Question

1. solve the triangle abc if a = 34 km, b = 20 km, and c = 18 km. show all the details in a logical order.

Explanation:

Step1: Identify the largest side

The largest side is \( a = 34\) km, so we first find the angle \( A \) opposite to it using the Law of Cosines: \( \cos A=\frac{b^{2}+c^{2}-a^{2}}{2bc}\)
Substitute \( a = 34\), \( b = 20\), \( c = 18\):
\(\cos A=\frac{20^{2}+18^{2}-34^{2}}{2\times20\times18}=\frac{400 + 324- 1156}{720}=\frac{724 - 1156}{720}=\frac{-432}{720}=- 0.6\)
So \( A=\arccos(-0.6)\approx126.87^{\circ}\)

Step2: Find angle \( B \) using Law of Sines

Law of Sines: \(\frac{\sin B}{b}=\frac{\sin A}{a}\)
\(\sin B=\frac{b\sin A}{a}=\frac{20\times\sin(126.87^{\circ})}{34}\)
\(\sin(126.87^{\circ})=\sin(180 - 53.13)^{\circ}=\sin53.13^{\circ}\approx0.8\)
\(\sin B=\frac{20\times0.8}{34}=\frac{16}{34}\approx0.4706\)
So \( B=\arcsin(0.4706)\approx28.07^{\circ}\)

Step3: Find angle \( C \)

Since the sum of angles in a triangle is \( 180^{\circ}\), \( C = 180^{\circ}-A - B\)
\( C=180 - 126.87-28.07 = 25.06^{\circ}\) (We can also verify using Law of Sines \(\frac{\sin C}{c}=\frac{\sin A}{a}\) to check the consistency)

Answer:

\( A\approx126.87^{\circ}\), \( B\approx28.07^{\circ}\), \( C\approx25.06^{\circ}\)