Sine function
This article is about a particular function from a subset of the real numbers to the real numbers. Information about the function, including its domain, range, and key data relating to graphing, differentiation, and integration, is presented in the article.
View a complete list of particular functions on this wiki
For functions involving angles (trigonometric functions, inverse trigonometric functions, etc.) we follow the convention that all angles are measured in radians. Thus, for instance, the angle ofis measured as
.
Contents
Definition
Unit circle definition
The sine function, denoted , is defined as follows.
Consider the unit circle centered at the origin, described as the following subset of the coordinate:
For a real number , we define
as follows:
- Start at the point
, which lies on the unit circle centered at the origin.
- Move a distance of
along the unit circle in the counter-clockwise direction (i.e., the motion begins in the first quadrant, with both coordinates positive).
- At the end, the
-coordinate of the point thus obtained is defined as
.
Triangle ratio definition (works for acute angles)
For an acute angle , i.e., for
in the open interval
,
can be defined as follows:
- Construct any right triangle with one of the acute angles equal to
.
-
is the ratio of the leg opposite to the angle
to the hypotenuse.
Key data
Item | Value |
---|---|
default domain | all real numbers, i.e., all of ![]() |
range | the closed interval ![]() |
period | ![]() ![]() |
mean value over a period | 0 |
local maximum values and points of attainment | local maximum value attained at all points of the form ![]() |
local minimum values and points of attainment | local minimum value attained at all points of the form ![]() |
points of inflection (both coordinates) | all points of the form ![]() ![]() |
important symmetries | odd function. More generally, half turn symmetry about all points of the form ![]() ![]() Also, mirror symmetry about all lines of the form ![]() |
first derivative | ![]() |
second derivative | ![]() |
sequence of derivatives | starting from first: ![]() |
first antiderivative | ![]() |
Identities
Type of identity | Identity in algebraic form |
---|---|
complementary angle | ![]() ![]() |
square relationship with cosine | ![]() |
angle sum sine formula | ![]() |
angle difference sine formula | ![]() |
product to sum conversion | ![]() |
sum to product conversion | ![]() |
double angle sine formula | ![]() |
double angle cosine formula | ![]() ![]() |
other symmetries | periodicity: ![]() anti-periodicity: ![]() odd: ![]() mirror symmetry about ![]() ![]() |
Related functions
Composition with other functions
Below are some composite functions of the form for suitable function
:
Function ![]() |
![]() ![]() |
![]() ![]() |
---|---|---|
reciprocal function | cosecant function ![]() ![]() |
sine of reciprocal function ![]() |
square function | sine-squared function ![]() |
sine of square function ![]() |
cube function | sine-cubed function ![]() |
sine of cube function ![]() |
positive power function ![]() ![]() |
positive power of sine function | sine of positive power function |
absolute value function | absolute value of sine function ![]() |
uninteresting |
positive part function | positive part of sine function ![]() |
uninteresting |
square root function | square root of sine function -- note that this makes sense only on a restricted domain | sine of square root function |
natural logarithm of absolute value | natural logarithm of absolute value of sine function ![]() |
sine of natural logarithm of absolute value ![]() |
Product with other functions
Function ![]() |
pointwise product of functions ![]() |
---|---|
reciprocal function | sinc function ![]() |
identity function | product of identity function and sine function ![]() |
exponential function | product of exponential function and sine function ![]() |
cosine function | ![]() ![]() |
Differentiation
First derivative
We deduce the formula from the limit:
Here's the full proof:
By the fact that limit is linear, the above limit can be rewritten as:
We now need to compute the two limits individually. Note first that both limits are independent of .
The first limit is:
We've thus expressed the limit as a product of limits where one of the factors goes to zero and the other goes to one, so the limit is zero.
The second limit is 1, as can be seen directly.
We thus get that the answer is:
This simplifies to
Second derivative
The derivative of is
, so we obtain:
Higher derivatives
The sequence of derivatives is periodic with period 4:
![]() |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
In particular, we obtain that for any nonnegative integer :
Equivalently, we also have:
In other words, differentiating the function times is equivalent to shifting the graph
to the left.
Integration
First antiderivative
We have:
Definite integrals
The mean value of over a period is 0. Thus:
Since is odd, the mean value over any interval symmetric about the origin is zero:
Also, the integral of on
and on
is 1 each, giving a mean value of
on these intervals:
Integration of transformed versions of function
We have, for , the following, using integration of linear transform of function:
Further, the mean value of over a period is 0.
Higher antiderivatives
The general expression for the second antiderivative is:
In general, the antiderivative is
or
, depending on the value of
mod 4. The general expression is the particular antiderivative plus an arbitrary polynomial of degree at most
.
Integration of product with polynomials
Using integration by parts, we know that knowledge of the first antiderivatives of
is sufficient to determine
via repeated application of integration by parts. Since the sine function can be antidifferentiated any number of times, this allows us to antidifferentiate any polynomial times the sine function.
For instance, the function , i.e., the product of identity function and sine function, can be integrated via knowledge of how to integrate the sine function twice:
Differential equations
The sine function and its transforms arises as the solution to many differential equations, including polynomial differential equations. Some of these are listed below.
Equation | Other forms | General solution |
---|---|---|
![]() |
![]() ![]() | |
![]() |
![]() |
See Verhulst process |
![]() |
![]() ![]() |
Power series and Taylor series
Computation of Taylor series
As noted above, we have that:
In particular, this means that:
Thus, the sequence of derivatives at zero (starting from ) is
.
The Taylor series is thus:
Taylor series equals power series
The sine function is a globally analytic function: the Taylor series for the sine function does in fact converge to the function everywhere. This can be proved in a number of ways. One method is to use that uniformly bounded derivatives implies analytic. Alternatively, we could note that satisfies a certain differential equation
, forcing it to be given by a power series.
Thus we have that:
Limit computations
Order of zero
We have the following limit:
Thus, the order of zero at 0 is 1 and the residue is 1.
Higher order limits
We have the limit:
The limit can be computed in either of two ways:
Name of method for computing the limit | Details |
---|---|
Using the L'Hopital rule | ![]() |
Using the power series | We have ![]() ![]() ![]() ![]() |