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.

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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.