How to derive cosine rule formula

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The Cosine Rule, also known as the Law of Cosines, is another key theorem in trigonometry, used primarily in non-right-angled triangles. It relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula is particularly useful for solving triangles when you know two sides and the included angle (SAS) or when you know all three sides (SSS).

The Cosine Rule is stated as:

c² = a² + b² - 2ab•cos(C)

Where:

a, b, and c are the lengths of the sides of the triangle,

C is the angle opposite side c .

Similarly, for the other angles, the rule can be written as:

a² = b² + c² - 2bc•cos(A)

b² = a² + c² - 2ac•cos(B)

The Cosine Rule is useful in two primary situations:

1. When you know two sides and the included angle (SAS) and want to find the third side.

2. When you know all three sides (SSS) and want to find an angle.

In the special case where the angle is 90° (right-angled triangle), the Cosine Rule simplifies to the Pythagorean Theorem.