Question

solve the right triangle.
image of right triangle with right angle at c, angle at b is 42.0897°, side ac = 50.851 cm, side bc labeled a, side ab labeled c
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 acute angles is \(90^\circ\). So \(A = 90^\circ - B\). Given \(B = 42.0897^\circ\), then \(A = 90 - 42.0897 = 47.9103^\circ\).

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

We know that \(\tan B=\frac{AC}{a}\), where \(AC = 50.851\) cm. So \(a=\frac{AC}{\tan B}\). \(\tan(42.0897^\circ)\approx0.9063\), then \(a = \frac{50.851}{0.9063}\approx56.100\) cm.

Step3: Find side c (hypotenuse)

We know that \(\sin B=\frac{AC}{c}\), so \(c=\frac{AC}{\sin B}\). \(\sin(42.0897^\circ)\approx0.6708\), then \(c=\frac{50.851}{0.6708}\approx75.800\) cm.

Answer:

\(A = 47.9103^\circ\)
\(a = 56.100\) cm
\(c = 75.800\) cm