To solve this problem using the sine rule:

The sine rule states: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

From the given diagram:

• $\frac{7}{\sin(44^\circ)} = \frac{x}{\sin(53^\circ)}$
• Rearrange for $x$: $x = \frac{7 \cdot \sin(53^\circ)}{\sin(44^\circ)}$

Now, calculate the values:

1. $\sin(53^\circ) \approx 0.7986$
2. $\sin(44^\circ) \approx 0.6947$

Substitute into the formula: $x = \frac{7 \cdot 0.7986}{0.6947} \approx \frac{5.5902}{0.6947} \approx 8.05$

The number that goes in the box is approximately 8.05.

Would you like further details or have additional questions?

Related Questions:

1. What is the sine rule, and how does it work for non-right triangles?
2. Why is the sine rule particularly useful for this triangle?
3. How do we handle rounding when using trigonometric values?
4. What is the importance of labeling angles and sides correctly in trigonometry problems?
5. How would this method differ if we had only one angle and two sides?

Tip:

Always double-check your angle-sum relationships in triangles to ensure consistent results!