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.