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

[tex]\[ 2 \cos \theta + \sqrt{2} = 0 \][/tex]

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

[tex]\[ \theta = \][/tex]

[tex]\[ \square \pi \][/tex]


Sagot :

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]