# nLab cosine

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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 $-\epsilon \lt \left(\sum_{i = 0}^{n} \frac{(-1)^i x^{2 i}}{(2 i)!}\right) - \cos(x) \lt \epsilon$.

## References

Last revised on November 26, 2022 at 04:37:01. See the history of this page for a list of all contributions to it.