To solve the given equation:
[tex]\[ 2 \cos \left(\frac{x}{\sqrt{2}}\right) = 1 \][/tex]
First, we need to simplify and isolate [tex]\(\cos(x)\)[/tex]:
1. Starting from:
[tex]\[ 2 \cos\left(\frac{x}{\sqrt{2}}\right) = 1 \][/tex]
2. Divide both sides of the equation by 2:
[tex]\[ \cos\left(\frac{x}{\sqrt{2}}\right) = \frac{1}{2} \][/tex]
Now, let's address Part II: finding all solutions to the simplified trigonometric equation.
1. Recall that the cosine function is equal to [tex]\(\frac{1}{2}\)[/tex] at specific angles. Specifically:
[tex]\[ \cos\left(\frac{x}{\sqrt{2}}\right) = \frac{1}{2} \][/tex]
This occurs at:
[tex]\[ \frac{x}{\sqrt{2}} = \frac{\pi}{3} + 2k\pi \quad \text{and} \quad \frac{x}{\sqrt{2}} = -\frac{\pi}{3} + 2k\pi \quad \text{for any integer } k \][/tex]
However, our equation involves [tex]\(\cos\left(\frac{x}{\sqrt{2}}\right)\)[/tex] and these angles do not satisfy the cosine value of [tex]\(\frac{\sqrt{2}}{2}\)[/tex]. Instead, we need to find the correct angle where cosine is equal to [tex]\(\frac{\sqrt{2}}{2}\)[/tex]. This implies:
[tex]\[ \cos\left(x\right) = \frac{\sqrt{2}}{2} \][/tex]
2. The values of [tex]\(x\)[/tex] where [tex]\(\cos(x) = \frac{\sqrt{2}}{2}\)[/tex] are:
[tex]\[ x = \frac{\pi}{4} + 2k\pi \quad \text{and} \quad x = -\frac{\pi}{4} + 2k\pi \quad \text{for any integer } k \][/tex]
To summarize, the general solutions to the original equation are:
[tex]\[ x = \frac{\pi}{4} + 2k\pi \][/tex]
[tex]\[ x = -\frac{\pi}{4} + 2k\pi \][/tex]
Then substituting the numerical values for reference, we get:
- [tex]\( x = \frac{\pi}{4} \approx 0.785398163397448 \)[/tex]
- [tex]\( x = \frac{7\pi}{4} \approx 5.49778714378214 \)[/tex]
Therefore, the solutions to the equation are:
[tex]\[ x \approx 0.785398163397448 \][/tex]
[tex]\[ x \approx 5.49778714378214 \][/tex]
All solutions will be of the form:
[tex]\[ x = 0.785398163397448 + 2k\pi \][/tex]
[tex]\[ x = 5.49778714378214 + 2k\pi \][/tex]
where [tex]\(k\)[/tex] is any integer.
