To prove that [tex]\(\cos \alpha + \cos 2\alpha + \cos 4\alpha = -\frac{1}{2}\)[/tex] given that [tex]\(7\alpha = 2\pi\)[/tex], let's follow these steps:
### Step 1: Simplify for [tex]\(\alpha\)[/tex]
First, solve for [tex]\(\alpha\)[/tex] from the given equation [tex]\(7\alpha = 2\pi\)[/tex].
[tex]\[
\alpha = \frac{2\pi}{7}
\][/tex]
### Step 2: Evaluate [tex]\(\cos \alpha\)[/tex]
We need to find the cosine of [tex]\(\alpha\)[/tex], where [tex]\(\alpha = \frac{2\pi}{7}\)[/tex].
[tex]\[
\cos \alpha \approx 0.6235
\][/tex]
### Step 3: Evaluate [tex]\(\cos 2\alpha\)[/tex]
Now, find the cosine of [tex]\(2\alpha\)[/tex], where [tex]\(2\alpha = 2 \cdot \frac{2\pi}{7} = \frac{4\pi}{7}\)[/tex].
[tex]\[
\cos 2\alpha \approx -0.2225
\][/tex]
### Step 4: Evaluate [tex]\(\cos 4\alpha\)[/tex]
Next, compute the cosine of [tex]\(4\alpha\)[/tex], where [tex]\(4\alpha = 4 \cdot \frac{2\pi}{7} = \frac{8\pi}{7}\)[/tex].
[tex]\[
\cos 4\alpha \approx -0.9010
\][/tex]
### Step 5: Sum the cosine values
Now that we have all the cosine values, let's sum them up:
[tex]\[
\cos \alpha + \cos 2\alpha + \cos 4\alpha \approx 0.6235 + (-0.2225) + (-0.9010)
\][/tex]
Simplifying this:
[tex]\[
0.6235 - 0.2225 - 0.9010 \approx -0.5000
\][/tex]
### Conclusion
Therefore, we have shown that:
[tex]\[
\cos \alpha + \cos 2\alpha + \cos 4\alpha = -0.5 = -\frac{1}{2}
\][/tex]
Thus, we have proved that [tex]\(\cos \alpha + \cos 2\alpha + \cos 4\alpha = -\frac{1}{2}\)[/tex] given that [tex]\(7\alpha = 2\pi\)[/tex].
