Consider the trigonometric equation:
[tex]\[
\sin \frac{x}{2} = \frac{\sqrt{3}}{2}
\][/tex]
We know from trigonometry that [tex]\(\sin \theta = \frac{\sqrt{3}}{2}\)[/tex] when [tex]\(\theta = 60^\circ\)[/tex] or [tex]\(\theta = 120^\circ\)[/tex].
So, we set:
[tex]\[
\frac{x}{2} = 60^\circ \quad \text{or} \quad \frac{x}{2} = 120^\circ
\][/tex]
To find [tex]\(x\)[/tex], we solve each equation:
1. [tex]\(\frac{x}{2} = 60^\circ\)[/tex]
[tex]\[
x = 2 \times 60^\circ = 120^\circ
\][/tex]
2. [tex]\(\frac{x}{2} = 120^\circ\)[/tex]
[tex]\[
x = 2 \times 120^\circ = 240^\circ
\][/tex]
Thus, the possible solutions for [tex]\(x\)[/tex] are [tex]\(120^\circ\)[/tex] and [tex]\(240^\circ\)[/tex].
Next, we compare these solutions with the given options:
- A. [tex]\(60^\circ\)[/tex]
- B. [tex]\(150^\circ\)[/tex]
- C. [tex]\(210^\circ\)[/tex]
- D. [tex]\(240^\circ\)[/tex]
Among the options provided, [tex]\(240^\circ\)[/tex] matches our solution.
Therefore, the correct answer is:
[tex]\[
\boxed{4}
\][/tex]
