The problem involves a right triangle with hypotenuse $9$ cm and one angle $55^\circ$. We are asked to find the length of the opposite side (denoted as $x$) and the adjacent side (denoted as $y$) to this angle.

1. Problem Description

The problem involves a right triangle with hypotenuse $9$ cm and one angle $55^\circ$. We are asked to find the length of the opposite side (denoted as $x$) and the adjacent side (denoted as $y$) to this angle.

2. Solution Steps

First, we find $x$, which is the side opposite to the $55^\circ$ angle. We use the sine function:

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

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

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

$x = 9 \cdot 0.81915 \approx 7.372$ cm.

Then, we find $y$, which is the side adjacent to the $55^\circ$ angle. The given solution appears to have mistakenly swapped the adjacent and opposite when calculating $y$ using the tangent function. The correct application of tangent is: $tan(\theta) = \frac{opposite}{adjacent}$. We already calculated the opposite side, $x = 7.372$. Therefore,

$tan(55^\circ) = \frac{7.372}{y}$

$y = \frac{7.372}{tan(55^\circ)}$

Since $tan(55^\circ) \approx 1.4281$,

$y = \frac{7.372}{1.4281} \approx 5.162$ cm.

Note: we can also solve for $y$ using cosine, and $cos(\theta) = \frac{adjacent}{hypotenuse}$

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

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

$y = 9 \cdot 0.57358 \approx 5.162$ cm.

3. Final Answer

$x = 9 sin(55^\circ) \approx 7.372$ cm

$y = \frac{7.372}{tan(55^\circ)} \approx 5.162$ cm