nLab cosine

Idea

See at trigonometric function.

Definitions

The cosine function is the function $\cos \;\colon\; \mathbb{R} \to \mathbb{R}$ from the real numbers to themselves which is characterized by the following equivalent conditions:

1. $\cos$ is the unique solution among smooth functions to the differential equation/initial value problem

   $$cos'' = -cos$$

   (where a prime indicates the derivative) subject to the initial conditions

   $$\begin{aligned} cos(0) &= 1 \\ cos'(0) & = 0 \,. \end{aligned}$$

2. (Euler's formula) $\cos$ is the real part of the exponential function with imaginary argument

   $$\begin{aligned} \cos(x) & = Re\left( \exp(i x) \right) \\ & = \frac{1}{2}\big( \exp(i x) + \exp(- i x)\big) \end{aligned} \,.$$

3. $\cos$ is the unique function given by the infinite series

$$\cos(x) \coloneqq \sum_{i = 0}^{\infty} \frac{(-1)^i x^{2 i}}{(2 i)!}$$

In arbitrary Archimedean ordered fields

In general, Archimedean ordered fields which are not sequentially Cauchy complete do not have an cosine function. Nevertheless, the cosine map is still guaranteed to be a partial function, because every Archimedean ordered field is a Hausdorff space and thus a sequentially Hausdorff space. Therefore an axiom could be added to Archimedean ordered fields $F$ to ensure that the cosine partial function is actually a total function:

Axiom of cosine function: For all elements $x \in F$, there exists a unique element $\cos(x) \in F$ such that for all positive elements $\epsilon \in F_+$, there exists a natural number $N \in \mathbb{N}$ such that for all natural numbers $n \in \mathbb{N}$, if $n \geq N$, then $-\epsilon \lt \left(\sum_{i = 0}^{n} \frac{(-1)^i x^{2 i}}{(2 i)!}\right) - \cos(x) \lt \epsilon$.

Related concepts

• periodic function

• trigonometry

• sine

• arccos

• exponential

• exponential ring

• Euler's formula

• trigonometric identity

References

• Wikipedia, Sine and cosine