Apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles.
Convert bearings into a solvable triangle
Problem
From \(A\), points \(B\) and \(C\) have bearings \(040°\) and \(110°\), with \(AB~=~12~\text{km}\) and \(AC~=~9~\text{km}\), as shown. Find interior angle \(BAC\), then use it to calculate \(BC\).
Convert directional bearings into the triangle's interior angle before applying any triangle law. When two bearings share the same north reference, their angular separation is found from the appropriate difference; at a changed viewpoint, account for the reversed direction first. Then classify the resulting side-angle data and solve the geometric triangle with full precision.
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