Deriving the Exponential Definitions of Sine and Cosine
Understanding the exponential definitions of sine and cosine is crucial for advanced mathematics and various scientific applications. This article will delve into how these definitions can be derived using Euler's formula and Taylor series, providing a comprehensive overview for both novice and seasoned mathematicians.
Euler's Formula: The Foundation
At the crux of our exploration is Euler's formula, which elegantly connects the exponential function with trigonometric functions:
[e^{ix} cos x i sin x]
This formula, where $i$ is the imaginary unit, forms the basis for understanding these trigonometric functions in the complex plane.
Step-by-Step Derivation
Step 1: Euler's Formula
We start with Euler's formula:
[e^{ix} cos x i sin x]
Step 2: Considering the Negative Exponent
Next, we consider the complex exponential with a negative argument:
[e^{-ix} cos(-x) i sin(-x)]
Using the properties of cosine and sine, specifically:
[cos(-x) cos x] [sin(-x) -sin x]We can rewrite the equation as:
[e^{-ix} cos x - i sin x]
Step 3: Adding and Subtracting the Equations
We now add and subtract the original equations to isolate sine and cosine:
Adding the Equations
[e^{ix} e^{-ix} (cos x i sin x) (cos x - i sin x) 2 cos x]
Thus, we obtain:
[cos x frac{e^{ix} e^{-ix}}{2}]
Subtracting the Equations
[e^{ix} - e^{-ix} (cos x i sin x) - (cos x - i sin x) 2i sin x]
Thus, we obtain:
[sin x frac{e^{ix} - e^{-ix}}{2i}]
Summary: The Exponential Definitions
The exponential definitions of sine and cosine are:
[cos x frac{e^{ix} e^{-ix}}{2}] [sin x frac{e^{ix} - e^{-ix}}{2i}]These definitions are fundamental in complex analysis and have various applications in mathematics, physics, and engineering.
Alternative Approach: Using Taylor Series
If you define the functions sin x and cos x as:
[cos x frac{e^{ix} e^{-ix}}{2}] [sin x frac{e^{ix} - e^{-ix}}{2i}]there's nothing left to derive. These are the definitions. However, if you define these trigonometric functions in any other way (e.g., the usual geometric relation between the sides of a rectangular triangle), then the expression:
[e^{ialpha} cos alpha i sin alpha]
must be proved. This proof can be accomplished using Taylor series or other methods.
Proof via Taylor Series
Assume we have already calculated:
[left. frac{dsin x}{dx} right|_{x0} cos 0 1] [left. frac{dcos x}{dx} right|_{x0} -sin 0 0]The successive derivatives of these functions are:
[left. frac{d^{2k}sin x}{dx^{2k}} right|_{x0} (-1)^k sin 0 0] [left. frac{d^{2k 1}sin x}{dx^{2k 1}} right|_{x0} (-1)^k cos 0 (-1)^k] [left. frac{d^{2k}cos x}{dx^{2k}} right|_{x0} (-1)^k cos 0 (-1)^k] [left. frac{d^{2k 1}cos x}{dx^{2k 1}} right|_{x0} -(-1)^k sin 0 0]The corresponding Taylor expansions of these functions around x0 are:
[sin x sum_{k0}^{infty} frac{(-1)^k x^{2k 1}}{(2k 1)!}] [cos x sum_{k0}^{infty} frac{(-1)^k x^{2k}}{(2k)!}]Alternatively, these Taylor series can be taken as starting points for the definitions of sin x and cos x.
Generalization to the Complex Domain
The exponential function e^x has all derivatives also equal to e^x and all have the value 1 at x0. Therefore, the Taylor series for the exponential around x0 is:
[e^x sum_{n0}^{infty} frac{x^n}{n!}]
The generalization of these functions to the complex domain is straightforward and the corresponding Taylor series are the same. All three series converge on the whole complex plane, meaning their radii of convergence are infinite. Let's calculate e^{ix} by using the Taylor series:
[e^{ix} sum_{n0}^{infty} frac{ix^n}{n!}]
Given that i^{2k} (-1)^k and i^{2k 1} (-1)^k i, we can separate the even and odd terms:
[e^{ix} sum_{k0}^{infty} frac{x^{2k}}{2k!} i sum_{k0}^{infty} frac{x^{2k 1}}{(2k 1)!} cos x i sin x]
By considering:
[e^{-ix} cos(-x) i sin(-x) cos x - i sin x]Conclusion
By understanding the exponential definitions of sine and cosine, we can appreciate the elegance and power of complex analysis. These definitions not only simplify many mathematical operations but also reveal deep connections between different branches of mathematics.