Exploring the Formula and Proof of the Cosine Rule

The cosine rule, also known as the law of cosines, is a fundamental formula in trigonometry that generalizes the Pythagorean theorem to non-right triangles. This article delves into the derivation of the cosine rule and provides a step-by-step guide to understanding its proof.

What is the Cosine Rule?

In a triangle ABC, the cosine rule states that the square of any side (say a) is equal to the sum of the squares of the other two sides (denoted as b and c) minus twice the product of these sides and the cosine of the included angle opposite to a. Mathematically, it can be expressed as:

[ a^2 b^2 c^2 - 2bc cos A ]

Similarly, the rules for sides b and c are:

[ b^2 a^2 c^2 - 2ac cos B ]

[ c^2 a^2 b^2 - 2ab cos C ]

Proof of the Cosine Rule

To derive the cosine rule, consider the triangle ABC with BD perpendicular to AC, splitting the triangle into two smaller right triangles BCD and ABD for side a.

Step 1: Analyzing Triangle BCD

[ cos C frac{CD}{a} implies CD a cos C ]

[ sin C frac{BD}{a} implies BD a sin C ]

Step 2: Analyzing Triangle ABD

[ c^2 AD^2 BD^2 ]

From the first step, we know that:

[ AD AB - BD b - a cos C ]

[ BD a sin C ]

Substituting these values into the equation for c2, we get:

[ c^2 (b - a cos C)^2 (a sin C)^2 ]

Expanding and simplifying this expression:

[ c^2 b^2 - 2ab cos C a^2 cos^2 C a^2 sin^2 C ]

[ c^2 b^2 - 2ab cos C a^2 (cos^2 C sin^2 C) ]

[ c^2 b^2 - 2ab cos C a^2 (1) ]

[ c^2 b^2 - 2ab cos C a^2 ]

[ c^2 a^2 b^2 - 2ab cos C ]

Application of the Cosine Rule

The cosine rule is a powerful tool in various geometric and trigonometric problems, extending the Pythagorean theorem to more complex scenarios. When A is 90 degrees, the cosine of A is 0, and the cosine rule simplifies to the Pythagorean theorem:

[ a^2 b^2 c^2 - 2bc cdot 0 ]

[ a^2 b^2 c^2 ]

This relationship aligns with the well-known Pythagorean theorem. Another interesting result from the cosine rule is the formula for cos 2A in terms of cosine:

[ cos 2A 2 cos^2 A - 1 ]

Inspiring a Proof for the Cosine Rule

To gain a deeper understanding of the cosine rule, one can attempt to prove it through geometric and trigonometric methods. Consider the triangle ABC with a perpendicular dropped from C to AB, meeting at D. If CAB is obtuse, D lies on BA extended. Using the Pythagorean theorem in triangles ADC and BDC, and eliminating the height CD, the formula can be derived. Here's a simplified version of the proof:

Step 1: Using the Perpendicular Drop

[ x b cos A ]

[ h b sin A ]

Step 2: Applying the Pythagorean Theorem

[ BD^2 CD^2 BC^2 ]

[ (b - x)^2 h^2 c^2 ]

[ (b - b cos A)^2 (b sin A)^2 c^2 ]

[ b^2 - 2b^2 cos A b^2 cos^2 A b^2 sin^2 A c^2 ]

[ b^2 (1 - 2 cos A cos^2 A sin^2 A) c^2 ]

[ b^2 (1 - 2 cos A 1) c^2 ]

[ b^2 (2 - 2 cos A) c^2 ]

[ c^2 b^2 b^2 - 2b^2 cos A ]

[ c^2 b^2 b^2 - 2b^2 cos A ]

[ c^2 b^2 - 2ab cos C ]

By completing these steps, you can see that the cosine rule is indeed derived from basic geometric and trigonometric principles.

The cosine rule is not just a tool for solving complex geometric problems; it is an essential concept that builds foundational knowledge in trigonometry. Understanding its derivation can help solidify your grasp of trigonometric principles and enhance problem-solving skills.