Table of Contents
The cosine addition formula is a fundamental identity in trigonometry that expresses the cosine of a sum of two angles in terms of the cosines and sines of the individual angles. Deriving this formula from first principles provides insight into the geometric and algebraic foundations of trigonometry.
Understanding the Geometric Setup
Consider two angles, α and β, and points on the unit circle corresponding to these angles. The goal is to find an expression for cos(α + β) using the coordinates of these points.
Coordinate Representation of Angles
On the unit circle, a point corresponding to an angle θ has coordinates (cos θ, sin θ). For angles α and β, the points are:
- Point for α: (cos α, sin α)
- Point for β: (cos β, sin β)
Using Rotation and Coordinates
To derive the addition formula, consider rotating the point for α by angle β. The rotation of a point (x, y) by β results in:
(x’, y’) = (x cos β – y sin β, x sin β + y cos β)
Applying this to the point for α, we get:
(cos(α + β), sin(α + β)) = (cos α cos β – sin α sin β, cos α sin β + sin α cos β)
The Cosine Addition Formula
From the rotated coordinates, the cosine of the sum is given by:
cos(α + β) = cos α cos β – sin α sin β
Conclusion
This derivation from first principles illustrates how the cosine addition formula naturally arises from geometric rotation on the unit circle. It is a key identity used in simplifying trigonometric expressions and solving equations involving angles.