Question

consider a triangle abc like the one below. suppose that a = 48°, c = 73°, and c = 8. (the figure is not drawn to scale.) solve the triangle. round your answers to the nearest tenth. if there is more than one solution, use the button labeled \or\.

Explanation:

Step1: Find angle \( B \)

Use the angle - sum property of a triangle (\( A + B + C=180^{\circ} \)).
Given \( A = 48^{\circ} \) and \( C = 73^{\circ} \), then \( B=180^{\circ}-(A + C)=180^{\circ}-(48^{\circ}+73^{\circ}) = 59^{\circ}\).

Step2: Use the Law of Sines (\(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\))

To find \( a \):
\(\frac{a}{\sin A}=\frac{c}{\sin C}\), so \(a=\frac{c\sin A}{\sin C}\).
Substitute \(c = 8\), \(A = 48^{\circ}\), \(C = 73^{\circ}\).
\(\sin48^{\circ}\approx0.743\), \(\sin73^{\circ}\approx0.956\).
\(a=\frac{8\times0.743}{0.956}=\frac{5.944}{0.956}\approx6.2\).
To find \( b \):
\(\frac{b}{\sin B}=\frac{c}{\sin C}\), so \(b=\frac{c\sin B}{\sin C}\).
Substitute \(c = 8\), \(B = 59^{\circ}\), \(C = 73^{\circ}\).
\(\sin59^{\circ}\approx0.857\), \(\sin73^{\circ}\approx0.956\).
\(b=\frac{8\times0.857}{0.956}=\frac{6.856}{0.956}\approx7.2\).

Answer:

\(a\approx6.2\), \(b\approx7.2\), \(B = 59^{\circ}\)