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.)


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}.


See also

Last revised on May 16, 2022 at 17:29:23. See the history of this page for a list of all contributions to it.