To solve the equation [tex]\(2 \sin (x) + 1 = 0\)[/tex], follow these steps:
1. Start with the equation:
[tex]\[2 \sin(x) + 1 = 0\][/tex]
2. Isolate [tex]\(\sin(x)\)[/tex]:
[tex]\[2 \sin(x) = -1\][/tex]
[tex]\[\sin(x) = -\frac{1}{2}\][/tex]
3. Determine the angles [tex]\(x\)[/tex] within the unit circle where [tex]\(\sin(x) = -\frac{1}{2}\)[/tex]. Specifically, we are interested in solutions for [tex]\(x\)[/tex] within the ranges given:
- In the range [tex]\(180^{\circ} \leq x \leq 270^{\circ}\)[/tex]
- In the range [tex]\(270^{\circ} \leq x \leq 360^{\circ}\)[/tex]
4. From trigonometric identities and the unit circle, we know that [tex]\(\sin(x) = -\frac{1}{2}\)[/tex] at the following specific angles within one period [tex]\(0^{\circ} \leq x \leq 360^{\circ}\)[/tex]:
- [tex]\(x = 210^{\circ}\)[/tex]
- [tex]\(x = 330^{\circ}\)[/tex]
5. Now, we identify which of these angles fall within the given ranges:
- For [tex]\(180^{\circ} \leq x \leq 270^{\circ}\)[/tex], the solution is:
[tex]\[x = 210^{\circ}\][/tex]
- For [tex]\(270^{\circ} \leq x \leq 360^{\circ}\)[/tex], the solution is:
[tex]\[x = 330^{\circ}\][/tex]
So the correct answers are:
If [tex]\(180^{\circ} \leq x \leq 270^{\circ}\)[/tex], then [tex]\(x = 210\)[/tex].
If [tex]\(270^{\circ} \leq x \leq 360^{\circ}\)[/tex], then [tex]\(x = 330\)[/tex].
