Answer:
[tex]\displaystyle x=-\frac{5\pi}{6}\; \;\&\;-\frac{\pi}{6}[/tex]
Step-by-step explanation:
We will solve the given equation for x with the domain [-π, π].
In the unit circle, the y-coordinate represents sin(θ). In our case, θ = x. We will use this unit circle, see the attached image, to find when the sin(x) equals negative one-half.
This equation has infinite solutions, however, the domain narrows it down to just a few.
We can also graph sin(x) and negative one-half, see attached. Their points of intersection are the solution.
Given:
[tex]\displaystyle sin(x) = -\frac{1}{2}[/tex]
Utilize the unit circle:
[tex]\displaystyle x=\frac{7\pi}{6} ,\frac{11\pi}{6}[/tex]
Find the equivalent angles within [-π, π] by subtracting 2π:
[tex]\displaystyle x=\frac{7\pi}{6}-2\pi ,\frac{11\pi}{6}-2\pi[/tex]
[tex]\displaystyle x=-\frac{5\pi}{6},-\frac{\pi}{6}[/tex]
