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This is a right triangle where you're asked to find the lengths of two sides, \( X \) and \( Y \), given an angle of \( 30^\circ \) and one side length of 2.

### Known:
- Opposite to the angle \( 30^\circ \): \( Y \) (hypotenuse)
- Adjacent to the angle \( 30^\circ \): \( X \)
- Opposite side (given): 2

We can use trigonometric ratios:

1. **For \( X \) (adjacent side):**
   \[
   \cos(30^\circ) = \frac{X}{Y}
   \]
   But we don't know \( Y \), so first we will calculate \( Y \) using the sine function.

2. **For \( Y \) (hypotenuse):**
   \[
   \sin(30^\circ) = \frac{2}{Y}
   \]
   Since \( \sin(30^\circ) = \frac{1}{2} \), we can solve for \( Y \):
   \[
   \frac{1}{2} = \frac{2}{Y} \quad \Rightarrow \quad Y = 4
   \]

3. **Now, for \( X \) (adjacent side):**
   \[
   \cos(30^\circ) = \frac{X}{4}
   \]
   \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \), so:
   \[
   \frac{\sqrt{3}}{2} = \frac{X}{4} \quad \Rightarrow \quad X = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3}
   \]
   Approximation:
   \[
   X \approx 2 \times 1.732 = 3.464
   \]

### Final Answers:
- \( X \approx 3.464 \)
- \( Y = 4 \)

Would you like further details or have any other questions?

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Here are 5 questions based on this problem:
1. How do you calculate other trigonometric ratios for different angles?
2. What is the exact value of \( \tan(30^\circ) \)?
3. How can you use the Pythagorean theorem to verify the results?
4. What are some applications of trigonometry in real life?
5. How would the solution change if the angle was \( 45^\circ \) instead of \( 30^\circ \)?

**Tip**: For quick reference, memorize key trigonometric values for standard angles like \( 30^\circ \), \( 45^\circ \), and \( 60^\circ \).

Find the lengths of X and Y in a right triangle where one side opposite the angle of 30° is given as 2.

Learn how to solve a right triangle with a 30° angle, where the opposite side is 2. This problem involves using the sine and cosine trigonometric ratios to find the hypotenuse and adjacent sides.

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