#### Answer

$B=41.7^{\circ}$

#### Work Step by Step

The law of sines in the case of this question is:
$\frac{b}{\sin B}=\frac{c}{\sin C}$
We substitute the values of $b,c$ and $C$ in the formula to find $B$:
$\frac{b}{\sin B}=\frac{c}{\sin C}$
$\frac{58.2}{\sin B}=\frac{68.3}{\sin 51.3}$
$\sin B\times68.3=58.2\times \sin 51.3$
$\sin B=\frac{58.2\times \sin 51.3}{68.3}$
We use a calculator to find the value of $\sin 51.3$:
$\sin B=\frac{58.2\times \sin 51.3}{68.3}$
$\sin B=\frac{58.2\times 0.78043}{68.3}$
$\sin B=0.66502$
$B=\sin^{-1} 0.66502$
$B=41.68\approx41.7$
Therefore, $B=41.7^{\circ}$.