The problem involves determining the value of $z$ in a right-angled triangle, where the given angle is $54^\circ$ and the opposite side length is 8 cm. The length $z$ is the adjacent side to the angle $54^\circ$.

To solve for $z$, we can use the trigonometric function tangent (tan), which relates the opposite side to the adjacent side in a right-angled triangle:

$\text{tan}(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Substituting the given values:

$\text{tan}(54^\circ) = \frac{8}{z}$

To solve for