To find all 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 [tex]\(\cos \theta\)[/tex] in the equation:
[tex]\[
2 \cos \theta + \sqrt{2} = 0
\][/tex]
[tex]\[
2 \cos \theta = -\sqrt{2}
\][/tex]
[tex]\[
\cos \theta = -\frac{\sqrt{2}}{2}
\][/tex]
2. Determine the angles [tex]\(\theta\)[/tex] for which [tex]\(\cos \theta = -\frac{\sqrt{2}}{2}\)[/tex]. The cosine function equals [tex]\(-\frac{\sqrt{2}}{2}\)[/tex] at the angles [tex]\(\theta = \frac{3\pi}{4}\)[/tex] and [tex]\(\theta = \frac{5\pi}{4}\)[/tex] within one period [tex]\([0, 2\pi)\)[/tex].
3. Therefore, the solutions within the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\(
\theta = \frac{3\pi}{4}, \frac{5\pi}{4}
\)[/tex]
Thus, in terms of [tex]\(\pi\)[/tex], the solutions are:
[tex]\[
\theta = 3\pi/4, 5\pi/4
\][/tex]
