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To solve for \( x \) and \( y \) in the right triangle with a 30° angle, we can use trigonometric ratios. Given the hypotenuse length of 42, we can use the following trigonometric properties:

1. **For \( y \):** \( y \) is the side opposite the 30° angle.
   - In a 30°-60°-90° triangle, the ratio of the opposite side to the hypotenuse is \( \sin(30^\circ) = \frac{1}{2} \).
   - Therefore, \( y = 42 \times \sin(30^\circ) = 42 \times \frac{1}{2} = 21 \).

2. **For \( x \):** \( x \) is the side adjacent to the 30° angle.
   - In a 30°-60°-90° triangle, the ratio of the adjacent side to the hypotenuse is \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \).
   - Therefore, \( x = 42 \times \cos(30^\circ) = 42 \times \frac{\sqrt{3}}{2} = 21\sqrt{3} \).

### Final Answer:
- \( x = 21\sqrt{3} \)
- \( y = 21 \)

Would you like more details or have any questions?

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Here are some related questions to explore:

1. How do you find the length of a side in any right triangle given an angle and the hypotenuse?
2. What are the properties of a 30°-60°-90° triangle?
3. How can you verify the solution using the Pythagorean theorem?
4. What are other real-world applications of 30°-60°-90° triangles?
5. How can trigonometric identities help in solving similar triangle problems?

**Tip:** In a 30°-60°-90° triangle, remember that the sides are in a specific ratio: 1 : \( \sqrt{3} \) : 2. This can simplify solving problems like this one without recalculating trigonometric values.

Find the value of x and y in a right triangle where the hypotenuse is 42, the angle is 30°, and the answers should be in simplest radical form.

Learn how to find the values of x and y in a 30°-60°-90° triangle with a hypotenuse of 42. This problem involves using trigonometric ratios and properties specific to 30°-60°-90° triangles, providing a step-by-step solution suitable for students in Grades 9-10.

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