https://file.aiquickdraw.com/mathbot/file/1727792779000ik2yry0p.jpg

The image shows a right-angled triangle with the following elements:

- One of the angles is \(30^\circ\).
- The hypotenuse has a length of \(15 \, \text{m}\).
- The sides opposite and adjacent to the \(30^\circ\) angle are labeled as \(y\) and \(x\), respectively.
- The other angle is labeled as \(\alpha\).

To find the unknowns \(x\), \(y\), and \(\alpha\), we can use trigonometric functions:

1. **For side \(x\)** (adjacent to the \(30^\circ\) angle):
   \[
   \cos(30^\circ) = \frac{x}{15}
   \]
   Solving for \(x\):
   \[
   x = 15 \cdot \cos(30^\circ) = 15 \cdot \frac{\sqrt{3}}{2} = 15 \cdot 0.866 = 12.99 \, \text{m}
   \]

2. **For side \(y\)** (opposite the \(30^\circ\) angle):
   \[
   \sin(30^\circ) = \frac{y}{15}
   \]
   Solving for \(y\):
   \[
   y = 15 \cdot \sin(30^\circ) = 15 \cdot 0.5 = 7.5 \, \text{m}
   \]

3. **For angle \(\alpha\)**:
   Since the triangle is a right triangle, the sum of the angles must equal \(180^\circ\). We know one angle is \(30^\circ\), and the right angle is \(90^\circ\), so:
   \[
   \alpha = 180^\circ - 90^\circ - 30^\circ = 60^\circ
   \]

Thus, the solutions are:
- \(x \approx 12.99 \, \text{m}\)
- \(y = 7.5 \, \text{m}\)
- \(\alpha = 60^\circ\)

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. What is the perimeter of this triangle?
2. What is the area of the triangle?
3. How do the trigonometric functions change if the angle \(30^\circ\) is replaced with a different angle?
4. Can you derive the value of \(\alpha\) using trigonometric functions instead of angle sums?
5. How do these calculations change if the hypotenuse is increased?

**Tip:** Always check if the triangle is a right-angled triangle before using trigonometric ratios, as they specifically apply to right-angled triangles.

Find the values of x, y, and α in a right-angled triangle where the hypotenuse is 15 m, one of the angles is 30°, and the opposite and adjacent sides to the 30° angle are labeled as y and x, respectively.

This guide explains how to solve a right-angled triangle problem where the hypotenuse is 15 m, one angle is 30°, and the opposite and adjacent sides are labeled. It covers trigonometric ratios and provides step-by-step solutions for finding the side lengths and the unknown angle α.

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