To solve the equation [tex]\( 1 + \sin \theta = \frac{\sqrt{3} + 2}{2} \)[/tex], we'll proceed step-by-step.
1. Isolate [tex]\(\sin \theta\)[/tex]:
Subtract 1 from both sides of the equation:
[tex]\[
\sin \theta = \frac{\sqrt{3} + 2}{2} - 1
\][/tex]
2. Simplify the right-hand side:
Simplify the expression on the right-hand side:
[tex]\[
\sin \theta = \frac{\sqrt{3} + 2}{2} - \frac{2}{2} = \frac{\sqrt{3} + 2 - 2}{2} = \frac{\sqrt{3}}{2}
\][/tex]
3. Set the equation for [tex]\(\sin \theta\)[/tex]:
Now we need to solve:
[tex]\[
\sin \theta = \frac{\sqrt{3}}{2}
\][/tex]
4. Find the angles that satisfy the equation:
We know from trigonometry that [tex]\(\sin \theta = \frac{\sqrt{3}}{2}\)[/tex] is true for:
[tex]\(\theta = \frac{\pi}{3}\)[/tex] and [tex]\(\theta = \frac{2\pi}{3} + 2k\pi\)[/tex]
However, we need to find solutions in the interval [tex]\( [0, 2\pi] \)[/tex]. Observing the symmetry and values on the unit circle:
[tex]\[
\theta = \frac{\pi}{3}, \quad \theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}
\][/tex]
[tex]\[
\theta = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}
\][/tex]
5. Verify the solutions and check intervals:
In the interval [tex]\( [0, 2\pi] \)[/tex], the solutions are:
[tex]\(\theta = \frac{\pi}{3}\)[/tex] and [tex]\(\theta = \frac{5\pi}{3}\)[/tex]
Thus, the correct answer is:
A. [tex]\(\frac{\pi}{3}, \frac{5 \pi}{3}\)[/tex]
