Problem

144. Aviation From city \( A \) to city \( B \), a plane flies 650 miles at a bearing of \( 48^{\circ} \). From city \( B \) to city \( C \), the plane flies 810 miles at a bearing of \( 115^{\circ} \). Find the distance from city \( A \) to city \( C \) and the bearing from city \( A \) to city \( C \).

Answer

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Answer

Bearing: \( B = \gamma + 48^\circ \)

Steps

Step 1 ：\( \alpha = 180^\circ - 48^\circ \)

Step 2 ：\( \beta = 115^\circ - \alpha \)

Step 3 ：Use Law of Cosines: \( AC = \sqrt{ 650^{2} + 810^{2} - 2 \times 650 \times 810 \times \cos{ \beta } } \)

Step 4 ：Use Law of Sines: \( \sin \gamma = \frac{810 \sin \beta}{AC} \)

Step 5 ：\( \gamma = \arcsin{(\sin \gamma)} \)

Step 6 ：Bearing: \( B = \gamma + 48^\circ \)

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