Answer
$b=458$
$c=877$
$B=31^{\circ}30'$
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 $b$, we use the formula $\tan\theta=\frac{a}{b}$.
Step 3: $\tan58.5^{\circ}=\frac{748}{b}$
Step 4: $b=\frac{748}{\tan58.5^{\circ}}$
Step 5: Using a calculator, $b\approx458.4$
Step 6: Rounding the answer to three significant figures, $b\approx458$
Step 7: To find $c$, we use the formula $\sin\theta=\frac{a}{c}$.
Step 8: $\sin 58.5^{\circ}=\frac{748}{c}$
Step 9: $c=\frac{748}{\sin 58.5^{\circ}}$
Step 10: Using a calculator, $c\approx877.3$
Step 11: Rounding the answer to three significant figures, $c\approx877$
Step 12: As $A+B=90^{\circ}$,
$B=90^{\circ}-58^{\circ}30'$
Step 13: Solving, $B=31^{\circ}30'$.
