10.2. Law of Cosines

For any triangle ΔABC (sides a, b, c; opposite angles A, B, C): • Law of Cosines: a² = b² + c² − 2bc cos(A) (and cyclically for b, c)

The Law of Cosines is a formula used in triangle geometry, particularly when dealing with triangles that are not right-angled. It relates the lengths of the sides of a triangle to the cosine of one of its angles. In the formula a² = b² + c² − 2bc cos(A), 'a' is the length of the side opposite angle A, and 'b' and 'c' are the lengths of the other two sides. The formula can be rearranged to solve for the cosine of an angle, which is useful when specific side lengths are known. Similar formulas exist for angles B and C by cyclically changing the letters.

Imagine you are trying to find the length of a rope (side 'a') that connects two points (A and B) from a third point (C) positioned at an angle. If you know the lengths of the other two sides (the two segments of the rope) and the angle between them, the Law of Cosines allows you to calculate how long the rope needs to be, much like finding the distance between two objects when you know how far they are from a third object and the angle at which they're placed.

Useful for solving non‑right triangles (e.g., ASA, SSA ambiguous etc.)

The Law of Cosines is particularly useful when dealing with non-right triangles, where traditional trigonometric relationships may not apply directly. It is applicable in cases such as when you have two sides and the angle between them (SAS), or two angles and a side (ASA). The law also can help resolve ambiguous cases in SSA (two sides and a non-included angle), where there might be two possible triangles that could fit the given information.

Consider a situation where you are trying to determine the angle at which a ladder rests against a wall. If you know the distance from the base of the ladder to the wall and the height at which the ladder touches the wall, you can treat this setup as a triangle. Since it’s not a right triangle, you can use the Law of Cosines to calculate the angle of the ladder against the wall, giving you practical insight into the optimal angle for stability.

Detailed Worked Examples

1. Applying Law of Cosines to find unknown side
2. Using Law of Sines in SSA case
3. Constructing circum‑ and incircles with compass/straightedge
4. Proof: concurrency of medians at centroid (via coordinates or area method)

The application of the Law of Cosines can be illustrated through several examples. For instance, if we want to find the length of side 'a' in triangle ABC, and we know the lengths of sides 'b' and 'c' along with angle A, we can directly substitute those values into the formula and solve for 'a'. Additionally, the examples can showcase how to use the Law of Sines in cases where more than one triangle is possible, like in SSA situations. The construction of circumcircles and incircles, as well as proofs involving triangle medians, also highlight the depth and connectivity of triangle properties.

Imagine planning a new park layout. You have two points where you want to build paths and you also wish to include a fountain at an angle with respect to these two paths. You'd need to figure out the lengths of the paths, incorporating the Law of Cosines might give you the best way to ensure the paths meet and the fountain fits within the design. This application of cosine metrics ensures the layout is geometrically sound.

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