftrightarrow\dfrac{\sqrt{53}}{2}\left(\dfrac{2}{\sqrt{53}}sin4x+\dfrac{7}{\sqrt{53}}cos4x\right)=-\dfrac{5}{2}\)

\(\Leftrightarrow sin\left(4x+arccos\dfrac{2}{\sqrt{53}}\right)=-\dfrac{5}{\sqrt{53}}\)

\(\Leftrightarrow\left[{}\begin{matrix}4x+arccos\dfrac{2}{\sqrt{53}}=arcsin\left(-\dfrac{5}{\sqrt{53}}\right)+k2\pi\\4x+arccos\dfrac{2}{\sqrt{53}}=\pi-arcsin\left(-\dfrac{5}{\sqrt{53}}\right)+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{1}{4}arccos\dfrac{2}{\sqrt{53}}+\dfrac{1}{4}arcsin\left(-\dfrac{5}{\sqrt{53}}\right)+\dfrac{k\pi}{2}\\x=\dfrac{\pi}{4}-\dfrac{1}{4}arccos\dfrac{2}{\sqrt{53}}-\dfrac{1}{4}arcsin\left(-\dfrac{5}{\sqrt{53}}\right)+\dfrac{k\pi}{2}\end{matrix}\right.\)

2.

\(2\sqrt{3}cos^2x+6sinx.cosx=3+\sqrt{3}\)

\(\Leftrightarrow\sqrt{3}\left(2cos^2x-1\right)+6sinx.cosx=3\)

\(\Leftrightarrow\sqrt{3}cos2x+3sin2x=3\)

\(\Leftrightarrow2\sqrt{3}\left(\dfrac{1}{2}cos2x+\dfrac{\sqrt{3}}{2}sin2x\right)=3\)

\(\Leftrightarrow cos\left(2x-\dfrac{\pi}{3}\right)=\dfrac{\sqrt{3}}{2}\)

\(\Leftrightarrow\left[{}\begin{matrix}2x-\dfrac{\pi}{3}=\dfrac{\pi}{6}+k2\pi\\2x-\dfrac{\pi}{3}=-\dfrac{\pi}{6}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{4}+k\pi\\x=\dfrac{\pi}{12}+k\pi\end{matrix}\right.\)

3.

\(3cos^24x+5sin^24x=2-2\sqrt{3}sin4x.cos4x\)

\(\Leftrightarrow4cos^24x+4sin^24x-cos^24x+sin^24x=2-2\sqrt{3}sin4x.cos4x\)

\(\Leftrightarrow4-cos8x=2-\sqrt{3}sin8x\)

\(\Leftrightarrow cos8x-\sqrt{3}sin8x=2\)

\(\Leftrightarrow\dfrac{1}{2}cos8x-\dfrac{\sqrt{3}}{2}sin8x=1\)

\(\Leftrightarrow cos\left(8x+\dfrac{\pi}{3}\right)=1\)

\(\Leftrightarrow8x+\dfrac{\pi}{3}=k2\pi\)

\(\Leftrightarrow x=-\dfrac{\pi}{24}+\dfrac{k\pi}{4}\)

ĐKXĐ: \(cos2x\ne\dfrac{1}{2}\Leftrightarrow x\ne\pm\dfrac{\pi}{6}+k\pi\)

\(\sqrt{3}sin^2x-2sinx.cosx-\sqrt{3}cos^2x=0\)

\(\Leftrightarrow-sin2x-\sqrt{3}\left(cos^2x-sin^2x\right)=0\)

\(\Leftrightarrow sin2x+\sqrt{3}cos2x=0\)

\(\Leftrightarrow\dfrac{1}{2}sin2x+\dfrac{\sqrt{3}}{2}cos2x=0\)

\(\Leftrightarrow sin\left(2x+\dfrac{\pi}{3}\right)=0\)

\(\Leftrightarrow2x+\dfrac{\pi}{3}=k\pi\)

\(\Leftrightarrow x=-\dfrac{\pi}{6}+\dfrac{k\pi}{2}\)

Nghiệm này bao gồm 2 họ nghiệm: \(\left[{}\begin{matrix}x=-\dfrac{\pi}{6}+k\pi\\x=\dfrac{\pi}{3}+k\pi\end{matrix}\right.\)

Do đó sau khi loại nghiệm theo ĐKXĐ ta được nghiệm của pt là: \(x=\dfrac{\pi}{3}+k\pi\)

Đáp án B

Điều kiện  x ≠ π 4 + k π 2 , k ∈ ℤ   .

1 = cos x cos x + 2 sin x + 3 sin x sin x + 2 sin 2 x

⇔ sin 2 x = cos 2 x + sin 2 x + 3 sin 2 x + 3 2 sin x

⇔ cos 2 x + 3 sin 2 x + 3 2 sin x = 0

⇔ 1 − sin 2 x + 3 sin 2 x + 3 2 sin x = 0

⇔ 2 sin 2 x + 3 2 sin x + 1 = 0

⇒ sin x = 10 − 3 2 4 do  − 1 ≤ sin x ≤ 1

Vậy có hai điểm biểu diễn nghiệm của phương trình đã cho trên đường tròn lượng giác.

a/ ĐKXĐ: \(sinx\ne-1\)

\(\Leftrightarrow\left(2sinx+1\right)\left(3cos4x+2sinx\right)+4cos^2x+1=8+8sinx\)

\(\Leftrightarrow6sinx.cos4x+4sin^2x+3cos4x+2sinx+4cos^2x+1=8+8sinx\)

\(\Leftrightarrow6sinx.cos4x+3cos4x-6sinx-3=0\)

\(\Leftrightarrow6sinx\left(cos4x-1\right)+3\left(cos4x-1\right)=0\)

\(\Leftrightarrow\left(6sinx+3\right)\left(cos4x-1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=-\frac{1}{2}\\cos4x-1=0\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}sinx=-\frac{1}{2}\\1-2sin^22x-1=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=-\frac{1}{2}\\sin^2x\left(1-sin^2x\right)=0\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}sinx=-\frac{1}{2}\\sin^2x\left(1+sinx\right)\left(1-sinx\right)=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=-\frac{1}{2}\\sinx=0\\sinx=1\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x=-\frac{\pi}{6}+k2\pi\\x=\frac{7\pi}{6}+k2\pi\\x=k\pi\\x=\frac{\pi}{2}+k2\pi\end{matrix}\right.\)

b/ ĐKXĐ: \(\left\{{}\begin{matrix}tanx\ne-1\\cosx\ne0\end{matrix}\right.\)

\(\Leftrightarrow\left(1+sinx+cos2x\right).\sqrt{2}sin\left(x+\frac{\pi}{4}\right)=cosx\left(1+\frac{sinx}{cosx}\right)\)

\(\Leftrightarrow\left(1+sinx+cos2x\right)\left(sinx+cosx\right)=cosx+sinx\)

\(\Leftrightarrow\left(cosx+sinx\right)\left(sinx+cos2x\right)=0\)

\(\Leftrightarrow sinx+cos2x=0\)

\(\Leftrightarrow-2sin^2x+sinx+1=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=1\left(l\right)\\sinx=-\frac{1}{2}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-\frac{\pi}{6}+k2\pi\\x=\frac{7\pi}{6}+k2\pi\end{matrix}\right.\)

ĐKXĐ: \(x\ne\dfrac{k\pi}{2}\)

\(\dfrac{\sqrt{3}}{cos^2x}+2+\dfrac{2}{sinx.cosx}-2\sqrt{3}=2\left(\dfrac{1}{tanx}+1\right)\)

\(\Leftrightarrow\sqrt{3}\left(1+tan^2x\right)+\dfrac{\dfrac{2}{cos^2x}}{\dfrac{sinx.cosx}{cos^2x}}+2-2\sqrt{3}=2\left(\dfrac{1}{tanx}+1\right)\)

\(\Leftrightarrow\sqrt{3}tan^2x+\dfrac{2\left(1+tan^2x\right)}{tanx}+2-\sqrt{3}=\dfrac{2}{tanx}+2\)

\(\Leftrightarrow\sqrt{3}tan^3x+2\left(1+tan^2x\right)-\sqrt{3}tanx=2\)

\(\Leftrightarrow\sqrt{3}tan^3x+2tan^2x-\sqrt{3}tanx=0\)

\(\Leftrightarrow...\)

a.

\(y=2\left(1-cos2x\right)-\dfrac{5}{2}sin2x+\dfrac{1}{2}+\dfrac{1}{2}cos2x+10\)

\(=-\dfrac{1}{2}\left(5sin2x+3cos2x\right)+\dfrac{25}{2}\)

\(=-\dfrac{\sqrt{34}}{2}\left(\dfrac{5}{\sqrt{34}}sin2x+\dfrac{3}{\sqrt{34}}cos2x\right)+\dfrac{25}{2}\)

Đặt \(\dfrac{5}{\sqrt{34}}=cosa\)

\(\Rightarrow y=-\dfrac{\sqrt{34}}{2}\left(sin2x.cosa+cos2x.sina\right)+\dfrac{25}{2}\)

\(=-\dfrac{\sqrt{34}}{2}sin\left(2x+a\right)+\dfrac{25}{2}\)

Do \(-1\le sin\left(2x+a\right)\le1\)

\(\Rightarrow\dfrac{25-\sqrt{34}}{2}\le y\le\dfrac{25+\sqrt{34}}{2}\)

b.

\(y=\dfrac{sin^2x-2sin2x+1}{3+sin^2x+2cos^2x}=\dfrac{2sin^2x-4sin2x+2}{6+2\left(sin^2x+cos^2x\right)+2cos^2x}\)

\(=\dfrac{1-cos2x-4sin2x+2}{8+1+cos2x}=\dfrac{3-4sin2x-cos2x}{9+cos2x}\)

\(\Rightarrow9y+y.cos2x=3-4sin2x-cos2x\)

\(\Rightarrow4sin2x+\left(y+1\right)cos2x=3-9y\)

Theo điều kiện có nghiệm của pt lượng giác bậc nhất:

\(4^2+\left(y+1\right)^2\ge\left(3-9y\right)^2\)

\(\Leftrightarrow80y^2-56y-8\le0\)

\(\Rightarrow\dfrac{7-\sqrt{89}}{20}\le y\le\dfrac{7+\sqrt{89}}{20}\)

c.

\(y=2sinx+3cosx+4\)

\(=\sqrt{13}\left(\dfrac{2}{\sqrt{13}}sinx+\dfrac{3}{\sqrt{13}}cosx\right)+4\)

Đặt \(\dfrac{2}{\sqrt{13}}=cosa\)

\(\Rightarrow y=\sqrt{13}\left(sinx.cosa+cosx.sina\right)+4\)

\(=\sqrt{13}sin\left(x+a\right)+4\)

Do \(-1\le sin\left(x+a\right)\le1\)

\(\Rightarrow-\sqrt{13}+4\le y\le\sqrt{13}+4\)