Question

consider a triangle abc like the one below. suppose that a = 41°, b = 65°, and c = 65. (the figure is not drawn to scale.) solve the triangle. round your answers to the nearest whole number. be sure to include the appropriate units in your answer.

Explanation:

Step1: Find angle \( C \)

The sum of angles in a triangle is \( 180^{\circ} \). So \( C=180^{\circ}-A - B \). Given \( A = 42^{\circ}\), \( B=65^{\circ}\), then \( C=180^{\circ}-42^{\circ}-65^{\circ}=73^{\circ}\)

Step2: Use the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). We know \( c = 65\), \(A = 42^{\circ}\), \(B = 65^{\circ}\), \(C=73^{\circ}\)

Find \( a \)

\(\frac{a}{\sin A}=\frac{c}{\sin C}\), so \(a=\frac{c\sin A}{\sin C}\). Substitute the values: \(a=\frac{65\times\sin42^{\circ}}{\sin73^{\circ}}\). Since \(\sin42^{\circ}\approx0.6691\), \(\sin73^{\circ}\approx0.9563\), then \(a=\frac{65\times0.6691}{0.9563}\approx45.5\)

Find \( b \)

\(\frac{b}{\sin B}=\frac{c}{\sin C}\), so \(b = \frac{c\sin B}{\sin C}\). Substitute the values: \(b=\frac{65\times\sin65^{\circ}}{\sin73^{\circ}}\). Since \(\sin65^{\circ}\approx0.9063\), \(\sin73^{\circ}\approx0.9563\), then \(b=\frac{65\times0.9063}{0.9563}\approx61.5\)

Answer:

\(C = 73^{\circ}\), \(a\approx45.5\), \(b\approx61.5\)