nLab sine








The sine function sin\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.


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

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

    sin=sin sin'' = -sin

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

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

    sin(x) =Im(exp(ix)) =12i(exp(ix)exp(ix)). \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\sin is the unique function which

    1. is continuous at 00

    2. satisfies the inequality

      1x 2sin(x)x1 1 - x^2 \;\leq\; \frac{\sin(x)}{x} \;\leq\; 1

      and the equation

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

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

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

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

Axiom of sine function: For all elements xFx \in F, there exists a unique element sin(x)F\sin(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+1(2i+1)!)sin(x)<ϵ-\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 F -algebra F[ϵ]/ϵ 2F[\epsilon]/\epsilon^2 by the equation f(x+ϵ)=f(x)+f(x)ϵf(x + \epsilon) = f(x) + f'(x) \epsilon:

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

In the dual numbers

The dual number real algebra 𝔻[ϵ]/ϵ 2\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+ϵ)=sin(x)+cos(x)ϵ\sin(x + \epsilon) = \sin(x) + \cos(x) \epsilon and cos(x+ϵ)=cos(x)sin(x)ϵ\cos(x + \epsilon) = \cos(x) - \sin(x) \epsilon with sin(0)=1\sin(0) = 1 and cos(0)=1\cos(0) = 1.

In constructive mathematics

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

i=0 n(1) ix 2i+1(2i+1)!\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 Cx \in \mathbb{R}_C. However, the sequence, by definition of C\mathbb{R}_C, does have a limit for all rational numbers xx \in \mathbb{Q}; this means that one could restrict the domain of the sine function to the rational numbers sin: C\sin:\mathbb{Q} \to \mathbb{R}_C, and define it in the usual manner:

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


Relation to other functions

See at trigonometric identity.


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


Discussion in constructive analysis:

See also

Last revised on February 8, 2023 at 11:26:03. See the history of this page for a list of all contributions to it.