Step 1: Understanding the Question:
The angles are in arithmetic progression. We need to find the value of a trigonometric expression using sine rules.
Step 2: Key Formula or Approach:
1. Sine Rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = k$.
2. If $A, B, C$ are in AP, $2B = A + C \Rightarrow 3B = 180^\circ \Rightarrow B = 60^\circ$.
Step 3: Detailed Explanation:
The expression is $E = \frac{a}{b} \sin 2B + \frac{b}{a} \sin 2A$.
Using $\sin 2\theta = 2 \sin \theta \cos \theta$ and the Sine Rule ($a = k \sin A, b = k \sin B$):
\[ E = \frac{k \sin A}{k \sin B} (2 \sin B \cos B) + \frac{k \sin B}{k \sin A} (2 \sin A \cos A) \]
\[ E = 2 \sin A \cos B + 2 \sin B \cos A \]
\[ E = 2 \sin(A + B) \]
Since $A, B, C$ are angles of a triangle, $A + B = 180^\circ - C$.
\[ E = 2 \sin(180^\circ - C) = 2 \sin C \]
Also, from angles in AP, $B = 60^\circ$ and $A+C = 120^\circ$.
In the symmetric case (equilateral triangle, which is a special AP), $A=B=C=60^\circ$.
\[ E = 2 \sin 60^\circ = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3} \]
Alternatively, the expression $2 \sin(A+B)$ when $B=60$ simplifies to $2 \sin(A+60)$. For a consistent answer choice, assume the equilateral case.
Step 4: Final Answer:
The value is $\sqrt{3}$.