The value of `a+b` such that the inequality `ale 5 cos theta+3cos (theta+(pi)/(3))+3le b ` holds true for all the real values of `theta` is (equality holds on both sides atleast once for real values of `theta`)

Answer

To solve the inequality \( a \leq 5 \cos \theta + 3 \cos \left( \theta + \frac{\pi}{3} \right) + 3 \leq b \) for all real values of \( \theta \), we need to find the values of \( a \) and \( b \) such that the inequality holds true and equality occurs at least once.

### Step-by-Step Solution:

1. **Express the cosine term using the cosine addition formula**:
\[ \cos \left( \theta + \frac{\pi}{3} \right) = \cos \theta \cos \frac{\pi}{3} - \sin \theta \sin \frac{\pi}{3} \]
Knowing that \( \cos \frac{\pi}{3} = \frac{1}{2} \) and \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \), we can substitute:
\[ \cos \left( \theta + \frac{\pi}{3} \right) = \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta \]

2. **Substituting back into the inequality**:
\[ 5 \cos \theta + 3 \left( \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta \right) + 3 \]
Simplifying this gives:
\[ 5 \cos \theta + \frac{3}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 3 \]
Combine the cosine terms:
\[ \left( 5 + \frac{3}{2} \right) \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 3 = \frac{13}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 3 \]

3. **Finding the maximum and minimum values**:
The expression \( \frac{13}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta \) can be rewritten in the form \( R \cos(\theta + \phi) \) where:
\[ R = \sqrt{\left( \frac{13}{2} \right)^2 + \left( -\frac{3\sqrt{3}}{2} \right)^2} \]
Calculate \( R \):
\[ R = \sqrt{\frac{169}{4} + \frac{27}{4}} = \sqrt{\frac{196}{4}} = \sqrt{49} = 7 \]

4. **Determining the range**:
The expression \( \frac{13}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta \) will vary from \( -7 \) to \( 7 \). Adding 3 to this gives:
\[ -7 + 3 \leq 5 \cos \theta + 3 \cos \left( \theta + \frac{\pi}{3} \right) + 3 \leq 7 + 3 \]
Thus, the range is:
\[ -4 \leq 5 \cos \theta + 3 \cos \left( \theta + \frac{\pi}{3} \right) + 3 \leq 10 \]

5. **Identifying values of \( a \) and \( b \)**:
From the inequality, we see that:
\[ a = -4 \quad \text{and} \quad b = 10 \]

6. **Calculating \( a + b \)**:
\[ a + b = -4 + 10 = 6 \]

### Final Answer:
The value of \( a + b \) is \( 6 \).