To solve 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 function:
[tex]\[
2 \cos \theta + \sqrt{2} = 0
\][/tex]
Subtract [tex]\(\sqrt{2}\)[/tex] from both sides:
[tex]\[
2 \cos \theta = -\sqrt{2}
\][/tex]
Divide both sides by [tex]\(2\)[/tex]:
[tex]\[
\cos \theta = -\frac{\sqrt{2}}{2}
\][/tex]
2. Identify the reference angle:
We need to find [tex]\(\theta\)[/tex] such that [tex]\(\cos \theta = -\frac{\sqrt{2}}{2}\)[/tex]. The value [tex]\(-\frac{\sqrt{2}}{2}\)[/tex] corresponds to the standard angles where the cosine value is [tex]\(\frac{\sqrt{2}}{2}\)[/tex], but in the second and third quadrants (where cosine is negative).
3. Determine the specific angles:
The angles that satisfy [tex]\(\cos \theta = -\frac{\sqrt{2}}{2}\)[/tex] are:
[tex]\[
\theta = \pi - \frac{\pi}{4} \quad \text{(Second quadrant)}
\][/tex]
and
[tex]\[
\theta = \pi + \frac{\pi}{4} \quad \text{(Third quadrant)}
\][/tex]
Simplifying these:
[tex]\[
\theta = \frac{3\pi}{4}
\][/tex]
and
[tex]\[
\theta = \frac{5\pi}{4}
\][/tex]
4. Write the solution in radians in terms of [tex]\(\pi\)[/tex]:
The solutions in the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
\theta = \frac{3\pi}{4}, \frac{5\pi}{4}
\][/tex]
Thus, the solutions to the equation [tex]\(2 \cos \theta + \sqrt{2} = 0\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
\theta = \frac{3\pi}{4}, \frac{5\pi}{4}
\][/tex]
