l: \(\Leftrightarrow2\cdot\left(cosx\cdot cos\left(\dfrac{pi}{3}\right)+sinx\cdot sin\left(\dfrac{pi}{3}\right)\right)+cosx+4sinx=2\)
\(\Leftrightarrow cosx+2\sqrt{3}\cdot sinx+cosx+4\cdot sinx=2\)
\(\Leftrightarrow sinx\left(2\sqrt{3}+4\right)+2cosx=2\)
=>\(\left(\sqrt{3}+2\right)\cdot sinx+cosx=1\)
\(\Leftrightarrow\dfrac{\sqrt{3}+2}{\sqrt{8+4\sqrt{3}}}\cdot sinx+\dfrac{1}{\sqrt{8+4\sqrt{3}}}\cdot cosx=\dfrac{1}{\sqrt{8+4\sqrt{3}}}\)
\(\Leftrightarrow sinx\cdot cosa+cosx\cdot sina=sina\)
=>sin(a+x)=sina
=>a+x=a+k2pi hoặc a+x=pi-a+k2pi
=>x=k2pi hoặc x=pi-2a+k2pi
m: \(\Leftrightarrow2\cdot2sin^2\left(x+\dfrac{pi}{6}\right)+sin2x=1\)
\(\Leftrightarrow2\left[1-cos\left(2x+\dfrac{pi}{3}\right)\right]+sin2x=1\)
=>\(2-2\left(cos2x\cdot cos\dfrac{pi}{3}-sin2x\cdot sin\left(\dfrac{pi}{3}\right)\right)+sin2x=1\)
\(\Leftrightarrow2-cos2x+\sqrt{3}\cdot sin2x+sin2x=1\)
=>\(sin2x\left(\sqrt{3}+1\right)-cos2x=-1\)
\(\Leftrightarrow\dfrac{sin2x\cdot\left(\sqrt{3}+1\right)}{\sqrt{5+2\sqrt{3}}}-\dfrac{cos2x\cdot1}{\sqrt{5+2\sqrt{3}}}=\dfrac{-1}{\sqrt{5+2\sqrt{3}}}\)
=>\(sin2x\cdot cosa-cos2x\cdot sina=-sina\)
=>sin(2x-a)=-sina=sin(-a)
=>2x-a=-a+k2pi hoặc 2x-a=pi+a+k2pi
=>x=kpi hoặc x=pi/2+a+kpi
