nLab cosine

Contents

Context

Trigonometry

Analysis

Algebra

Contents

Idea

See at trigonometric function.

Definitions

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

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

    cos=cos cos'' = -cos

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

    cos(0) =1 cos(0) =0. \begin{aligned} cos(0) &= 1 \\ cos'(0) & = 0 \,. \end{aligned}
  2. (Euler's formula) cos\cos is the real part of the exponential function with imaginary argument

    cos(x) =Re(exp(ix)) =12(exp(ix)+exp(ix)). \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\cos is the unique function given by the infinite series

cos(x) i=0 (1) ix 2i(2i)!\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 FF to ensure that the cosine partial function is actually a total function:

Axiom of cosine function: For all elements xFx \in F, there exists a unique element cos(x)F\cos(x) \in F such that for all positive elements ϵF +\epsilon \in F_+, there exists a natural number NN \in \mathbb{N} such that for all natural numbers nn \in \mathbb{N}, if nNn \geq N, then ϵ<( i=0 n(1) ix 2i(2i)!)cos(x)<ϵ-\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.