To solve the equation [tex]\(2 \sin \theta - \sqrt{3} = 0\)[/tex] for [tex]\(\theta\)[/tex], let's follow these steps:
1. Rearrange the equation to solve for [tex]\(\sin \theta\)[/tex]:
[tex]\[
2 \sin \theta - \sqrt{3} = 0
\][/tex]
To isolate [tex]\(\sin \theta\)[/tex], add [tex]\(\sqrt{3}\)[/tex] to both sides of the equation:
[tex]\[
2 \sin \theta = \sqrt{3}
\][/tex]
Next, divide both sides by 2:
[tex]\[
\sin \theta = \frac{\sqrt{3}}{2}
\][/tex]
2. Identify the angles [tex]\(\theta\)[/tex] that satisfy [tex]\(\sin \theta = \frac{\sqrt{3}}{2}\)[/tex] within the range [tex]\([0, 2\pi)\)[/tex]:
From trigonometric knowledge, we know that [tex]\(\sin \theta = \frac{\sqrt{3}}{2}\)[/tex] at two specific angles in the unit circle:
- [tex]\(\theta = \frac{\pi}{3}\)[/tex]
- [tex]\(\theta = \frac{2\pi}{3}\)[/tex]
3. List all solutions within the specified range:
The angles [tex]\(\theta\)[/tex] that satisfy the equation [tex]\(2 \sin \theta - \sqrt{3} = 0\)[/tex] in the range [tex]\([0, 2\pi)\)[/tex] are:
- [tex]\(\theta = \frac{\pi}{3}\)[/tex]
- [tex]\(\theta = \frac{2\pi}{3}\)[/tex]
Thus, the solutions to the equation [tex]\(2 \sin \theta - \sqrt{3} = 0\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
\boxed{\frac{\pi}{3} \text{ and } \frac{2\pi}{3}}
\][/tex]
In numerical form, these solutions are approximately:
[tex]\[
\boxed{1.0471975511965976 \text{ and } 2.0943951023931953}
\][/tex]
