Apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles.
Use the Law of Cosines to find a missing side when two sides and the included angle are given
Problem
\(\text{Two}~\text{paths}\) are \(6~\text{km}\) and \(9~\text{km}\) long with an included angle of \(50°\), as shown. Write a Law of Cosines equation for the distance \(d\) between their endpoints.
Recognize the two known sides and the angle between them as an SAS pattern. The unknown closing side must be paired with that included angle because they are opposite each other. Substitute those roles into the Law of Cosines carefully, keeping the square on the unknown and the minus sign on the cosine correction.
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