`cos ^(3) 110 ^(@) + cos ^(3) 10^(@) + cos ^(3) 130^(@)=`

Options

A. `3/4`
B. `3/8`
C. `(3sqrt3)/(8)`
D. `(3sqrt3)/(4)`

Answer

To solve the problem \( \cos^3(110^\circ) + \cos^3(10^\circ) + \cos^3(130^\circ) \), we will use the identity for the sum of cubes and some trigonometric identities.

### Step 1: Identify the values
Let:
- \( a = \cos(110^\circ) \)
- \( b = \cos(10^\circ) \)
- \( c = \cos(130^\circ) \)

### Step 2: Check if \( a + b + c = 0 \)
We will first check if \( a + b + c = 0 \):
\[ \cos(110^\circ) + \cos(10^\circ) + \cos(130^\circ) \]
Using the identity \( \cos(180^\circ - x) = -\cos(x) \):
\[ \cos(110^\circ) = -\cos(70^\circ) \quad \text{and} \quad \cos(130^\circ) = -\cos(50^\circ) \]
Thus:
\[ \cos(110^\circ) + \cos(130^\circ) = -\cos(70^\circ) - \cos(50^\circ) \]
Now, using the cosine addition formula:
\[ \cos(110^\circ) + \cos(10^\circ) + \cos(130^\circ) = -\cos(70^\circ) + \cos(10^\circ) - \cos(50^\circ) \]
Calculating these values confirms that they indeed sum to zero.

### Step 3: Use the identity for the sum of cubes
Since \( a + b + c = 0 \), we can use the identity:
\[ a^3 + b^3 + c^3 = 3abc \]
Thus:
\[ \cos^3(110^\circ) + \cos^3(10^\circ) + \cos^3(130^\circ) = 3 \cdot \cos(110^\circ) \cdot \cos(10^\circ) \cdot \cos(130^\circ) \]

### Step 4: Calculate \( abc \)
Now we need to find \( abc \):
\[ abc = \cos(110^\circ) \cdot \cos(10^\circ) \cdot \cos(130^\circ) \]
Using the cosine of angles:
\[ \cos(130^\circ) = \cos(180^\circ - 50^\circ) = -\cos(50^\circ) \]
Thus:
\[ abc = \cos(110^\circ) \cdot \cos(10^\circ) \cdot (-\cos(50^\circ)) \]

### Step 5: Simplify using trigonometric identities
Using the product-to-sum identities:
\[ \cos(110^\circ) \cdot \cos(10^\circ) = \frac{1}{2} [\cos(100^\circ) + \cos(120^\circ)] \]
Calculating \( \cos(100^\circ) \) and \( \cos(120^\circ) \):
\[ \cos(100^\circ) = -\sin(10^\circ), \quad \cos(120^\circ) = -\frac{1}{2} \]
Thus:
\[ abc = \frac{1}{2}[-\sin(10^\circ) - \frac{1}{2}] \cdot (-\cos(50^\circ) \]
Calculating this gives a final value.

### Step 6: Final calculation
Using the known values:
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
Thus:
\[ \cos^3(110^\circ) + \cos^3(10^\circ) + \cos^3(130^\circ) = 3 \cdot \frac{\sqrt{3}}{8} \]

### Final Answer
The final answer is:
\[ \frac{3\sqrt{3}}{8} \]