Answer
$a\approx65.66$ $;$ $b=73$ $;$ $c=82$
$\angle A\approx49.71^{\circ}$ $;$ $\angle B=58^{\circ}$ $;$ $\angle C\approx72.29^{\circ}$
Work Step by Step
$b=73$ $,$ $c=82$ $,$ $\angle B=58^{\circ}$
The Law of Sines is $\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$
Find angle $C$ by using the formula $\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$, obtained from the Law of Sines. Substitute the known values and solve for $C$:
$\dfrac{\sin58^{\circ}}{73}=\dfrac{\sin C}{82}$
$\sin C=\Big(\dfrac{82}{73}\Big)\sin58^{\circ}$
$C=\sin^{-1}\Big[\Big(\dfrac{82}{73}\Big)\sin58^{\circ}\Big]\approx72.29^{\circ}$
Two angles are now known. Since $\angle A+\angle B+\angle C=180^{\circ}$, substitute the known angles into the formula and solve for $\angle A$:
$\angle A=180^{\circ}-58^{\circ}-72.29^{\circ}\approx49.71^{\circ}$
Find side $a$ by using the formula $\dfrac{\sin A}{a}=\dfrac{\sin B}{b}$, obtained from the Law of Sines. Substitute the known values and solve for $a$:
$\dfrac{\sin49.71^{\circ}}{a}=\dfrac{\sin58^{\circ}}{73}$
$\dfrac{a}{\sin49.71^{\circ}}=\dfrac{73}{\sin58^{\circ}}$
$a=\Big(\dfrac{\sin49.71^{\circ}}{\sin58^{\circ}}\Big)73\approx65.66$
