Question

solve the right triangle abc, with c = 90°. a = 56.3°, c = 18.5 ft b = \\(\square\\)° (simplify your answer. type an integer or a decimal. round to the nearest tenth as needed.) a = \\(\square\\) ft (simplify your answer. type an integer or a decimal. round to the nearest tenth as needed.) b = \\(\square\\) ft (simplify your answer. type an integer or a decimal. round to the nearest tenth as needed.)

Explanation:

Step1: Find angle B

In a right triangle, the sum of angles is \(180^\circ\), and \(C = 90^\circ\), \(A = 56.3^\circ\). So \(B=180^\circ - 90^\circ - 56.3^\circ\)
\(B = 33.7^\circ\)

Step2: Find side a (opposite angle A)

Using sine function: \(\sin A=\frac{a}{c}\), so \(a = c\times\sin A\)
\(a = 18.5\times\sin(56.3^\circ)\)
\(\sin(56.3^\circ)\approx0.832\), so \(a\approx18.5\times0.832\approx15.4\)

Step3: Find side b (adjacent to angle A)

Using cosine function: \(\cos A=\frac{b}{c}\), so \(b = c\times\cos A\)
\(b = 18.5\times\cos(56.3^\circ)\)
\(\cos(56.3^\circ)\approx0.554\), so \(b\approx18.5\times0.554\approx10.2\)

Answer:

\(B = 33.7^\circ\)
\(a \approx 15.4\) ft
\(b \approx 10.2\) ft