\(\begin{array}{l}a)\;sin2x + cos3x = 0\\ \Leftrightarrow cos\left( {\frac{\pi }{2} - 2x} \right) + cos3x = 0\\ \Leftrightarrow cos\left( {\frac{\pi }{2} - 2x} \right) =  - cos3x\\ \Leftrightarrow cos\left( {\frac{\pi }{2} - 2x} \right) = cos\left( {\pi  - 3x} \right)\\ \Leftrightarrow \left[ \begin{array}{l}\frac{\pi }{2} - 2x = \pi  - 3x + k2\pi \\\frac{\pi }{2} - 2x =  - \pi  + 3x + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = \frac{\pi }{2} + k2\pi \\x = \frac{{3\pi }}{{10}} + k\frac{{2\pi }}{5}\end{array} \right.\left( {k \in \mathbb{Z}} \right)\end{array}\)

\(\begin{array}{l}b)\;sinx.cosx = \frac{{\sqrt 2 }}{4}\\ \Leftrightarrow \frac{1}{2}\;sin2x = \frac{{\sqrt 2 }}{4}\\ \Leftrightarrow sin2x = \frac{{\sqrt 2 }}{2} = sin\left( {\frac{\pi }{4}} \right)\\ \Leftrightarrow \left[ \begin{array}{l}2x = \frac{\pi }{4} + k2\pi \\2x = \pi  - \frac{\pi }{4} + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = \frac{\pi }{8} + k\pi \\x = \frac{{3\pi }}{8} + k\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right)\end{array}\)

\(\begin{array}{l}c)\;sinx + sin2x = 0\\ \Leftrightarrow sinx =  - sin2x\\ \Leftrightarrow sinx = sin( - 2x)\\ \Leftrightarrow \left[ \begin{array}{l}x =  - 2x + k2\pi \\x = \pi  + 2x + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = k\frac{{2\pi }}{3}\\x =  - \pi  + k2\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right)\end{array}\)

Hướng dẫn giải:

Chọn D.

 không là nghiệm của phương trình

Chia 2 vế phương trình cho cos²x ta được

a: =>sin2x+2*(1-cos2x)/2=2

=>sin2x-cos2x=1

=>căn 2*sin(2x-pi/4)=1

=>2x-pi/4=pi/4+k2pi hoặc 2x-pi/4=3/4pi+k2pi

=>x=pi/4+kpi hoặc x=pi/2+kpi

b: =>2*(1+cos2x)/2+1/2*sin2x-1/2(1-cos2x)=0

=>1+cos2x+1/2*sin2x-1/2+1/2cos2x=0

=>1/2*sin2x+3/2*cos2x=-1/2

=>sin(2x+a)=-cos(a)=cos(pi-a)

=>sin(2x+a)=sin(-pi/2+a)

=>2x+a=-pi/2+a+k2pi hoặc 2x+a=3/2pi-a+k2pi

=>x=-pi/4+kpi hoặc x=3/4pi-a+kpi

1.

\(\Leftrightarrow2sinx.cosx=0\)

\(\Leftrightarrow sin2x=0\)

\(\Leftrightarrow2x=k\pi\)

\(\Leftrightarrow x=\frac{k\pi}{2}\)

2.

\(\Leftrightarrow\frac{\sqrt{2}}{2}sinx+\frac{\sqrt{2}}{2}cosx=-\frac{\sqrt{2}}{2}\)

\(\Leftrightarrow sin\left(x+\frac{\pi}{4}\right)=-\frac{\sqrt{2}}{2}\)

\(\Leftrightarrow\left[{}\begin{matrix}x+\frac{\pi}{4}=-\frac{\pi}{4}+k2\pi\\x+\frac{\pi}{4}=\frac{5\pi}{4}+k2\pi\end{matrix}\right.\)

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

3.

\(\Leftrightarrow2.\left(2sinx.cosx\right)cos2x=0\)

\(\Leftrightarrow2sin2x.cos2x=0\)

\(\Leftrightarrow sin4x=0\)

\(\Leftrightarrow4x=k\pi\)

\(\Leftrightarrow x=\frac{k\pi}{4}\)

4.

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

\(\Leftrightarrow\left[{}\begin{matrix}sinx=1\\sinx=3>1\left(l\right)\end{matrix}\right.\)

\(\Leftrightarrow x=\frac{\pi}{2}+k2\pi\)

5.

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

\(\Leftrightarrow sin2x.sin\frac{\pi}{4}+cos2x.cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\)

\(\Leftrightarrow sin\left(2x+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}\)

\(\Leftrightarrow\left[{}\begin{matrix}2x+\frac{\pi}{4}=\frac{\pi}{4}+k2\pi\\2x+\frac{\pi}{4}=\frac{3\pi}{4}+k2\pi\end{matrix}\right.\)

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

6.

\(\Leftrightarrow2sin2x=-1\)

\(\Leftrightarrow sin2x=-\frac{1}{2}\)

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

\(\Leftrightarrow\left[{}\begin{matrix}x=-\frac{\pi}{12}+k\pi\\x=\frac{7\pi}{12}+k\pi\end{matrix}\right.\)

Câu 1:

\(cos^2\) gì nhỉ?

Câu 2: đề không hợp lý \(\sqrt{3}sin9x\) là \(\sqrt{3}cos9x\) có lý hơn

\(\Leftrightarrow3sin3x-4sin^33x+\sqrt{3}sin9x=1\)

\(\Leftrightarrow sin9x+\sqrt{3}sin9x=1\)

\(\Leftrightarrow sin9x=\frac{1}{\sqrt{3}+1}\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{1}{9}arcsin\left(\frac{1}{\sqrt{3}+1}\right)+\frac{k2\pi}{9}\\x=\frac{\pi}{9}-\frac{1}{9}arcsin\left(\frac{1}{\sqrt{3}+1}\right)+\frac{k2\pi}{9}\end{matrix}\right.\)

Nghiệm nhìn rất ngớ ngẩn nếu đề đúng

3.

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

\(\Leftrightarrow sin\left(2x-\frac{\pi}{6}\right)=\frac{1}{2}\)

\(\Leftrightarrow\left[{}\begin{matrix}2x-\frac{\pi}{6}=\frac{\pi}{6}+k2\pi\\2x-\frac{\pi}{6}=\frac{5\pi}{6}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow...\)