To calculate the length $x$ using the sine rule, we can apply the following formula:

$\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$

Where:

• $a$ and $A$ represent one side of the triangle and its opposite angle.
• $b$ and $B$ represent another side and its opposite angle.

In this case:

• $a = 9$ cm (the given side).
• $A = 57^\circ$ (the angle opposite the given side).
• $B = 40^\circ$ (the angle opposite the unknown side $x$).

We need to solve for $x$, so:

$\frac{x}{\sin(40^\circ)} = \frac{9}{\sin(57^\circ)}$

Now let's calculate $x$.The length of $x$, calculated using the sine rule, is approximately $6.9 \, \text{cm}$ to 1 decimal place.

Would you like more details or have any questions?

Here are 5 related questions to consider:

1. How is the sine rule derived from trigonometric principles?
2. What are other scenarios where the sine rule is applicable?
3. How would you solve this if two sides and an angle were given?
4. How does the cosine rule compare to the sine rule?
5. What happens if the angles add up to more or less than $180^\circ$?

Tip: Always ensure that the angles in a triangle sum up to $180^\circ$ when using trigonometric rules in non-right-angled triangles.