Introduction to the Sine Function

Defining the sine function

The sine function is one of the oldest mathematical functions. It was used in ancient Greece and India, and in 1140, R. de Chesters translated Abu Ja'far Muhammed ibn Musa al‐Khwarizme's works and used the word "sine" (in Latin, "sinus").

The classical definition of the sine function for real arguments is: "the sine of an angle in a right‐angle triangle is the ratio of the length of the opposite leg to the length of the hypotenuse." This description of is valid for when the triangle is nondegenerate. This approach to sine can be expanded to arbitrary real values of if consideration is given to the arbitrary point in the ,‐Cartesian plane and is defined as the ratio , assuming that α is the value of the angle between the positive direction of the ‐axis and the direction from the origin to the point .

The following formula can also be used as a definition of the sine function:

This series converges for all finite numbers .

A quick look at the sine function

Here is a graphic of the sine function for real values of its argument .

Representation through more general functions

The function is a particular case of more complicated mathematical functions. For example, it is a special case of the generalized hypergeometric function with the parameter at , multiplied by :

It is also a particular case of the Bessel function with the parameter , multiplied by :

Other Bessel functions can also be expressed through sine functions for similar values of the parameter:

Struve functions can also degenerate into the sine function for similar values of the parameter:

But the function is also a degenerate case of the doubly periodic Jacobi elliptic functions when their second parameter is equal to or :

Finally, the function is the particular case of another class of functions—the Mathieu functions:

Definition of the sine function for a complex argument

In the complex ‐plane, the function is defined using the exponential function in the points and through the formula:

The key role in this definition of belongs to the famous Euler formula connecting the exponential, the sine, and the cosine functions:

Changing to , the Euler formula becomes:

Taking the difference of the preceding formulas and dividing by 2ⅈ gives the following result:

Here are two graphics showing the real and imaginary parts of the sine function over the complex plane.

The best-known properties and formulas for the sine function

Values in points

Students usually learn the following basic table of sine function values for special points of the circle:

General characteristics

For real values of argument , the values of are real.

In the points , the values of are algebraic. In several cases they can even be rational numbers, 0, or 1. Here are some examples:

The values of can be expressed using only square roots if and is a product of a power of 2 and distinct Fermat primes {3, 5, 17, 257, …}.

The function is an entire analytical function of that is defined over the whole complex ‐plane and does not have branch cuts and branch points. It has an essential singular point at . It is a periodic function with the real period :

The function is an odd function with mirror symmetry:

Differentiation

The derivatives of have simple representations using either the function or the function:

Ordinary differential equation

The function satisfies the simplest possible linear differential equation with constant coefficients:

The complete solution of this equation can be represented as a linear combination of and with arbitrary constant coefficients and :

The function also satisfies first-order nonlinear differential equations:

Series representation

The function has a simple series expansion at the origin that converges in the whole complex ‐plane:

For real this series can be interpreted as the imaginary part of the series expansion for the exponential function :

Product representation

The following famous infinite product representation for clearly illustrates that at :

Indefinite integration

Indefinite integrals of expressions that contain the sine function can sometimes be expressed using elementary functions. However, special functions are frequently needed to express the results even when the integrands have a simple form (if they can be evaluated in closed form). Here are some examples:

The last integral cannot be evaluated in closed form using the known classical special functions for arbitrary values of parameters and .

Definite integration

Definite integrals that contain the sine function are sometimes simple. For example, the famous Dirichlet and Fresnel integrals have the following values:

Some special functions can be used to evaluate more complicated definite integrals. For example, elliptic integrals and gamma functions are needed to express the following integrals:

Integral transforms

Integral transforms of expressions that contain the sine function may not be classically convergent, but they can be interpreted in a generalized functions (distributions) sense. For example, the exponential Fourier transform of the sine function does not exist in the classical sense, but it can be expressed using the Dirac delta function:

Among other integral transforms of the sine function, the best known are the Fourier cosine and sine transforms, the Laplace, the Mellin, the Hilbert, and the Hankel transforms:

Finite summation

The following finite sums from the sine can be expressed using trigonometric functions:

Infinite summation

The following infinite sums can be expressed using elementary functions:

Finite products

The following finite products from the sine can be expressed using trigonometric functions:

Infinite products

The following infinite product can be expressed using the sine function:

Addition formulas

The sine of a sum can be represented by the rule: "the sine of a sum is equal to the sum of the sine multiplied by the cosine and the cosine multiplied by the sine." A similar rule is valid for the sine of the difference:

Multiple arguments

In the case of multiple arguments , , , …, the function can be represented as the finite sum of terms that include powers of the sine and cosine:

The function can also be represented as the finite product including the sine of the linear argument of :

Half-angle formulas

The sine of the half‐angle can be represented by the following simple formula that is valid in some vertical strips:

To make this formula correct for all complex , a complicated prefactor is needed:

where contains the unit step, real part, imaginary part, and the floor functions.

Sums of two direct functions

The sum of two sine functions can be described by the rule: "the sum of the sines is equal to the doubled cosine of the half‐difference multiplied by the sine of the half‐sum." A similar rule is valid for the difference of two sines:

Products involving the direct function

The product of two sine functions and the product of the sine and cosine have the following representations:

Powers of the direct function

The integer powers of the sine functions can be expanded as finite sums of cosine (or sine) functions with multiple arguments. These sums include binomial coefficients:

The previous formulas can be joined into the following one:

Inequalities

The most famous inequalities for the sine function are:

Relations with its inverse function

There are simple relations between the function and its inverse function :

The second formula is valid at least in the vertical strip . Outside of this strip a much more complicated relation (that contains the unit step, real part, and the floor functions) holds:

Representations through other trigonometric functions

Sine and cosine functions are connected by a very simple formula that contains the linear function in the argument:

Another famous formula, connecting and , is reflected in the well‐known Pythagorean theorem:

The last restriction on can be removed, but the formula will get a complicated coefficient that contains the unit step, real part, imaginary part, and the floor function and c(z) ⩵ 1:

The sine function can also be represented using other trigonometric functions by the following formulas:

Representations through hyperbolic functions

The sine function has representations using the hyperbolic functions:

Applications

The sine function is used throughout mathematics, the exact sciences, and engineering.