Question

use the triangle shown on the right and the given information to solve the triangle.
b = 8, a = 10°, find a, c, and b
a = □ (round to two decimal places as needed.)

Explanation:

Step1: Find angle \( B \)

In a right - triangle, \( A + B+90^{\circ}=180^{\circ} \). Given \( A = 10^{\circ} \), then \( B=180^{\circ}-90^{\circ}-10^{\circ}=80^{\circ} \).

Step2: Find side \( a \)

We know that \(\tan A=\frac{a}{b}\). Given \( b = 8\) and \(A = 10^{\circ}\), then \(a=b\tan A\). Substitute \(b = 8\) and \(A = 10^{\circ}\) into the formula: \(a = 8\times\tan(10^{\circ})\). Since \(\tan(10^{\circ})\approx0.1763\), \(a=8\times0.1763 = 1.41\).

Step3: Find side \( c \)

We know that \(\cos A=\frac{b}{c}\), then \(c=\frac{b}{\cos A}\). Substitute \(b = 8\) and \(A = 10^{\circ}\) into the formula. Since \(\cos(10^{\circ})\approx0.9848\), \(c=\frac{8}{0.9848}\approx8.12\).

Answer:

\(a = 1.41\), \(c\approx8.12\), \(B = 80^{\circ}\)