Question

Help question attached multiple choice

Answer 1

`x = (\pi)/(2) , (7\pi)/(6) , (3\pi)/(2) , (11\pi)/(6)`

Step-by-step 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) = - (1)/(2)`

Using the unit circle,

`\cos(x) = 0`

when

`x = (\pi)/(2)`

and

`x = (3\pi)/(2)`

We know

`\sin(y) = (1)/(2)`

when

`y= (\pi)/(6)`

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

`x = \pi + (\pi)/(6) = (7\pi)/(6)`

`x = 2\pi - (\pi)/(6) = (11\pi)/(6)`

Hence the solutions are:

`x = (\pi)/(2) , (7\pi)/(6) , (3\pi)/(2) , (11\pi)/(6)`