To solve for $x$ in this triangle using the given information, we can apply the sine rule since the triangle is non-right-angled. The sine rule states:

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

Where:

• $a$, $b$, and $c$ are the lengths of the sides of the triangle,
• $A$, $B$, and $C$ are the angles opposite those sides.

In the given triangle:

• One side $b = 12 \, \text{cm}$,
• The angle opposite this side is $B = 86^\circ$,
• Another angle $A = 42^\circ$ is given, and we are solving for $x$, the side opposite to this angle.

We use the sine rule:

$\frac{x}{\sin 42^\circ} = \frac{12}{\sin 86^\circ}$

Steps:

1. Calculate $\sin 42^\circ$ and $\sin 86^\circ$.
2. Rearrange to solve for $x$.

$x = \frac{12 \times \sin 42^\circ}{\sin 86^\circ}$

Let’s calculate the value of $x$.

$\sin 42^\circ \approx 0.6691$ $\sin 86^\circ \approx 0.9986$

Now substitute these values:

$x = \frac{12 \times 0.6691}{0.9986} \approx \frac{8.0292}{0.9986} \approx 8.04 \, \text{cm}$

Final Answer:

The value of $x$ is approximately $8.0 \, \text{cm}$ (correct to 1 decimal place).

Would you like further details or clarification? Here are some related questions:

1. What is the sine rule used for solving triangles?
2. How can we use the cosine rule in non-right-angled triangles?
3. What is the general form of the sine and cosine rules?
4. Can the sine rule be used if no angles are known in the triangle?
5. How do you determine which rule to apply (sine vs. cosine) in a non-right triangle?

Tip: Always ensure that your angles and sides correspond properly when using trigonometric rules to avoid calculation errors.