The value of `cos 20^(@) + Cos 100^(@) + cos 140^(@)` is

Options

A. `1/2`
B. `(1)/ (sqrt(3))`
C. `sqrt(3)`
D. 0

Answer

To find the value of \( \cos 20^\circ + \cos 100^\circ + \cos 140^\circ \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \cos 20^\circ + \cos 100^\circ + \cos 140^\circ \]

### Step 2: Use the cosine addition formula
We can rewrite \( \cos 100^\circ \) using the identity \( \cos(90^\circ + \theta) = -\sin(\theta) \):
\[ \cos 100^\circ = \cos(90^\circ + 10^\circ) = -\sin 10^\circ \]
So, we can rewrite the expression as:
\[ \cos 20^\circ - \sin 10^\circ + \cos 140^\circ \]

### Step 3: Combine \( \cos 20^\circ \) and \( \cos 140^\circ \)
Next, we can combine \( \cos 20^\circ \) and \( \cos 140^\circ \) using the cosine addition formula:
\[ \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
Let \( A = 20^\circ \) and \( B = 140^\circ \):
\[ \cos 20^\circ + \cos 140^\circ = 2 \cos\left(\frac{20^\circ + 140^\circ}{2}\right) \cos\left(\frac{20^\circ - 140^\circ}{2}\right) \]

### Step 4: Calculate the averages
Calculating the averages:
\[ \frac{20^\circ + 140^\circ}{2} = \frac{160^\circ}{2} = 80^\circ \]
\[ \frac{20^\circ - 140^\circ}{2} = \frac{-120^\circ}{2} = -60^\circ \]
Thus, we have:
\[ \cos 20^\circ + \cos 140^\circ = 2 \cos 80^\circ \cos(-60^\circ) \]

### Step 5: Use the cosine values
We know that \( \cos(-\theta) = \cos(\theta) \) and \( \cos 60^\circ = \frac{1}{2} \):
\[ \cos 80^\circ = \sin 10^\circ \quad (\text{since } \cos(90^\circ - x) = \sin x) \]
So we can substitute:
\[ \cos 20^\circ + \cos 140^\circ = 2 \sin 10^\circ \cdot \frac{1}{2} = \sin 10^\circ \]

### Step 6: Combine with \( -\sin 10^\circ \)
Now, we substitute back into our expression:
\[ \sin 10^\circ - \sin 10^\circ = 0 \]

### Conclusion
Thus, the value of \( \cos 20^\circ + \cos 100^\circ + \cos 140^\circ \) is:
\[ \boxed{0} \]