The problem involves solving a triangle given the following information: $a = 7.82$ cm, $b = 14.35$ cm, and angle $B = 115^{\circ} 120'' = 117^{\circ}$. First, the angle A is calculated using the Law of Sines. Second, the angle C is calculated. Then, the side c is calculated using Law of Sines.

1. Problem Description

The problem involves solving a triangle given the following information:

$a = 7.82$ cm, $b = 14.35$ cm, and angle $B = 115^{\circ} 120'' = 117^{\circ}$.

First, the angle A is calculated using the Law of Sines.

Second, the angle C is calculated. Then, the side c is calculated using Law of Sines.

2. Solution Steps

First, we use the Law of Sines to find angle $A$:

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

$\frac{7.82}{\sin A} = \frac{14.35}{\sin 117^{\circ}}$

$\sin A = \frac{7.82 \sin 117^{\circ}}{14.35}$

$A = \arcsin(\frac{7.82 \sin 117^{\circ}}{14.35})$

$A = \arcsin(0.4874)$

$A \approx 29.16^{\circ}$

The handwritten notes shows A as approximately equal to $28.1^{\circ}$. The value obtained in the image is $A = 28.11306998$, so it has been rounded to $28.1^{\circ}$.

Second, we calculate angle $C$ using the fact that the sum of angles in a triangle is $180^{\circ}$:

$A + B + C = 180^{\circ}$

$C = 180^{\circ} - A - B$

$C = 180^{\circ} - 28.1^{\circ} - 117^{\circ}$

$C = 34.9^{\circ}$

Next, we calculate side $c$ using the Law of Sines:

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

$\frac{c}{\sin 34.9^{\circ}} = \frac{14.35}{\sin 117^{\circ}}$

$c = \frac{14.35 \sin 34.9^{\circ}}{\sin 117^{\circ}}$

$c = \frac{14.35(0.5721)}{0.891}$

$c \approx 9.21$ cm

3. Final Answer

$A = 28.1^{\circ}$

$C = 34.9^{\circ}$

$c = 9.21$ cm