nLab sine

Idea

The sine function $\sin$ is one of the basic trigonometric functions.

It may be thought of as assigning to any angle the distance to a chosen axis of the point on the unit circle with that angle to that axis.

Definitions

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

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

   $$sin'' = -sin$$

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

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

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

   $$\begin{aligned} \sin(x) & = Im\left( \exp(i x) \right) \\ & = \frac{1}{2 i}\left( \exp(i x) - \exp(- i x)\right) \end{aligned} \,.$$

3. $\sin$ is the unique function which

   1. is continuous at $0$

   2. satisfies the inequality

      $$1 - x^2 \;\leq\; \frac{\sin(x)}{x} \;\leq\; 1$$

      and the equation

      $$\sin(3 x) \;=\; 3 \sin(x) - 4 (\sin(x))^3$$

   (see Trimble: Characterization of sine. Some additional discussion at the nForum.)

4. $\sin$ is the unique function given by the infinite series

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

In arbitrary Archimedean ordered fields

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

Axiom of sine function: For all elements $x \in F$, there exists a unique element $\sin(x) \in F$ 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 $-\epsilon \lt \left(\sum_{i = 0}^{n} \frac{(-1)^i x^{2 i + 1}}{(2 i + 1)!}\right) - \sin(x) \lt \epsilon$.

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

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

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 sine function to the rational numbers $\sin:\mathbb{Q} \to \mathbb{R}_C$, and define it in the usual manner:

• For all rational numbers $x \in \mathbb{Q}$, there exists a unique modulated Cantor real number $\sin(x) \in \mathbb{R}_C$ 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 $-\epsilon \lt \sum_{i = 0}^{n} \frac{(-1)^i x^{2 i + 1}}{(2 i + 1)!} - \sin(x) \lt \epsilon$.

Properties

Relation to other functions

See at trigonometric identity.

Roots

The roots of the sine function, hence the argument where its value is zero, are the integer multiples of pi $\pi \in \mathbb{R}$.

Related concepts

• periodic function

• cosine

• arcsin

• hyperbolic sine, hyperbolic cosine

• exponential, exponential ring, Euler's formula

• trigonometric identity

• pi, circle constant

References

Discussion in constructive analysis:

• Errett Bishop, §7 in: Foundations of Constructive Analysis, McGraw-Hill (1967)

• Errett Bishop, Douglas Bridges §7 in: Constructive Analysis, Grundlehren der mathematischen Wissenschaften 279, Springer (1985) [doi:10.1007/978-3-642-61667-9]

See also

• Wikipedia, Sine and cosine

• Todd Trimble, Characterization of sine