#### Answer

$a=638$
$b=391$
$B=31^{\circ}30'$

#### Work Step by Step

Step 1: Converting the angle to decimal degrees;
$58^{\circ}30'=58\frac{30}{60}^{\circ}=58.5^{\circ}$
Step 2: To find $a$, we use the formula $\sin\theta=\frac{a}{c}$.
Step 3: $\sin58.5^{\circ}=\frac{a}{748}$
Step 4: $a=748\times\sin58.5^{\circ}$
Step 5: Using a calculator, $a\approx637.7$
Step 6: Rounding the answer to three significant degrees, $a\approx638$
Step 7: To find $b$, we use the formula $\cos\theta=\frac{b}{c}$.
Step 8: $\cos58.5^{\circ}=\frac{b}{748}$
Step 9: $b=748\times\cos58.5^{\circ}$
Step 10: Using a calculator, $b\approx390.8$
Step 11: Rounding the answer to three significant degrees, $b\approx391$
Step 12: As $A+B=90^{\circ}$,
$B=90^{\circ}-58^{\circ}30'$
Step 13: Solving, $B=31^{\circ}30'$