The problem involves finding the length of the side $x$ of a triangle using the Law of Sines, which states:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Given:

• Side $a = 23$, opposite angle $A = 54^\circ$
• Side $x$, opposite angle $B = 16^\circ$

We apply the formula:

$\frac{x}{\sin(16^\circ)} = \frac{23}{\sin(54^\circ)}$

Solving for $x$:

Rearranging the equation:

$x = \frac{23 \cdot \sin(16^\circ)}{\sin(54^\circ)}$

Calculations:

1. Find $\sin(16^\circ)$ and $\sin(54^\circ)$:

   • $\sin(16^\circ) \approx 0.2756$
   • $\sin(54^\circ) \approx 0.8090$

2. Substitute these values: $x = \frac{23 \cdot 0.2756}{0.8090} \approx \frac{6.3388}{0.8090} \approx 7.8348$

Final Answer:

$x = 7.8348 \, \text{(rounded to 4 decimal places)}$

Would you like more details or have further questions?

Questions for further exploration:

1. How does the Law of Sines relate to the Law of Cosines?
2. What is the significance of knowing the sum of angles in a triangle?
3. Can this approach be used for non-acute triangles?
4. What happens if the triangle is a right triangle—how would you approach solving for unknown sides differently?
5. How can this method be applied in real-world scenarios, such as navigation or construction?

Tip: Always ensure angle measurements are in the correct unit (degrees or radians) before performing trigonometric calculations.