The problem presented is about a triangle $\triangle ABC$, where:

• $\angle A = 60^\circ$
• $\angle B = 45^\circ$
• $BC = 12 \, \text{cm}$

We are asked to find the length of side $AC$.

Approach:

1. Use the Law of Sines to solve for side $AC$. The Law of Sines states: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ Where $a, b, c$ are the lengths of sides opposite to angles $A, B, C$ respectively.

2. First, determine angle $C$ using the fact that the sum of angles in a triangle is $180^\circ$: $\angle C = 180^\circ - 60^\circ - 45^\circ = 75^\circ$

3. Using the Law of Sines, express the ratio for side $AC$ as: $\frac{AC}{\sin B} = \frac{BC}{\sin C}$ Therefore, $AC = \frac{BC \cdot \sin B}{\sin C}$ Substitute the known values: $AC = \frac{12 \cdot \sin 45^\circ}{\sin 75^\circ}$

4. Now, let's compute the values for $\sin 45^\circ$ and $\sin 75^\circ$.

• $\sin 45^\circ = \frac{\sqrt{2}}{2}$
• $\sin 75^\circ = \sin(90^\circ - 15^\circ) = \cos 15^\circ$, and using trigonometric identities, $\cos 15^\circ = \frac{\sqrt{6} + \sqrt{2}}{4}$.

I will now compute the final value for $AC$.The length of side $AC$ is approximately $8.78 \, \text{cm}$.

Looking at the answer choices, none of them are a direct match for this value. However, this result might suggest a simplification based on different expressions involving radicals. Let's explore further details to find the best fit if needed.

Would you like to go deeper into the calculations or explore any other related details?

Here are 5 questions that expand on this topic:

1. How can we apply the Law of Cosines in this problem?
2. What are some common trigonometric identities that help simplify expressions involving angles?
3. How does changing the angles in a triangle affect its side lengths?
4. What is the significance of the Law of Sines in non-right triangles?
5. Can we use a different method, like the Law of Cosines, to verify the result?

Tip: For triangles, remember that trigonometric identities like $\sin(A + B) = \sin(A) \cos(B) + \cos(A) \sin(B)$ are useful when simplifying angle expressions.