Part 2/5:
This formula is a generalization of the Pythagorean theorem and holds true for all triangle types, including obtuse and acute triangles. When angle C is a right angle, cosine of C equals zero, thus reverting to the familiar Pythagorean theorem:
[
c^2 = a^2 + b^2
]
Proof of the Law of Cosines
To prove the Law of Cosines, we can start by constructing a triangle with one vertex angle labeled as C and corresponding opposite side c. By drawing a perpendicular from the vertex angle C to the baseline (side a) and labeling the lengths appropriately, we can break the triangle into two right-angled triangles.
Let:
The lengths of the triangle be labeled as a, b, and c.
The height from angle C be represented as y.