Answer
$a=75$ $;$ $b=100$ $;$ $c\approx142.504$
$\angle A=30^{\circ}$ $;$ $\angle B\approx41.81^{\circ}$ $;$ $\angle C\approx108.19^{\circ}$
Work Step by Step
$a=75$ $,$ $b=100$ $,$ $\angle A=30^{\circ}$
The Law of Sines is $\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$
Find the angle $B$ by using the formula $\dfrac{\sin A}{a}=\dfrac{\sin B}{b}$, obtained from the Law of Sines. Substitute the known values into the formula and solve for $B$:
$\dfrac{\sin 30^{\circ}}{75}=\dfrac{\sin B}{100}$
$\sin B=\Big(\dfrac{100}{75}\Big)\sin30^{\circ}$
$B=\sin^{-1}\Big[\Big(\dfrac{100}{75}\Big)\sin30^{\circ}\Big]\approx41.81^{\circ}$
Two angles are now known. Since $\angle A+\angle B+\angle C=180^{\circ}$, substitute the known angles into the formula and solve for $\angle C$:
$\angle C=180^{\circ}-30^{\circ}-41.81^{\circ}\approx108.19^{\circ}$
Find the side $c$ by using the formula $\dfrac{\sin A}{a}=\dfrac{\sin C}{c}$, obtained from the Law of Sines. Substitute the known values and solve for $c$:
$\dfrac{\sin30^{\circ}}{75}=\dfrac{\sin108.19^{\circ}}{c}$
$\dfrac{75}{\sin30^{\circ}}=\dfrac{c}{\sin108.19^{\circ}}$
$c=\Big(\dfrac{\sin108.19^{\circ}}{\sin30^{\circ}}\Big)75\approx142.504$
