To solve the equation [tex]\(\sin \theta - 4 = -3\)[/tex] for [tex]\(\theta\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex], follow these steps:
1. Start with the given equation:
[tex]\[
\sin \theta - 4 = -3
\][/tex]
2. Add 4 to both sides to simplify:
[tex]\[
\sin \theta = -3 + 4
\][/tex]
[tex]\[
\sin \theta = 1
\][/tex]
3. Next, determine the values of [tex]\(\theta\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] where [tex]\(\sin \theta = 1\)[/tex]. We know that the sine of an angle reaches a maximum value of 1 at specific points on the unit circle.
4. Recall that:
[tex]\[
\sin(\frac{\pi}{2}) = 1
\][/tex]
5. Therefore, [tex]\(\theta = \frac{\pi}{2}\)[/tex] is a solution since it lies within the given interval [tex]\([0, 2\pi)\)[/tex] and satisfies the condition [tex]\(\sin \theta = 1\)[/tex].
6. In the interval [tex]\([0, 2\pi)\)[/tex], the sine function reaches 1 only at [tex]\(\theta = \frac{\pi}{2}\)[/tex].
So, the solution to the equation [tex]\(\sin \theta - 4 = -3\)[/tex] in the interval [tex]\( [0, 2\pi) \)[/tex] is:
[tex]\[
\theta = \frac{\pi}{2}
\][/tex]
Thus, the answer is:
[tex]\[
\theta = \frac{\pi}{2}
\][/tex]
