Step 1: Find the third angle of the triangle. The sum of angles in a triangle is $180^\circ$. Let the unknown angle be $\theta$. $$ \theta = 180^\circ - (60^\circ + 30^\circ) $$ $$ \theta = 180^\circ - 90^\circ $$ $$ \theta = 90^\circ $$ This means the triangle is a right-angled triangle. Step 2: Identify the sides and angles in relation to the right-angled triangle. Let the angle opposite side $x$ be $30^\circ$. Let the angle opposite the side $8$ cm be $90^\circ$. This means $8$ cm is the hypotenuse. We need to find $x$, which is the side opposite the $30^\circ$ angle. Step 3: Use the sine function to find $x$. The sine of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the hypotenuse. $$ \sin(\text{angle}) = \frac{\text{opposite}}{\text{hypotenuse}} $$ For the $30^\circ$ angle: $$ \sin(30^\circ) = \frac{x}{8} $$ We know that $\sin(30^\circ) = \frac{1}{2}$. $$ \frac{1}{2} = \frac{x}{8} $$ Step 4: Solve for $x$. Multiply both sides by $8$: $$ x = 8 \times \frac{1}{2} $$ $$ x = 4 $$ The unit for $x$ is cm. The value of $x$ is $4$ cm. $$ x = \boxed{\text{4 cm}} $$