To find the solutions of the equation [tex]\(2 \cos \theta - \sqrt{2} = 0\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex], follow these steps:
1. Isolate the cosine term:
[tex]\[
2 \cos \theta = \sqrt{2}
\][/tex]
2. Solve for [tex]\(\cos \theta\)[/tex] by dividing both sides by 2:
[tex]\[
\cos \theta = \frac{\sqrt{2}}{2}
\][/tex]
3. Recognize the standard angles where the cosine value is [tex]\(\frac{\sqrt{2}}{2}\)[/tex]:
The cosine function [tex]\(\cos \theta = \frac{\sqrt{2}}{2}\)[/tex] corresponds to angles [tex]\(\theta\)[/tex] where:
[tex]\[
\theta = \frac{\pi}{4} + 2k\pi \quad \text{or} \quad \theta = 2\pi - \frac{\pi}{4} + 2k\pi
\][/tex]
where [tex]\(k\)[/tex] is any integer, since [tex]\(\cos(\theta) = \cos(-\theta)\)[/tex].
4. Simplify the angles within the interval [tex]\([0, 2\pi)\)[/tex]:
[tex]\[
\theta = \frac{\pi}{4}
\][/tex]
[tex]\[
\theta = 2\pi - \frac{\pi}{4} = \frac{8\pi}{4} - \frac{\pi}{4} = \frac{7\pi}{4}
\][/tex]
5. Verify that these angles are within the given interval [tex]\(0 \leq \theta < 2\pi\)[/tex]:
Both [tex]\(\frac{\pi}{4}\)[/tex] and [tex]\(\frac{7\pi}{4}\)[/tex] lie within the interval [tex]\([0, 2\pi)\)[/tex].
Therefore, the solutions to the equation [tex]\(2 \cos \theta - \sqrt{2} = 0\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
\theta = \frac{\pi}{4}, \frac{7\pi}{4}
\][/tex]
