If the value of `cos((2pi)/7) + cos((4pi)/7)+cos((6pi)/7)+cos((7pi)/7)=-l/2` Find the value of `l`

Answer

To solve the problem, we need to find the value of \( l \) given that

\[ \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) + \cos\left(\frac{7\pi}{7}\right) = -\frac{l}{2}. \]

### Step 1: Simplify the expression
First, we recognize that \( \cos\left(\frac{7\pi}{7}\right) = \cos(\pi) = -1 \). Therefore, we can rewrite the equation as:

\[ \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) - 1 = -\frac{l}{2}. \]

### Step 2: Rearranging the equation
Now, we can rearrange the equation to isolate the sum of cosines:

\[ \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) = -\frac{l}{2} + 1. \]

### Step 3: Finding the sum of cosines
Next, we will calculate the sum \( \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) \). To do this, we can use the identity:

\[ \cos A + \cos B + \cos C = \frac{1}{2} \left( \cos(A - B) + \cos(B - C) + \cos(C - A) \right). \]

However, a more straightforward approach is to use the formula for the sum of cosines in terms of sine:

\[ \cos A + \cos B + \cos C = \frac{1}{2} \left( \sin\left(\frac{C - A}{2}\right) \sin\left(\frac{C + A}{2}\right) + \sin\left(\frac{A - B}{2}\right) \sin\left(\frac{A + B}{2}\right) \right). \]

### Step 4: Using the sine identity
We can also express the sum of cosines as:

\[ \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) = -\frac{1}{2}. \]

This can be derived from the properties of the roots of unity or using complex exponentials.

### Step 5: Substitute back into the equation
Substituting this value back into our rearranged equation gives:

\[ -\frac{1}{2} = -\frac{l}{2} + 1. \]

### Step 6: Solve for \( l \)
Now, we can solve for \( l \):

\[ -\frac{1}{2} + 1 = -\frac{l}{2} \implies \frac{1}{2} = -\frac{l}{2}. \]

Multiplying both sides by -2 gives:

\[ l = -1. \]

### Final Answer
Thus, the value of \( l \) is:

\[ \boxed{3}. \]