The given problem shows a triangle $\triangle LKJ$ where:

• $LJ = 4.7 \, \text{cm}$,
• $LK = 9.2 \, \text{cm}$,
• $\angle JKL = 78^\circ$,
• $\angle KLJ = x^\circ$ (to be calculated).

We can use the sine rule to solve for angle $x$. The sine rule is given as:

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

In this case, we will apply it to sides and angles of $\triangle LKJ$:

• $a = 9.2 \, \text{cm}$ (opposite to angle $\angle JKL = 78^\circ$),
• $b = 4.7 \, \text{cm}$ (opposite to angle $x^\circ$).

Thus, applying the sine rule:

$\frac{9.2}{\sin 78^\circ} = \frac{4.7}{\sin x}$

We can solve this equation for $\sin x$:

$\sin x = \frac{4.7 \cdot \sin 78^\circ}{9.2}$

Now, calculate:

$\sin 78^\circ \approx 0.9781$

$\sin x = \frac{4.7 \times 0.9781}{9.2} = \frac{4.59707}{9.2} \approx 0.4997$

Now, take the inverse sine to find $x$:

$x = \sin^{-1}(0.4997) \approx 29.98^\circ$

Rounded to one decimal place:

$x \approx 30.0^\circ$

Thus, the value of $x$ is approximately $30.0^\circ$.

Would you like more details on any of the steps? Here are some additional questions related to the problem:

1. How is the sine rule applied when all angles are known but only one side?
2. What is the cosine rule, and how does it differ from the sine rule?
3. How do you calculate the area of a triangle using angles and sides?
4. What would happen if the angle was obtuse instead of acute?
5. How does rounding affect the accuracy of trigonometric calculations?

Tip: Always double-check your angle unit (degrees or radians) when using a calculator to avoid errors.