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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.

Find the exact values of x and y given that s = 9 and the triangle includes a 30° angle.

Learn how to find the exact values of the sides of a right triangle using trigonometric functions. This problem involves using sine and cosine to calculate the sides opposite and adjacent to a 30° angle, with the hypotenuse given as 9.

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