`cos 35^(@) + cos 85^(@) + cos 155^(@) =`

Options

A. 0
B. `1// sqrt( 3)`
C. `1 // sqrt(2)`
D. `cos 275^(@)`

Answer

To solve the equation \( \cos 35^\circ + \cos 85^\circ + \cos 155^\circ \), we will follow these steps:

### Step 1: Combine the first two cosine terms
We will use the cosine addition formula:
\[ \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]
Let \( A = 35^\circ \) and \( B = 85^\circ \).

### Step 2: Calculate \( A + B \) and \( A - B \)
\[ A + B = 35^\circ + 85^\circ = 120^\circ \]
\[ A - B = 35^\circ - 85^\circ = -50^\circ \]

### Step 3: Substitute into the formula
Now, substituting into the formula:
\[ \cos 35^\circ + \cos 85^\circ = 2 \cos\left(\frac{120^\circ}{2}\right) \cos\left(\frac{-50^\circ}{2}\right) \]
\[ = 2 \cos(60^\circ) \cos(-25^\circ) \]

### Step 4: Evaluate \( \cos(60^\circ) \) and \( \cos(-25^\circ) \)
We know that:
\[ \cos(60^\circ) = \frac{1}{2} \]
And since cosine is an even function:
\[ \cos(-25^\circ) = \cos(25^\circ) \]

### Step 5: Substitute back
Now substituting these values back:
\[ \cos 35^\circ + \cos 85^\circ = 2 \cdot \frac{1}{2} \cdot \cos(25^\circ) = \cos(25^\circ) \]

### Step 6: Add the third term
Now we add \( \cos 155^\circ \):
\[ \cos 35^\circ + \cos 85^\circ + \cos 155^\circ = \cos(25^\circ) + \cos(155^\circ) \]

### Step 7: Simplify \( \cos(155^\circ) \)
We can express \( \cos(155^\circ) \) as:
\[ \cos(155^\circ) = \cos(180^\circ - 25^\circ) = -\cos(25^\circ) \]

### Step 8: Combine the terms
Now we can combine:
\[ \cos(25^\circ) + (-\cos(25^\circ)) = 0 \]

### Final Answer
Thus, the final value is:
\[ \cos 35^\circ + \cos 85^\circ + \cos 155^\circ = 0 \]