Question

solve the right triangle shown in the figure. b = 16.5°, b = 31.5 a = ° (round to the nearest tenth as needed.) a ≈ (round to the nearest hundredth as needed.) c ≈ (round to the nearest hundredth as needed.)

Explanation:

Step1: Find angle A

In a right - triangle, the sum of the two non - right angles is 90°. So $A = 90^{\circ}-B$. Given $B = 16.5^{\circ}$, then $A=90 - 16.5=73.5^{\circ}$.

Step2: Find side a

We know that $\tan B=\frac{b}{a}$. Rearranging for $a$, we get $a=\frac{b}{\tan B}$. Substituting $B = 16.5^{\circ}$ and $b = 31.5$, we have $\tan(16.5^{\circ})\approx0.296$. So $a=\frac{31.5}{0.296}\approx106.42$.

Step3: Find side c

We know that $\sin B=\frac{b}{c}$. Rearranging for $c$, we get $c=\frac{b}{\sin B}$. Substituting $B = 16.5^{\circ}$ and $b = 31.5$, we have $\sin(16.5^{\circ})\approx0.284$. So $c=\frac{31.5}{0.284}\approx110.92$.

Answer:

$A = 73.5^{\circ}$
$a\approx106.42$
$c\approx110.92$