The question involves solving for the unknown side $x$ in the triangle using the Law of Sines. Let's extract the information and solve it step by step:

Given Information:

1. $\angle Y = 55^\circ$
2. $\angle Z = 47^\circ$
3. Side $YZ = 60 \, \text{cm}$.
4. Unknown: $x$, the length of side $XY$.

Steps to Solve:

Step 1: Find $\angle X$

The sum of angles in a triangle is $180^\circ$.
$\angle X = 180^\circ - \angle Y - \angle Z = 180^\circ - 55^\circ - 47^\circ = 78^\circ.$

Step 2: Apply the Law of Sines

The Law of Sines states: $\frac{\sin \angle Y}{x} = \frac{\sin \angle Z}{60}.$

Rearranging for $x$: $x = \frac{60 \cdot \sin \angle Y}{\sin \angle Z}.$

Step 3: Calculate the Values

Using a scientific calculator:

• $\sin 55^\circ \approx 0.8192$,
• $\sin 47^\circ \approx 0.7314$.

Substitute these into the equation: $x = \frac{60 \cdot 0.8192}{0.7314} \approx \frac{49.152}{0.7314} \approx 67.2 \, \text{cm}.$

Final Answer:

The unknown side $x$ is approximately 67.2 cm.

Would you like further clarification on any step or need assistance with related problems? Here are some related questions you might explore:

1. How do you calculate the area of a triangle using trigonometry?
2. Can we solve this problem using the Law of Cosines instead of the Law of Sines?
3. What is the significance of the Law of Sines in non-right triangles?
4. How does the angle measurement affect side length in a triangle?
5. How would you solve this if the angles were measured in radians?

Tip: Always ensure your calculator is set to the correct mode (degrees or radians) based on the problem's requirements.