ANSWER
[tex]x = \frac{\pi}{2} , \frac{7\pi}{6} , \frac{3\pi}{2} , \frac{11\pi}{6} [/tex]
EXPLANATION
The given trigonometric equation is
[tex] \cos(x) + 2 \cos(x) \sin(x) = 0[/tex]
We factor cos(x) to get:
[tex] \cos(x) (1 + 2 \sin(x) ) = 0[/tex]
Apply the zero product property to obtain:
[tex] \cos(x) = 0 \: or \: 1 + 2 \sin(x) = 0[/tex]
[tex] \cos(x) = 0 \: or \: \sin(x) = - \frac{1}{2} [/tex]
Using the unit circle,
[tex] \cos(x) = 0[/tex]
when
[tex]x = \frac{\pi}{2} [/tex]
and
[tex]x = \frac{3\pi}{2} [/tex]
We know
[tex] \sin(y) = \frac{1}{2} [/tex]
when
[tex]y= \frac{\pi}{6} [/tex]
The sine function is negative in the third and fourth quadrants.
[tex]x = \pi + \frac{\pi}{6} = \frac{7\pi}{6} [/tex]
[tex]x = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6} [/tex]
Hence the solutions are:
[tex]x = \frac{\pi}{2} , \frac{7\pi}{6} , \frac{3\pi}{2} , \frac{11\pi}{6} [/tex]
