We are given a right triangle with a hypotenuse $s = 9$ and an angle of $30^\circ$. We need to find the values of $x$ (the opposite side to the $30^\circ$ angle) and $y$ (the adjacent side to the $30^\circ$ angle).

We can use trigonometric functions to find the values of $x$ and $y$.

1. Finding $x$ (opposite side):

We use the sine function: $\sin(30^\circ) = \frac{x}{s}$ Since $\sin(30^\circ) = \frac{1}{2}$, we have: $\frac{1}{2} = \frac{x}{9}$ Solving for $x$: $x = \frac{9}{2} = 4.5$

2. Finding $y$ (adjacent side):

We use the cosine function: $\cos(30^\circ) = \frac{y}{s}$ Since $\cos(30^\circ) = \frac{\sqrt{3}}{2}$, we have: $\frac{\sqrt{3}}{2} = \frac{y}{9}$ Solving for $y$: $y = 9 \times \frac{\sqrt{3}}{2} = \frac{9\sqrt{3}}{2}$

Therefore, the exact values are: $x = 4.5$ $y = \frac{9\sqrt{3}}{2}$

Would you like any further details or explanations?

Relative Questions:

1. How do you use trigonometric ratios to solve for unknown sides in a right triangle?
2. What are the sine and cosine values for other common angles like $45^\circ$ and $60^\circ$?
3. Can you solve for the area of the triangle using the sides we found?
4. How would you solve this problem if a different angle was given, such as $45^\circ$?
5. What is the relationship between the sides of a 30°-60°-90° triangle?

Tip:

In a 30°-60°-90° triangle, the side opposite the 30° angle is always half of the hypotenuse, while the side opposite the 60° angle is $\frac{\sqrt{3}}{2}$ times the hypotenuse.