Question

calculate the side lengths a and b to two decimal places.
a. $a = 5.45$ and $b = 6.37$
b. $a = 11.08$ and $b = 12.94$
c. $a = 11.08$ and $b = 13.54$
d. $a = 6.37$ and $b = 5.45$
e. $a = 12.94$ and $b = 11.08$

Explanation:

Step1: Find angle C

Triangle angle sum is $180^\circ$.
$\angle C = 180^\circ - 58^\circ - 82^\circ = 40^\circ$

Step2: Use Law of Sines for side a

Relate a, AB, $\angle A$, $\angle C$.
$\frac{a}{\sin 58^\circ} = \frac{8.4}{\sin 40^\circ}$
$a = \frac{8.4 \times \sin 58^\circ}{\sin 40^\circ}$
$a \approx \frac{8.4 \times 0.8480}{0.6428} \approx 12.94$

Step3: Use Law of Sines for side b

Relate b, AB, $\angle B$, $\angle C$.
$\frac{b}{\sin 82^\circ} = \frac{8.4}{\sin 40^\circ}$
$b = \frac{8.4 \times \sin 82^\circ}{\sin 40^\circ}$
$b \approx \frac{8.4 \times 0.9903}{0.6428} \approx 11.08$

Answer:

E. a = 12.94 and b = 11.08