To solve the equation [tex]\( 3 \sin x = \sin x + 1 \)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex], let's go through the steps methodically:
1. Start by simplifying the given equation:
[tex]\[
3 \sin x = \sin x + 1
\][/tex]
2. Subtract [tex]\(\sin x\)[/tex] from both sides to isolate the sine term:
[tex]\[
3 \sin x - \sin x = 1
\][/tex]
3. This simplifies to:
[tex]\[
2 \sin x = 1
\][/tex]
4. Divide both sides of the equation by 2 to solve for [tex]\(\sin x\)[/tex]:
[tex]\[
\sin x = \frac{1}{2}
\][/tex]
5. We need to find the values of [tex]\(x\)[/tex] for which [tex]\(\sin x = \frac{1}{2}\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex].
6. Recall the unit circle values for sine. The sine function equals [tex]\(\frac{1}{2}\)[/tex] at two angles in one full rotation (0 to [tex]\(2\pi\)[/tex]):
- [tex]\( x = \frac{\pi}{6} \)[/tex]
- [tex]\( x = \frac{5\pi}{6} \)[/tex]
Therefore, the solutions to the equation [tex]\(3 \sin x = \sin x + 1\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
x = \frac{\pi}{6} \quad \text{and} \quad x = \frac{5\pi}{6}
\][/tex]
In decimal terms, these are approximately:
[tex]\[
\frac{\pi}{6} \approx 0.5236 \quad \text{and} \quad \frac{5\pi}{6} \approx 2.6180
\][/tex]
