Question

Find all solutions and solutions in interval[0,2pi)

Answer 1

$\bf sin(2\theta)=2sin(\theta)cos(\theta)\\\\ -------------------------------\\\\ sin(2x)-\sqrt{3}cos(x)=0\implies 2sin(x)cos(x)-cos(x)\sqrt{3}=0 \\\\\\ \stackrel{\stackrel{common}{factor}}{cos(x)}[2sin(x)-\sqrt{3}]=0\\\\ -------------------------------$

$\bf cos(x)=0\implies \measuredangle x= \begin{cases} \frac{\pi }{2}\\\\ \frac{3\pi }{2} \end{cases}\\\\ -------------------------------\\\\ 2sin(x)-\sqrt{3}=0\implies 2sin(x)=\sqrt{3}\implies sin(x)=\cfrac{\sqrt{3}}{2} \\\\\\ \measuredangle x= \begin{cases} \frac{\pi }{3}\\\\ \frac{2\pi }{3} \end{cases}$