Question

solve the right triangle abc, with c = 90°. b = 63.9°, b = 113 in. a = ° (simplify your answer. type an integer or a decimal. round to the nearest tenth as needed.) a = in. (simplify your answer. type an integer or a decimal. round to the nearest tenth as needed.) c = in. (simplify your answer. type a whole number.)

Explanation:

Step1: Find angle A

In a right - triangle, the sum of the interior angles is 180°. Given \(C = 90^{\circ}\) and \(B=63.9^{\circ}\), we use the formula \(A + B + C=180^{\circ}\). So \(A=180^{\circ}-90^{\circ}-63.9^{\circ}\).
\[A = 26.1^{\circ}\]

Step2: Find side a

We know that \(\tan B=\frac{b}{a}\). Rearranging for \(a\), we get \(a = \frac{b}{\tan B}\). Substituting \(b = 113\) in and \(B = 63.9^{\circ}\), we have \(a=\frac{113}{\tan(63.9^{\circ})}\).
\[a=\frac{113}{2.047}\approx55.2\]

Step3: Find side c

We know that \(\sin B=\frac{b}{c}\). Rearranging for \(c\), we get \(c=\frac{b}{\sin B}\). Substituting \(b = 113\) in and \(B = 63.9^{\circ}\), we have \(c=\frac{113}{\sin(63.9^{\circ})}\).
\[c=\frac{113}{0.898}\approx126\]

Answer:

\(A = 26.1^{\circ}\)
\(a\approx55.2\) in.
\(c = 126\) in.