Table of Contents
The cosine addition and subtraction formulas are fundamental in trigonometry, allowing us to simplify expressions involving angles. These formulas are:
Cosine Addition Formula: cos(A + B) = cos A cos B – sin A sin B
Cosine Subtraction Formula: cos(A – B) = cos A cos B + sin A sin B
Proof of the Cosine Addition Formula
One common proof uses the unit circle and coordinates. Consider two angles, A and B, on the unit circle. The coordinates of a point at angle A are (cos A, sin A), and similarly for B.
Using the rotation of coordinate axes, the formula can be derived by expressing cos(A + B) as the x-coordinate of a rotated point. The derivation involves the following steps:
- Express cos(A + B) in terms of sine and cosine using the rotation identity.
- Apply the sum of angles in sine and cosine functions.
- Simplify to arrive at the formula: cos A cos B – sin A sin B.
Proof of the Cosine Subtraction Formula
This proof is similar to the addition formula but considers the angle subtraction. It uses the same rotation and coordinate approach.
By analyzing the unit circle and applying the cosine and sine sum identities, we find:
- The expression for cos(A – B) simplifies to cos A cos B + sin A sin B.
- This confirms the subtraction formula.
Alternative Proofs
Other proofs include using the Law of Cosines in triangles or Euler’s formula with complex exponentials. These approaches also lead to the same identities, demonstrating their fundamental nature in mathematics.
Applications of These Formulas
The cosine addition and subtraction formulas are essential in solving trigonometric equations, analyzing wave patterns, and in Fourier analysis. They simplify the process of working with multiple angles and are foundational in physics and engineering.