+(sin(1e-3*PI*X/WS*1e3)/H+sin(1e-3*PI*Y/HS*1e3)/W)*0.45e3

The given expression is:

\[
\left( \frac{\sin\left(\frac{10^{-3} \pi X}{WS} \cdot 10^3\right)}{H} + \frac{\sin\left(\frac{10^{-3} \pi Y}{HS} \cdot 10^3\right)}{W} \right) \cdot 0.45 \cdot 10^3
\]

Let’s simplify it step by step.

### Step-by-step simplification:

1. **Break down the trigonometric terms:**

   - The first part involves the sine function:

   \[
   \frac{\sin\left(\frac{10^{-3} \pi X}{WS} \cdot 10^3\right)}{H} = \frac{\sin\left(\frac{\pi X}{WS}\right)}{H}
   \]

   - Similarly, for the second part:

   \[
   \frac{\sin\left(\frac{10^{-3} \pi Y}{HS} \cdot 10^3\right)}{W} = \frac{\sin\left(\frac{\pi Y}{HS}\right)}{W}
   \]

2. **Combine the two simplified terms:**

   The expression now becomes:

   \[
   \left( \frac{\sin\left(\frac{\pi X}{WS}\right)}{H} + \frac{\sin\left(\frac{\pi Y}{HS}\right)}{W} \right) \cdot 0.45 \cdot 10^3
   \]

3. **Factor out the constants:**

   Finally, multiplying by the constant \( 0.45 \cdot 10^3 = 450 \), we get:

   \[
   450 \cdot \left( \frac{\sin\left(\frac{\pi X}{WS}\right)}{H} + \frac{\sin\left(\frac{\pi Y}{HS}\right)}{W} \right)
   \]

This is the simplified form of the expression.

---

Do you have any questions or would you like further details?

Here are some related questions for deeper understanding:
1. How can you calculate the value of this expression for specific \( X \), \( Y \), \( WS \), and \( HS \)?
2. What is the significance of the trigonometric terms in this equation?
3. How would the result change if \( H \) and \( W \) were equal?
4. What role does the constant \( 0.45 \times 10^3 \) play in the scaling of the final result?
5. How would the graph of this function look for different values of \( WS \) and \( HS \)?

**Tip:** Sine functions are periodic, meaning they repeat their values over regular intervals. This property is often used in wave-like patterns.

Learn how to simplify a trigonometric expression involving sine functions and constants. This problem covers key mathematical concepts such as trigonometry and algebra, making it ideal for students in Grades 10-12.

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