To solve for the unknown angles in triangle $ABC$ given that:

• $B = 158.2^\circ$
• $c = 8.4$
• $b = 14.8$

we can proceed as follows:

1. Check for Validity: Since the angle $B$ is very large (close to $180^\circ$), we first verify that the triangle is valid. In a triangle, the sum of all angles must equal $180^\circ$. This leaves only $21.8^\circ$ for angles $A$ and $C$, combined. This may limit the possible configurations of the triangle.

2. Use the Law of Sines: We can apply the Law of Sines to find one of the other angles.

   The Law of Sines states: $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

   Rearranging to find $\sin C$: $\sin C = \frac{c \cdot \sin B}{b}$

   Substituting in the values: $\sin C = \frac{8.4 \cdot \sin(158.2^\circ)}{14.8}$

3. Calculate $\sin(158.2^\circ)$: $\sin(158.2^\circ) \approx 0.3746$ Plugging this into the equation: $\sin C = \frac{8.4 \cdot 0.3746}{14.8} \approx 0.2128$

4. Find $C$: Now we take the inverse sine to find $C$: $C \approx \arcsin(0.2128) \approx 12.3^\circ$

5. Calculate Angle $A$: Since $A + B + C = 180^\circ$, $A = 180^\circ - B - C = 180^\circ - 158.2^\circ - 12.3^\circ \approx 9.5^\circ$

Summary

The angles in triangle $ABC$ are approximately:

• $A \approx 9.5^\circ$
• $B = 158.2^\circ$
• $C \approx 12.3^\circ$

Would you like more details on any of these steps, or do you have any questions?

Here are five related questions for further exploration:

1. How does the Law of Sines apply to obtuse triangles?
2. What other methods could be used to solve for unknown angles in a triangle?
3. How does the triangle change if angle $B$ were an acute angle?
4. What happens if the side lengths $b$ and $c$ are different?
5. How would we solve for side $a$ in this triangle?

Tip: Remember, the sum of the angles in any triangle is always $180^\circ$. This fact can help you check the validity of your answers.