https://file.aiquickdraw.com/mathbot/file/1730835178507c017xzmq.jpg

To solve this problem using the sine rule, we need to match the provided information in the question to find the missing value.

### Sine Rule Recap
The sine rule states that for any triangle:
\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
\]
where \( a \), \( b \), and \( c \) are the lengths of the sides opposite to angles \( A \), \( B \), and \( C \), respectively.

### Given Data
In this case:
- We have a triangle with sides of 7 cm, 8 cm, and 10 cm, and angles of \( 53^\circ \) and \( 44^\circ \).
- We are asked to find the length in the box in the given equation:
  \[
  \frac{7}{\sin(44^\circ)} = \frac{\text{missing value}}{\sin(53^\circ)}
  \]

### Solving for the Missing Value
Rearrange the equation to solve for the missing value:
\[
\text{missing value} = \frac{7 \cdot \sin(53^\circ)}{\sin(44^\circ)}
\]

1. Calculate \( \sin(53^\circ) \) and \( \sin(44^\circ) \).
   - \( \sin(53^\circ) \approx 0.7986 \)
   - \( \sin(44^\circ) \approx 0.6947 \)

2. Substitute these values:
   \[
   \text{missing value} = \frac{7 \cdot 0.7986}{0.6947} \approx \frac{5.5902}{0.6947} \approx 8.05
   \]

### Answer
The number that goes in the box is approximately **8.05**.

Would you like more details on this solution or have any questions?

### Related Questions to Explore Further
1. How is the sine rule derived, and why does it work in any triangle?
2. How can we use the sine rule to find unknown angles instead of side lengths?
3. What is the cosine rule, and how does it differ from the sine rule?
4. Can the sine rule be applied to right triangles, and if so, how does it simplify?
5. How would you solve this problem if you only had the side lengths and no angles?

### Tip
When using trigonometric values, it’s essential to ensure accuracy by using a calculator or accurate tables, especially in applications needing high precision.

Using the sine rule, write down the number that goes in the box to complete the equation below.

Learn how to use the sine rule to solve for a missing side in a triangle. This example involves a triangle with sides of 7 cm, 8 cm, and 10 cm, and angles of 53° and 44°. The solution includes using the sine rule formula and calculating approximate trigonometric values.

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