Answer
$b = 731 \hspace{1pt} m$
$A = 15.6^\circ$
$C = 12.9^\circ$
Work Step by Step
Using the law of Cosines:
$$b^2 = a^2+c^2 -2ac \cos{B} \\ b^2 = (412)^2+(342)^2 - 2\times 412 \times 342 \cos{151.5^\circ} \\ b^2 = 534361$$
$$\therefore b = \boxed{731 \hspace{1pt} m}$$
Using the law of Sines:
$$\dfrac{\sin{B}}{b} = \dfrac{\sin{A}}{a} = \dfrac{\sin{C}}{c}$$
$$\sin{A} = \dfrac{a\times \sin{B}}{b}= \dfrac{412 \times \sin{151.5^\circ}}{731} = 0.2689 $$
$$\therefore A = \boxed{15.6^\circ}$$
$$\sin{C} = \dfrac{c\times \sin{B}}{b}= \dfrac{342 \times \sin{151.5^\circ}}{731} = 0.2232$$
$$\therefore C = \boxed{12.9^\circ}$$
