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

[tex]\[2 \sin \theta - \sqrt{2} = 0\][/tex]

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


Sagot :

To find all solutions of the equation [tex]\( 2 \sin \theta - \sqrt{2} = 0 \)[/tex] within the interval [tex]\( [0, 2\pi) \)[/tex], follow these steps:

1. Isolate the sine function:

[tex]\(2 \sin \theta - \sqrt{2} = 0\)[/tex]

Add [tex]\(\sqrt{2}\)[/tex] to both sides:

[tex]\(2 \sin \theta = \sqrt{2}\)[/tex]

Divide both sides by 2:

[tex]\(\sin \theta = \frac{\sqrt{2}}{2}\)[/tex]

2. Recognize standard values:

The value [tex]\(\frac{\sqrt{2}}{2}\)[/tex] is a known value of the sine function. From the unit circle, we know that:

[tex]\(\sin \theta = \frac{\sqrt{2}}{2}\)[/tex] at:

[tex]\(\theta = \frac{\pi}{4}\)[/tex] (for the first quadrant) and

[tex]\(\theta = \frac{3\pi}{4}\)[/tex] (for the second quadrant).

3. State the solutions:

Therefore, the solutions of the equation [tex]\( 2 \sin \theta - \sqrt{2} = 0 \)[/tex] within the interval [tex]\( [0, 2\pi) \)[/tex] are:

[tex]\(\theta = \frac{\pi}{4}, \frac{3\pi}{4}\)[/tex]

So, the solutions in radians are:
[tex]\[ \boxed{\frac{\pi}{4}, \frac{3\pi}{4}} \][/tex]