Question

solve the right triangle shown in the figure. b = 15.1°, b = 35.5 a = 74.9° (round to the nearest tenth as needed.) a ≈ 132.07 (round to the nearest hundredth as needed.) c ≈ 136.28 (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 $B = 15.1^{\circ}$, then $A=90^{\circ}- 15.1^{\circ}=74.9^{\circ}$.

Step2: Find side a

We know that $\tan B=\frac{b}{a}$. So $a=\frac{b}{\tan B}$. Substituting $B = 15.1^{\circ}$ and $b = 35.5$, we have $a=\frac{35.5}{\tan(15.1^{\circ})}$. Since $\tan(15.1^{\circ})\approx0.269$, then $a=\frac{35.5}{0.269}\approx132.07$.

Step3: Find side c

We know that $\sin B=\frac{b}{c}$. So $c=\frac{b}{\sin B}$. Substituting $B = 15.1^{\circ}$ and $b = 35.5$, we have $c=\frac{35.5}{\sin(15.1^{\circ})}$. Since $\sin(15.1^{\circ})\approx0.261$, then $c=\frac{35.5}{0.261}\approx136.28$.

Answer:

$A = 74.9^{\circ}$, $a\approx132.07$, $c\approx136.28$