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To solve for \( x \) in this right triangle, we can apply trigonometric principles since one of the angles is given.

### Step-by-Step Solution:

1. **Given Information:**
   - The hypotenuse \( 12 \, \text{cm} \)
   - Angle \( 40^\circ \)
   - Opposite side to the angle is \( x \, \text{cm} \).

2. **Using Sine Function:**
   Since we know the opposite side \( x \) and the hypotenuse, we can use the sine function:
   \[
   \sin(40^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{x}{12}
   \]

3. **Solving for \( x \):**
   \[
   x = 12 \times \sin(40^\circ)
   \]

4. **Calculating the value:**
   Let me calculate this value.

The value of \( x \) is approximately \( 7.71 \, \text{cm} \) when rounded to two decimal places.

### Final Answer:
- **Simplified radical form**: There is no simplified radical form since the sine of \( 40^\circ \) doesn't simplify neatly into a radical.
- **Decimal answer**: \( 7.71 \, \text{cm} \).

Would you like more details or have any further questions?

### Relative Questions:
1. How would you solve for the other side of the triangle using trigonometry?
2. How can you verify the result using the Pythagorean theorem?
3. What are the other trigonometric functions that could help in this problem?
4. How does the sine function relate to other trigonometric ratios?
5. How would the solution change if the angle were larger or smaller?

### Tip:
In right triangles, always check which sides are given (opposite, adjacent, or hypotenuse) to choose the correct trigonometric ratio.

Solve for x in the right triangle with a hypotenuse of 12 cm and an angle of 40 degrees.

Learn how to solve for the unknown side of a right triangle using trigonometric principles, specifically the sine function. Given a hypotenuse of 12 cm and an angle of 40 degrees, this problem demonstrates how to calculate the opposite side using sin(40°) = opposite/hypotenuse.

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