The problem is to find the lengths of the sides $x$ and $y$ of a right triangle, given that the hypotenuse is $9$ cm and one of the angles is $55^\circ$. The side $x$ is opposite to the $55^\circ$ angle, and the side $y$ is adjacent to the $55^\circ$ angle.

1. Problem Description

The problem is to find the lengths of the sides $x$ and $y$ of a right triangle, given that the hypotenuse is $9$ cm and one of the angles is $55^\circ$. The side $x$ is opposite to the $55^\circ$ angle, and the side $y$ is adjacent to the $55^\circ$ angle.

2. Solution Steps

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

To find $x$, we can use the sine function:

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

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

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

To find $y$, we can use the cosine function:

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

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

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

Using a calculator, we find:

$\sin(55^\circ) \approx 0.819$

$\cos(55^\circ) \approx 0.574$

Therefore:

$x \approx 9 \cdot 0.819 \approx 7.371$

$y \approx 9 \cdot 0.574 \approx 5.166$

3. Final Answer

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

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