What is `cos20^@ + cos 100^@ + cos 140^@` equal to?

Options

A. 2
B. 1
C. `1/2`
D. 0

Answer

To solve the expression \( \cos 20^\circ + \cos 100^\circ + \cos 140^\circ \), we can use the cosine addition formula. Let's break it down step by step.

### Step 1: Group the cosines
We can group the first two terms together:
\[ \cos 20^\circ + \cos 100^\circ + \cos 140^\circ \]

### Step 2: Use the cosine addition formula
The cosine addition formula states that:
\[ \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]
We will apply this to \( \cos 20^\circ + \cos 100^\circ \).

Let \( A = 20^\circ \) and \( B = 100^\circ \):
\[ \cos 20^\circ + \cos 100^\circ = 2 \cos\left(\frac{20^\circ + 100^\circ}{2}\right) \cos\left(\frac{20^\circ - 100^\circ}{2}\right) \]

### Step 3: Calculate the averages
Calculating the averages:
\[ \frac{20^\circ + 100^\circ}{2} = \frac{120^\circ}{2} = 60^\circ \]
\[ \frac{20^\circ - 100^\circ}{2} = \frac{-80^\circ}{2} = -40^\circ \]

### Step 4: Substitute back into the equation
Now substituting back into our equation:
\[ \cos 20^\circ + \cos 100^\circ = 2 \cos(60^\circ) \cos(-40^\circ) \]

### Step 5: Evaluate \( \cos(60^\circ) \)
We know that:
\[ \cos(60^\circ) = \frac{1}{2} \]
Thus:
\[ \cos 20^\circ + \cos 100^\circ = 2 \cdot \frac{1}{2} \cdot \cos(-40^\circ) = \cos(-40^\circ) = \cos(40^\circ) \]

### Step 6: Add \( \cos 140^\circ \)
Now we have:
\[ \cos 20^\circ + \cos 100^\circ + \cos 140^\circ = \cos(40^\circ) + \cos(140^\circ) \]
Using the cosine addition formula again:
\[ \cos(40^\circ) + \cos(140^\circ) = 2 \cos\left(\frac{40^\circ + 140^\circ}{2}\right) \cos\left(\frac{40^\circ - 140^\circ}{2}\right) \]

### Step 7: Calculate the averages again
Calculating the averages:
\[ \frac{40^\circ + 140^\circ}{2} = \frac{180^\circ}{2} = 90^\circ \]
\[ \frac{40^\circ - 140^\circ}{2} = \frac{-100^\circ}{2} = -50^\circ \]

### Step 8: Substitute back into the equation
Substituting back:
\[ \cos(40^\circ) + \cos(140^\circ) = 2 \cos(90^\circ) \cos(-50^\circ) \]

### Step 9: Evaluate \( \cos(90^\circ) \)
We know that:
\[ \cos(90^\circ) = 0 \]
Thus:
\[ 2 \cdot 0 \cdot \cos(-50^\circ) = 0 \]

### Final Result
Therefore, the final result is:
\[ \cos 20^\circ + \cos 100^\circ + \cos 140^\circ = 0 \]