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 determine the range of the expression \( f(\theta) = 5 \cos \theta + 3 \cos \left( \theta + \frac{\pi}{3} \right) + 3 \).

### Step 1: Expand the cosine term
Using the cosine addition formula, we can expand \( \cos \left( \theta + \frac{\pi}{3} \right) \):
\[ \cos \left( \theta + \frac{\pi}{3} \right) = \cos \theta \cos \frac{\pi}{3} - \sin \theta \sin \frac{\pi}{3} \]
Substituting the values \( \cos \frac{\pi}{3} = \frac{1}{2} \) and \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \), we get:
\[ \cos \left( \theta + \frac{\pi}{3} \right) = \cos \theta \cdot \frac{1}{2} - \sin \theta \cdot \frac{\sqrt{3}}{2} \]
Thus, we can rewrite \( f(\theta) \):
\[ f(\theta) = 5 \cos \theta + 3 \left( \frac{1}{2} \cos \theta - \frac{\sqrt{3}}{2} \sin \theta \right) + 3 \]
\[ = 5 \cos \theta + \frac{3}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 3 \]
\[ = \left( 5 + \frac{3}{2} \right) \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 3 \]
\[ = \frac{13}{2} \cos \theta - \frac{3\sqrt{3}}{2} \sin \theta + 3 \]

### Step 2: Identify the coefficients
Now we can identify the coefficients:
- \( A = \frac{13}{2} \)
- \( B = -\frac{3\sqrt{3}}{2} \)

### Step 3: Calculate the range of the trigonometric expression
The range of the expression \( A \cos \theta + B \sin \theta \) is given by:
\[ \text{Range} = \left[-\sqrt{A^2 + B^2}, \sqrt{A^2 + B^2}\right] \]
Calculating \( A^2 + B^2 \):
\[ A^2 = \left(\frac{13}{2}\right)^2 = \frac{169}{4} \]
\[ B^2 = \left(-\frac{3\sqrt{3}}{2}\right)^2 = \frac{27}{4} \]
\[ A^2 + B^2 = \frac{169}{4} + \frac{27}{4} = \frac{196}{4} = 49 \]
Thus,
\[ \sqrt{A^2 + B^2} = \sqrt{49} = 7 \]
The range of \( A \cos \theta + B \sin \theta \) is:
\[ [-7, 7] \]

### Step 4: Adjust the range with the constant term
Adding 3 to the entire range:
\[ [-7 + 3, 7 + 3] = [-4, 10] \]

### Step 5: Identify values of \( a \) and \( b \)
From the range \( [-4, 10] \), we have:
- \( a = -4 \)
- \( b = 10 \)

### Step 6: Calculate \( a + b \)
\[ a + b = -4 + 10 = 6 \]

### Final Answer
The value of \( a + b \) is \( \boxed{6} \).