To solve the equation [tex]\(-\sqrt{2} + 3 \cos(\alpha) = \cos(\alpha)\)[/tex] for [tex]\(\alpha\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex], follow these steps:
1. Simplify the Equation:
Start by simplifying the given equation. Combine like terms involving [tex]\(\cos(\alpha)\)[/tex]:
[tex]\[
-\sqrt{2} + 3 \cos(\alpha) = \cos(\alpha)
\][/tex]
2. Isolate the [tex]\(\cos(\alpha)\)[/tex] Term:
Subtract [tex]\(\cos(\alpha)\)[/tex] from both sides of the equation:
[tex]\[
-\sqrt{2} + 3 \cos(\alpha) - \cos(\alpha) = 0
\][/tex]
Simplify the expression:
[tex]\[
-\sqrt{2} + 2 \cos(\alpha) = 0
\][/tex]
3. Solve for [tex]\(\cos(\alpha)\)[/tex]:
Add [tex]\(\sqrt{2}\)[/tex] to both sides to isolate the cosine term:
[tex]\[
2 \cos(\alpha) = \sqrt{2}
\][/tex]
Now, divide both sides by 2:
[tex]\[
\cos(\alpha) = \frac{\sqrt{2}}{2}
\][/tex]
4. Determine the Solutions:
The value [tex]\(\frac{\sqrt{2}}{2}\)[/tex] corresponds to specific angles [tex]\(\alpha\)[/tex] where the cosine function equals [tex]\(\frac{\sqrt{2}}{2}\)[/tex]. These angles in the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
\alpha = \frac{\pi}{4} \quad \text{and} \quad \alpha = \frac{7\pi}{4}
\][/tex]
5. Write the Final Answer:
Thus, the solutions for [tex]\(\alpha\)[/tex] that satisfy the given equation within the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
\alpha = \frac{\pi}{4}, \frac{7\pi}{4}
\][/tex]
Therefore, the exact simplified answers in radian units are:
[tex]\[
\alpha = \frac{\pi}{4}, \frac{7\pi}{4}
\][/tex]
