To find the value of angle $B$ in a non-right-angled triangle, we can use the Law of Sines, which states:

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

Given:

• Side $a = 8.4$
• Side $b = 10.5$
• Angle $A = 22.3^\circ$

We want to find angle $B$. According to the Law of Sines, we have:

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

Substituting the known values, we get:

$\frac{8.4}{\sin 22.3^\circ} = \frac{10.5}{\sin B}$

To find $\sin B$, rearrange the equation:

$\sin B = \frac{10.5 \sin 22.3^\circ}{8.4}$

Let's calculate $\sin B$ and then use the inverse sine function to find $B$.The value of angle $B$ is approximately $28.3^\circ$ when rounded to one decimal place.

Would you like a detailed explanation, or do you have any questions about the solution?

Here are some related questions you might find interesting:

1. How can we find the third side $c$ of the triangle using the Law of Cosines?
2. What is the Law of Cosines, and how is it applied in non-right-angled triangles?
3. How do you find the remaining angle $C$ of the triangle?
4. What is the importance of knowing all three angles in a triangle?
5. How can the area of this triangle be calculated?
6. What are some real-life applications of the Law of Sines?
7. How does the Law of Sines help in solving problems involving oblique triangles?
8. Can the Law of Sines be used to solve right-angled triangles as well?

Tip: When using the Law of Sines, always check if the calculated sine value leads to a valid angle.