Find the value of `cos20^(@)+cos100^(@)+cos140^(@):`

Options

A. `-1`
B. 0
C. 1
D. none of these

Answer

To find the value of \( \cos 20^\circ + \cos 100^\circ + \cos 140^\circ \), we can use the cosine addition formula and properties of cosine.

### Step-by-Step Solution:

1. **Group the Cosines**:
We can group the terms in pairs. Notice that \( \cos 100^\circ \) and \( \cos 140^\circ \) can be expressed in relation to \( \cos 20^\circ \):
\[ \cos 100^\circ = \cos(180^\circ - 80^\circ) = -\cos 80^\circ \]
\[ \cos 140^\circ = \cos(180^\circ - 40^\circ) = -\cos 40^\circ \]

2. **Rewrite the Expression**:
Now we can rewrite the original expression:
\[ \cos 20^\circ + \cos 100^\circ + \cos 140^\circ = \cos 20^\circ - \cos 80^\circ - \cos 40^\circ \]

3. **Use the Cosine Addition Formula**:
We can use the formula for the sum of cosines:
\[ \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
Let's apply this to \( \cos 20^\circ + \cos 140^\circ \):
\[ \cos 20^\circ + \cos 140^\circ = 2 \cos\left(\frac{20^\circ + 140^\circ}{2}\right) \cos\left(\frac{20^\circ - 140^\circ}{2}\right) \]
\[ = 2 \cos(80^\circ) \cos(-60^\circ) \]
Since \( \cos(-\theta) = \cos(\theta) \):
\[ = 2 \cos(80^\circ) \cos(60^\circ) \]
Knowing that \( \cos(60^\circ) = \frac{1}{2} \):
\[ = 2 \cos(80^\circ) \cdot \frac{1}{2} = \cos(80^\circ) \]

4. **Combine the Results**:
Now we can substitute this back into our expression:
\[ \cos 20^\circ + \cos 100^\circ + \cos 140^\circ = \cos(80^\circ) - \cos(80^\circ) = 0 \]

### Final Answer:
Thus, the value of \( \cos 20^\circ + \cos 100^\circ + \cos 140^\circ \) is \( 0 \).