Question

Help question attached multiple choice

Answer 1

ANSWER

$x = \frac{\pi}{2} , \frac{7\pi}{6} , \frac{3\pi}{2} , \frac{11\pi}{6}$

EXPLANATION

The given trigonometric equation is

$\cos(x) + 2 \cos(x) \sin(x) = 0$

We factor cos(x) to get:

$\cos(x) (1 + 2 \sin(x) ) = 0$

Apply the zero product property to obtain:

$\cos(x) = 0 \: or \: 1 + 2 \sin(x) = 0$

$\cos(x) = 0 \: or \: \sin(x) = - \frac{1}{2}$

Using the unit circle,

$\cos(x) = 0$

when

$x = \frac{\pi}{2}$

and

$x = \frac{3\pi}{2}$

We know

$\sin(y) = \frac{1}{2}$

when

$y= \frac{\pi}{6}$

The sine function is negative in the third and fourth quadrants.

$x = \pi + \frac{\pi}{6} = \frac{7\pi}{6}$

$x = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}$

Hence the solutions are:

$x = \frac{\pi}{2} , \frac{7\pi}{6} , \frac{3\pi}{2} , \frac{11\pi}{6}$