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

Options

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

Answer

To solve the problem \( \cos^3(10^\circ) + \cos^3(110^\circ) + \cos^3(130^\circ) \), we can use the identity for the sum of cubes of cosines.

### Step-by-Step Solution:

1. **Identify the angles**: We have three angles: \( 10^\circ \), \( 110^\circ \), and \( 130^\circ \).

2. **Use the identity**: We can use the identity:
\[ \cos^3 x + \cos^3(120^\circ - x) + \cos^3(120^\circ + x) = \frac{3}{4} \cos(3x) \]
In our case, we can set \( x = 10^\circ \).

3. **Substituting into the identity**: According to the identity:
\[ \cos^3(10^\circ) + \cos^3(110^\circ) + \cos^3(130^\circ) = \frac{3}{4} \cos(3 \times 10^\circ) \]

4. **Calculate \( \cos(30^\circ) \)**: We know that:
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]

5. **Substituting back into the equation**: Now substituting \( \cos(30^\circ) \) into our equation:
\[ \cos^3(10^\circ) + \cos^3(110^\circ) + \cos^3(130^\circ) = \frac{3}{4} \cdot \frac{\sqrt{3}}{2} \]

6. **Simplifying the expression**: This simplifies to:
\[ = \frac{3\sqrt{3}}{8} \]

7. **Final answer**: Therefore, the final answer is:
\[ \cos^3(10^\circ) + \cos^3(110^\circ) + \cos^3(130^\circ) = \frac{3\sqrt{3}}{8} \]