The problem is to find the values of $x$ and $y$ in the given right triangle. The hypotenuse has length $9$ cm, one angle is $55^\circ$, $x$ is the side opposite to the $55^\circ$ angle, and $y$ is the side adjacent to the $55^\circ$ angle.

1. Problem Description

The problem is to find the values of $x$ and $y$ in the given right triangle. The hypotenuse has length $9$ cm, one angle is $55^\circ$, $x$ is the side opposite to the $55^\circ$ angle, and $y$ is the side adjacent to the $55^\circ$ angle.

2. Solution Steps

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

Since we know the hypotenuse and want to find the side opposite the $55^\circ$ angle, we can use the sine function:

$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

$\sin(55^\circ) = \frac{x}{9}$

$x = 9 \sin(55^\circ)$

Since we know the hypotenuse and want to find the side adjacent to the $55^\circ$ angle, we can use the cosine function:

$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

$\cos(55^\circ) = \frac{y}{9}$

$y = 9 \cos(55^\circ)$

Using a calculator, we find that $\sin(55^\circ) \approx 0.819$ and $\cos(55^\circ) \approx 0.574$.

Therefore, $x = 9 \times 0.819 \approx 7.371$ and $y = 9 \times 0.574 \approx 5.166$.

3. Final Answer

$x = 9 \sin(55^\circ)$

$y = 9 \cos(55^\circ)$