Question

1) Nghiệm của phương trình \(\sin x-\sqrt{3}\cos x=0\)
2) Phương trình sin x + cos x = \(\sqrt{2}\sin5x\) có nghiệm ?
3) Phương trình \(\sqrt{3}.\sin3x+\cos3x=-1\) là?

Answer 1

1: \(\Leftrightarrow sinx\cdot\dfrac{1}{2}-\dfrac{\sqrt{3}}{2}\cdot cosx=0\)

=>\(sin\left(x-\dfrac{pi}{3}\right)=0\)

=>x-pi/3=kpi

=>x=kpi+pi/3

3: =>sin 3x*căn 3/2+1/2*cos3x=-1/2

=>sin(3x+pi/6)=-1/2

=>3x+pi/6=-pi/6+k2pi hoặc 3x+pi/6=7/6pi+k2pi

=>3x=-1/3pi+k2pi hoặc 3x=pi+k2pi

=>x=-1/9pi+k2pi/3 hoặc x=pi/3+k2pi/3

Answer 2

2.

\(sinx+cosx=\sqrt{2}sin5x\)

\(\Leftrightarrow\sqrt{2}sin\left(x+\dfrac{\pi}{4}\right)=\sqrt{2}sin5x\)

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

\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{16}+\dfrac{k\pi}{2}\\x=-\dfrac{\pi}{4}+k2\pi\end{matrix}\right.\)