Formal Definition of the Sine Function Without Geometry

The sine function, often denoted as sin x, is one of the fundamental trigonometric functions. Traditionally, it is defined geometrically or through power series expansions. However, in this article, we will explore a formal definition of the sine function using Euler's formula and a differential equation. This approach provides a robust framework for understanding the sine function without relying on geometric interpretations.

Definition Using Euler's Formula

Euler's formula states that for any real number x:

eix cos x i sin x

From this, we can isolate sin x in terms of the exponential function. Rearranging the formula, we get:

sin x (eix - e-ix) / (2i)

Properties of the Definition

Periodicity

This definition shows that sin x is periodic with a period of 2π. This can be seen by noting that:

eix 2π eix

Odd Function

It is clear from the expression that sin(-x) -sin x, which confirms that sin x is an odd function.

Values at Key Points

We can derive specific values:

sin(0) 0 sin(π/2) 1 sin(π) 0 sin(3π/2) -1 sin(2π) 0

Formal Definition Using Differential Equations

Another way to formally define the sine function is as the unique solution on the real numbers of the differential equation:

y'' y 0

with the initial conditions:

y(0) 0 y'(0) 1

This perspective emphasizes the analytical and differential properties of the sine function.

A third definition, as proposed by Arun Iljak, is:

sin x (eix - e-ix) / (2i)

This definition does not rely on constructing a right-angled triangle, especially if the angle is not real.

Conclusion

In conclusion, the sine function can be formally defined using a variety of approaches, including Euler's formula and differential equations. This approach grounds the sine function in complex analysis and provides a robust framework for further exploration of its properties.