The value of ` sum _(k = 1)^(6) ( sin "" ( 2 pi k)/( 7) - i cos "" ( 2 pi k)/( 7))` is

Options

A. `-1`
B. 0
C. `-i`
D. i

Answer

To solve the given problem, we need to evaluate the sum:

\[ S = \sum_{k=1}^{6} \left( \sin\left(\frac{2\pi k}{7}\right) - i \cos\left(\frac{2\pi k}{7}\right) \right) \]

### Step 1: Rewrite the expression
We can rewrite the expression inside the summation using Euler's formula, where \( e^{ix} = \cos(x) + i\sin(x) \). Thus, we can express \( \sin \) and \( \cos \) in terms of \( e^{ix} \):

\[ \sin\left(\frac{2\pi k}{7}\right) = \frac{e^{i\frac{2\pi k}{7}} - e^{-i\frac{2\pi k}{7}}}{2i} \]
\[ \cos\left(\frac{2\pi k}{7}\right) = \frac{e^{i\frac{2\pi k}{7}} + e^{-i\frac{2\pi k}{7}}}{2} \]

Substituting these into the sum gives:

\[ S = \sum_{k=1}^{6} \left( \frac{e^{i\frac{2\pi k}{7}} - e^{-i\frac{2\pi k}{7}}}{2i} - i \cdot \frac{e^{i\frac{2\pi k}{7}} + e^{-i\frac{2\pi k}{7}}}{2} \right) \]

### Step 2: Simplify the expression
Now, we can combine the terms inside the summation:

\[ S = \sum_{k=1}^{6} \left( \frac{1}{2i}(e^{i\frac{2\pi k}{7}} - e^{-i\frac{2\pi k}{7}}) - \frac{i}{2}(e^{i\frac{2\pi k}{7}} + e^{-i\frac{2\pi k}{7}}) \right) \]

This simplifies to:

\[ S = \sum_{k=1}^{6} \left( \frac{1}{2i} e^{i\frac{2\pi k}{7}} - \frac{i}{2} e^{i\frac{2\pi k}{7}} - \frac{1}{2i} e^{-i\frac{2\pi k}{7}} - \frac{i}{2} e^{-i\frac{2\pi k}{7}} \right) \]

### Step 3: Combine like terms
Combining the terms gives:

\[ S = \sum_{k=1}^{6} \left( \left(\frac{1}{2i} - \frac{i}{2}\right)e^{i\frac{2\pi k}{7}} - \left(\frac{1}{2i} + \frac{i}{2}\right)e^{-i\frac{2\pi k}{7}} \right) \]

### Step 4: Recognize the geometric series
The terms \( e^{i\frac{2\pi k}{7}} \) for \( k = 1, 2, \ldots, 6 \) represent the 7th roots of unity (excluding \( k=0 \)). The sum of all 7th roots of unity is zero:

\[ 1 + e^{i\frac{2\pi}{7}} + e^{i\frac{4\pi}{7}} + e^{i\frac{6\pi}{7}} + e^{i\frac{8\pi}{7}} + e^{i\frac{10\pi}{7}} + e^{i\frac{12\pi}{7}} = 0 \]

Thus, we have:

\[ e^{i\frac{2\pi}{7}} + e^{i\frac{4\pi}{7}} + e^{i\frac{6\pi}{7}} + e^{i\frac{8\pi}{7}} + e^{i\frac{10\pi}{7}} + e^{i\frac{12\pi}{7}} = -1 \]

### Step 5: Calculate the final sum
Substituting back into our expression for \( S \):

\[ S = -i \cdot (-1) = i \]

### Conclusion
Thus, the final value of the sum is:

\[ \boxed{i} \]