In a triangle, when two angles and one side are given, we can use the Law of Sines to find the length of an unknown side.

Given:

• $\angle A = 40^\circ$
• $\angle B = 60^\circ$
• $a = 8 \, \text{cm}$

We need to find the length of side $b$, which is opposite $\angle B$.

Step 1: Find the third angle $\angle C$

Since the sum of all angles in a triangle is 180°: $\angle C = 180^\circ - \angle A - \angle B$ $\angle C = 180^\circ - 40^\circ - 60^\circ = 80^\circ$

Step 2: Use the Law of Sines

The Law of Sines states: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$ We can use this to find $b$: $\frac{b}{\sin(B)} = \frac{a}{\sin(A)}$ $b = a \times \frac{\sin(B)}{\sin(A)}$

Step 3: Calculate the length of side $b$

Substitute the known values: $b = 8 \times \frac{\sin(60^\circ)}{\sin(40^\circ)}$ Now we calculate the sines: $\sin(60^\circ) \approx 0.866 \quad \text{and} \quad \sin(40^\circ) \approx 0.643$ $b = 8 \times \frac{0.866}{0.643} \approx 8 \times 1.347 \approx 10.78 \, \text{cm}$

Thus, the length of side $b$ is approximately 10.78 cm.

Would you like more details or have any questions?

Here are some related questions to expand on:

1. How do you derive the Law of Sines?
2. What happens when all angles and one side are known in a triangle?
3. Can the Law of Sines be used for all types of triangles?
4. What is the difference between the Law of Sines and the Law of Cosines?
5. How do you calculate the area of a triangle using trigonometry?

Tip: The Law of Sines is particularly useful for solving non-right triangles when you know two angles and one side (AAS or ASA cases).