Use the law of cosines to solve the given problems. The GPS...

Use the law of cosines to solve the given problems. The GPS of a plane over Lake Erie indicates that the plane's position is $190 \mathrm{km}$ from Detroit and $110 \mathrm{km}$ from London, Ontario, which are known to be $160 \mathrm{km}$ apart. What is the angle between the plane's directions to Detroit and London?

Use the law of cosines to solve the given problems.
The GPS of a plane over Lake Erie indicates that the plane's position is $190 \mathrm{km}$ from Detroit and $110 \mathrm{km}$ from London, Ontario, which are known to be $160 \mathrm{km}$ apart. What is the angle between the plane's directions to Detroit and London?

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Key Concepts

Law of Cosines

The Law of Cosines is a key mathematical principle used to relate the lengths of the sides of any triangle with the cosine of one of its angles. It generalizes the Pythagorean theorem to triangles that are not right-angled, allowing one to calculate a side when two sides and the included angle are known, or to determine an angle when all three sides are known.

Triangle Geometry

Triangle geometry involves the study of the relationships between the sides and angles in a triangle. Understanding these relationships, including the rule that the sum of interior angles is constant, is essential for solving problems where some measurements are known and others must be determined using methods such as the Law of Cosines.

Inverse Cosine Function

The inverse cosine function, often written as arccos, is used to compute the measure of an angle when its cosine value is known. This function is crucial when the Law of Cosines yields a cosine value for an angle, allowing one to obtain the corresponding angle measurement by taking the arccosine.

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