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.