[tex]\bf 2sin(\theta )cos(\theta )+\sqrt{3}cos(\theta )=0\implies \stackrel{\textit{common factor}}{cos(\theta )[2sin(\theta )+\sqrt{3}]=0} \\\\[-0.35em] ~\dotfill\\\\ cos(\theta )=0\implies \theta =cos^{-1}(0)\implies \theta = \begin{cases} \frac{\pi }{2}\\\\ \frac{3\pi }{2} \end{cases} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\bf 2sin(\theta )+\sqrt{3}=\implies 2sin(\theta )=-\sqrt{3}\implies sin(\theta )=-\cfrac{\sqrt{3}}{2} \\\\\\ \theta =sin^{-1}\left( -\cfrac{\sqrt{3}}{2} \right)\implies \theta= \begin{cases} \frac{4\pi }{3}\\\\ \frac{5\pi }{3} \end{cases}[/tex]
Step-by-step answer:
Given equation:
2sin(theta)cos(theta) + sqrt(3)*cos(theta) = 0 ...........................(1)
Solve for theta for 0<=theta<=2pi.
Factor out cos(theta), we get
cos(theta) * ( 2sin(theta) + sqrt(3) ) = 0
By the zero product theorem, we can conclude
cos(theta) = 0 ...................................(2)
OR
2sin(theta) + sqrt(3) = 0 ................. (3)
Solving (2)
cos(theta) = 0 has solutions pi/2 or 3pi/2 from the cosing curve.
Solving (3)
2sin(theta) + sqrt(3) = 0 =>
sin(theta) = -sqrt(3)/2
which has solutions 4pi/3 or 5pi/3.
So the solutions to equation (1) are
S={pi/2, 4pi/3, 3pi/2, 5pi/3}
