The maximum value of `5 sin theta+3 sin (theta + pi/3) + 3` is -

Explanation

To find the maximum value of the expression \( 5 \sin \theta + 3 \sin \left( \theta + \frac{\pi}{3} \right) + 3 \), we can follow these steps:

### Step 1: Use the sine addition formula
We start by applying the sine addition formula for \( \sin \left( \theta + \frac{\pi}{3} \right) \):
\[ \sin \left( \theta + \frac{\pi}{3} \right) = \sin \theta \cos \frac{\pi}{3} + \cos \theta \sin \frac{\pi}{3} \]
Using the values \( \cos \frac{\pi}{3} = \frac{1}{2} \) and \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \), we can rewrite the expression:
\[ \sin \left( \theta + \frac{\pi}{3} \right) = \sin \theta \cdot \frac{1}{2} + \cos \theta \cdot \frac{\sqrt{3}}{2} \]

### Step 2: Substitute back into the original expression
Now, substituting this back into the original expression:
\[ 5 \sin \theta + 3 \left( \frac{1}{2} \sin \theta + \frac{\sqrt{3}}{2} \cos \theta \right) + 3 \]
This simplifies to:
\[ 5 \sin \theta + \frac{3}{2} \sin \theta + \frac{3\sqrt{3}}{2} \cos \theta + 3 \]
Combining the sine terms:
\[ \left( 5 + \frac{3}{2} \right) \sin \theta + \frac{3\sqrt{3}}{2} \cos \theta + 3 = \frac{13}{2} \sin \theta + \frac{3\sqrt{3}}{2} \cos \theta + 3 \]

### Step 3: Identify the maximum value of the sine and cosine terms
The expression \( A \sin \theta + B \cos \theta \) has a maximum value of \( \sqrt{A^2 + B^2} \). Here, \( A = \frac{13}{2} \) and \( B = \frac{3\sqrt{3}}{2} \).

Calculating \( A^2 + B^2 \):
\[ A^2 = \left( \frac{13}{2} \right)^2 = \frac{169}{4} \]
\[ B^2 = \left( \frac{3\sqrt{3}}{2} \right)^2 = \frac{27}{4} \]
Adding these:
\[ A^2 + B^2 = \frac{169}{4} + \frac{27}{4} = \frac{196}{4} = 49 \]

### Step 4: Calculate the maximum value
Thus, the maximum value of \( A \sin \theta + B \cos \theta \) is:
\[ \sqrt{49} = 7 \]

### Step 5: Add the constant term
Finally, we add the constant term from our expression:
\[ \text{Maximum value} = 7 + 3 = 10 \]

### Conclusion
The maximum value of \( 5 \sin \theta + 3 \sin \left( \theta + \frac{\pi}{3} \right) + 3 \) is \( \boxed{10} \).