Question

solve the right triangle. a = □° (round to four decimal places as needed.) a = □ cm (round to three decimal places as needed.) c = □ cm (round to three decimal places 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 $B = 38.0898^{\circ}$, then $A=90^{\circ}- 38.0898^{\circ}=51.9102^{\circ}$.

Step2: Find side a

We know that $\tan B=\frac{b}{a}$, where $b = 55.855$ cm. So, $a=\frac{b}{\tan B}$. Substituting the values, $\tan(38.0898^{\circ})\approx0.7899$, and $a=\frac{55.855}{0.7899}\approx70.711$ cm.

Step3: Find side c

We know that $\sin B=\frac{b}{c}$, so $c=\frac{b}{\sin B}$. Since $\sin(38.0898^{\circ})\approx0.6173$, then $c=\frac{55.855}{0.6173}\approx90.483$ cm.

Answer:

$A = 51.9102^{\circ}$
$a = 70.711$ cm
$c = 90.483$ cm