Question

solve the right triangle.

diagram: right triangle with right angle at c, angle at b is 40.0899°, side ac = 54.859 cm, hypotenuse c, side bc = a, angle at a is a

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 angles is \(180^\circ\), and \(\angle C = 90^\circ\), \(\angle B = 40.0899^\circ\). So \(\angle A=180^\circ - 90^\circ - 40.0899^\circ\)
\(\angle A = 49.9101^\circ\)

Step2: Find side a (opposite to angle B, adjacent to angle A)

We know \(\tan B=\frac{AC}{a}\), where \(AC = 54.859\) cm, \(\angle B = 40.0899^\circ\). So \(a=\frac{AC}{\tan B}\)
\(\tan(40.0899^\circ)\approx0.8462\), then \(a=\frac{54.859}{0.8462}\approx64.830\) cm

Step3: Find side c (hypotenuse)

We know \(\sin B=\frac{AC}{c}\), so \(c=\frac{AC}{\sin B}\)
\(\sin(40.0899^\circ)\approx0.6447\), then \(c=\frac{54.859}{0.6447}\approx85.092\) cm

Answer:

\(A = 49.9101^\circ\)
\(a = 64.830\) cm
\(c = 85.092\) cm