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

There is another axiom which uses the fact that derivatives of functions are well defined in the ordered local Artin $F$-algebra $F[\epsilon]/\epsilon^2$ by the equation $f(x + \epsilon) = f(x) + f'(x) \epsilon$:

Axiom of sine and cosine functions: Let $F[\epsilon]/\epsilon^2$ be the ordered local Artin $F$-algebra, with non-zero non-positive non-negative nilpotent element $\epsilon \in F[\epsilon]/\epsilon^2$ where $\epsilon^2 = 0$ and canonical $F$-algebra homomorphism $h:F \to F[x]/x^2$. There exists unique functions $\sin:F \to F$ and $\cos:F \to F$ and functions $\sin':F[\epsilon]/\epsilon^2 \to F[\epsilon]/\epsilon^2$ and $\cos':F[\epsilon]/\epsilon^2 \to F[\epsilon]/\epsilon^2$ such that for every element $x \in F$, $h(\sin(x)) = \sin'(h(x))$, $h(\cos(x)) = \cos'(h(x))$, $\sin'(x + \epsilon) = \sin'(x) + \cos'(x) \epsilon$, $\cos'(x + \epsilon) = \cos'(x) - \sin'(x) \epsilon$, $\sin'(0) = 0$, and $\cos'(0) = 1$.

In the dual numbers

The dual number real algebra $\mathbb{D} \coloneqq \mathbb{R}[\epsilon]/\epsilon^2$ has a notion of sine and cosine function, which are the solutions to the system of functional equations $\sin(x + \epsilon) = \sin(x) + \cos(x) \epsilon$ and $\cos(x + \epsilon) = \cos(x) - \sin(x) \epsilon$ with $\sin(0) = 1$ and $\cos(0) = 1$.

In constructive mathematics

In classical mathematics, one could prove that the modulated Cantor real numbers $\mathbb{R}_C$ are sequentially Cauchy complete and equivalent to the HoTT book real numbers $\mathbb{R}_H$. However, in constructive mathematics, the above cannot be proven; while the HoTT book real numbers $\mathbb{R}_H$ are still sequentially Cauchy complete, the modulated Cantor real numbers $\mathbb{R}_C$ in general cannot be proven to be sequentially Cauchy complete. In particular, this means that the sequence

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

does not have a limit for all modulated Cantor real numbers $x \in \mathbb{R}_C$. However, the sequence, by definition of $\mathbb{R}_C$, does have a limit for all rational numbers $x \in \mathbb{Q}$; this means that one could restrict the domain of the cosine function to the rational numbers $\cos:\mathbb{Q} \to \mathbb{R}_C$, and define it in the usual manner:

• For all rational numbers $x \in \mathbb{Q}$, there exists a unique modulated Cantor real number $\cos(x) \in \mathbb{R}_C$ such that for all positive rational numbers $\epsilon \in \mathbb{Q}_+$, there exists a natural number $N \in \mathbb{N}$ such that for all natural numbers $n \in \mathbb{N}$, if $n \geq N$, then $-\epsilon \lt \sum_{i = 0}^{n} \frac{(-1)^i x^{2 i}}{(2 i)!} - \cos(x) \lt \epsilon$.

References

Discussion in constructive analysis: