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The use of \(\sin(45^\circ)\) twice in part (b) of the problem refers to two different force components being resolved along two different directions.

Here’s the explanation:

1. **Vertical equilibrium**: 
   The first equation involves the forces in the vertical direction, where \(\sin(45^\circ)\) is used to resolve the upward component of the force \(S\). The equation \[ V - mg + S \cdot \sin(45^\circ) = 0 \] describes the balance of vertical forces:
   - \(V\): Vertical reaction at point \(A\),
   - \(mg\): Weight of the object (acting downwards),
   - \(S \cdot \sin(45^\circ)\): Vertical component of the force \(S\), acting upwards at 45 degrees.

2. **Horizontal equilibrium**:
   The second time \(\sin(45^\circ)\) appears is in the **moment equation** about point \(A\). The force \(S\) has a horizontal component that creates a moment at point \(A\):
   \[ 26 \cdot mg - 40 \cdot S \cdot \sin(45^\circ) = 0 \]
   This equation ensures that the system is in rotational equilibrium. Here, \(S \cdot \sin(45^\circ)\) represents the force creating the moment about point \(A\).

In summary, the same angle \(45^\circ\) applies to two different components:
- In the **vertical direction**, for vertical force balance.
- In the **moment balance** equation, for calculating the moment about point \(A\).

Do you want more details on the individual components or the solution? Here are some questions that may help deepen understanding:

1. Why is \(\cos(45^\circ)\) not used in the vertical force equation?
2. What role does the moment equation play in solving this problem?
3. How does the horizontal force component \(S \cdot \cos(45^\circ)\) affect the solution?
4. What assumptions are made about the force \(S\) and its angle?
5. How would changing the angle \(45^\circ\) affect the equilibrium conditions?

**Tip**: When dealing with force vectors, it's essential to decompose them into components along the horizontal and vertical axes, especially when they are not aligned directly along those axes.

Explore the use of sin(45°) in statics problems, specifically when analyzing force equilibrium and moments. This problem involves breaking down forces into components for vertical and rotational equilibrium. Suitable for university-level engineering students.

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