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The question asks which of the options is **incorrect** for solving \( \sin(x) = 0.5 \) for \( x \).

Let's analyze each option:

1. \( x = \arcsin(0.5) \):  
   This is correct because \( \arcsin \) is the inverse of the sine function. If \( \sin(x) = 0.5 \), then \( x = \arcsin(0.5) \).

2. \( x = \sin^{-1}(0.5) \):  
   This is another correct way to represent \( \arcsin(0.5) \), since \( \sin^{-1} \) and \( \arcsin \) both refer to the inverse sine function.

3. \( x = \frac{0.5}{\sin} \):  
   This is incorrect. This expression does not make sense because you cannot divide by \( \sin \) as if it were a number or variable. The sine function cannot be treated in this way. This is the **incorrect** option.

4. \( x = \arcsin\left(\frac{1}{2}\right) \):  
   This is correct because \( \frac{1}{2} \) is equivalent to 0.5, so this is just another way of writing \( \arcsin(0.5) \).

### Conclusion:  
The incorrect option is **\( x = \frac{0.5}{\sin} \)**.

Would you like further details or explanations? Here are some related questions to explore:

1. What are the general solutions for \( \sin(x) = 0.5 \) on a given interval?
2. How does the arcsine function behave in different quadrants?
3. What is the difference between \( \arcsin(x) \) and \( \sin^{-1}(x) \)?
4. Can you apply similar techniques to solve \( \cos(x) = 0.5 \)?
5. What are the ranges of the sine and arcsine functions?

**Tip:** Always remember that inverse functions "undo" the original trigonometric functions but have restricted ranges!

¿Cuál de los siguientes despejes para 'x' es INCORRECTO si: \( \sin(x) = 0.5 \)?

Discover the correct method to solve the equation sin(x) = 0.5. Analyze multiple inverse trigonometric representations and identify which option is incorrect. This detailed breakdown covers key trigonometric concepts and inverse functions. Ideal for high school students studying trigonometry.

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