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Find all solutions of the equation in the interval [tex]\([0, 2\pi)\)[/tex].

[tex]\[ \sin \theta - 4 = -3 \][/tex]

Write your answer in radians in terms of [tex]\(\pi\)[/tex].
If there is more than one solution, separate them with commas.

[tex]\(\theta =\)[/tex] [tex]\(\square \pi\)[/tex], ....

Sagot :

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]