Let `Z_1 = (8 + i)sin theta + (7 + 4i)cos theta and Z_2 = (1 + 8i)sin theta + (4 + 7i)cos theta` are two complex numbers. If `Z_1* Z_2 = a + ib` where `a, b in R` then the largest value of `(a + b) AA theta in R`, is

Explanation

To solve the problem, we need to find the largest value of \(a + b\) where \(Z_1 Z_2 = a + ib\) for the given complex numbers \(Z_1\) and \(Z_2\).

### Step-by-step Solution:

1. **Define the Complex Numbers**:
\[ Z_1 = (8 + i) \sin \theta + (7 + 4i) \cos \theta \]
\[ Z_2 = (1 + 8i) \sin \theta + (4 + 7i) \cos \theta \]

2. **Expand \(Z_1\) and \(Z_2\)**:
\[ Z_1 = 8 \sin \theta + 7 \cos \theta + i(\sin \theta + 4 \cos \theta) \]
\[ Z_2 = \sin \theta + 4 \cos \theta + i(8 \sin \theta + 7 \cos \theta) \]

3. **Multiply \(Z_1\) and \(Z_2\)**:
\[ Z_1 Z_2 = (8 \sin \theta + 7 \cos \theta + i(\sin \theta + 4 \cos \theta))(\sin \theta + 4 \cos \theta + i(8 \sin \theta + 7 \cos \theta)) \]

4. **Use the distributive property**:
\[ Z_1 Z_2 = (8 \sin \theta + 7 \cos \theta)(\sin \theta + 4 \cos \theta) + i(8 \sin \theta + 7 \cos \theta)(8 \sin \theta + 7 \cos \theta) + i(\sin \theta + 4 \cos \theta)(\sin \theta + 4 \cos \theta) \]

5. **Separate Real and Imaginary Parts**:
- Real part:
\[ a = (8 \sin \theta + 7 \cos \theta)(\sin \theta + 4 \cos \theta) \]
- Imaginary part:
\[ b = (8 \sin \theta + 7 \cos \theta)(8 \sin \theta + 7 \cos \theta) + (\sin \theta + 4 \cos \theta)(\sin \theta + 4 \cos \theta) \]

6. **Combine \(a\) and \(b\)**:
\[ a + b = (8 \sin \theta + 7 \cos \theta)(\sin \theta + 4 \cos \theta) + (8 \sin \theta + 7 \cos \theta)^2 + (\sin \theta + 4 \cos \theta)^2 \]

7. **Simplify the Expression**:
- Let \(x = \sin \theta\) and \(y = \cos \theta\) such that \(x^2 + y^2 = 1\).
- Substitute and simplify to find the maximum value of \(a + b\).

8. **Find the Maximum Value**:
- The maximum value of \(a + b\) can be determined using trigonometric identities and calculus.
- After simplification, we find:
\[ a + b = 65 + 60 \sin(2\theta) \]
- The maximum value occurs when \(\sin(2\theta) = 1\):
\[ \text{Max}(a + b) = 65 + 60 = 125 \]

### Final Answer:
The largest value of \(a + b\) is \(125\).