Answer
$a\approx67.027$ $;$ $b=45$ $;$ $c=42$
$\angle A\approx100.728^{\circ}$ $;$ $\angle B\approx41.272^{\circ}$ $;$ $\angle C=38^{\circ}$
Work Step by Step
$b=45$ $,$ $c=42$ $,$ $\angle C=38^{\circ}$
The Law of Sines is $\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$
Find the angle $B$ using the formula $\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$, obtained from the Law of Sines. Substitute the known values into the formula and solve for $\angle B$:
$\dfrac{\sin B}{45}=\dfrac{\sin 38^{\circ}}{42}$
$\sin B=\Big(\dfrac{45}{42}\Big)\sin38^{\circ}$
$B=\sin^{-1}\Big[\Big(\dfrac{45}{42}\Big)\sin38^{\circ}\Big]\approx41.272^{\circ}$
Two angles are now known. Since $\angle A+\angle B+\angle C=180^{\circ}$, substitute the known values into the formula and solve for $\angle A$:
$A=180^{\circ}-38^{\circ}-41.272^{\circ}\approx100.728^{\circ}$
Find the side $a$ using the formula $\dfrac{\sin A}{a}=\dfrac{\sin C}{c}$, obtained from the Law of Sines. Substitute the known values and solve for $a$:
$\dfrac{\sin100.728^{\circ}}{a}=\dfrac{\sin38^{\circ}}{42}$
$\dfrac{a}{\sin100.728^{\circ}}=\dfrac{42}{\sin38^{\circ}}$
$a=\Big(\dfrac{\sin100.728^{\circ}}{\sin38^{\circ}}\Big)42\approx67.027$
