In mathematics, the trigonometric functions are functions of an angle,
important when studying triangles and modeling periodic phenomena. They are
commonly defined as ratios of two sides of a right triangle containing the
angle, and can equivalently be defined as the lengths of various line segments
from a unit circle. More modern definitions express them as infinite series or
as solutions of certain differential equations, allowing their extension to
positive and negative values and even to complex numbers. All of these
approaches will be presented below.
In modern usage, there are six basic
trigonometric functions, which are tabulated below along with equations relating
them to one another. (Especially in the case of the last four, these relations
are often taken as the definitions of those functions, but one can equally
define them geometrically or by other means and derive the relations.)
Function | Abbreviation | Relation |
Sine | sin | |
Cosine | cos | |
Tangent | tan | |
Cotangent | cot | |
Secant | sec | |
Cosecant | csc (or cosec) |
A few other functions were common historically (and appeared in the earliest
tables), but are now little-used, such as:
* versed sine (versin = 1 −
cos)
* exsecant (exsec = sec − 1).
Many more relations between these
functions are listed in the article about trigonometric identities.
Contents
1 History 2 Right triangle definitions 2.1 Mnemonics 2.2 Slope definitions 3 Unit-circle definitions 4 Series definitions 4.1 Relationship to exponential function 5 Definitions via differential equations 5.1 The significance of radians 6 Other definitions 7 Computation 8 Inverse functions 9 Identities 10 Properties and applications 10.1 Law of sines 10.2 Law of cosines 10.3 Law of tangents 11 References |
History
The earliest systematic study of trigonometric functions and tabulation of
their values was performed by Hipparchus of Nicaea (180-125 BC), who tabulated
the lengths of circle arcs (angle A times radius r) with the lengths of the
subtending chords (2r sin(A/2)). Later, Ptolemy (2nd century) expanded upon this
work in his Almagest, deriving addition/subtraction formulas for the equivalent
of sin(A + B) and cos(A + B). Ptolemy also derived the equivalent of the
half-angle formula sin2(A/2) = (1 − cos(A))/2, allowing him to create tables
with any desired accuracy. Neither the tables of Hipparchus nor of Ptolemy have
survived to the present day.
The next significant development of
trigonometry was in India, in the works known as the Siddhantas (4th5th
century), which first defined the sine as the modern relationship between half
an angle and half a chord. The Siddhantas also contained the earliest surviving
tables of sine values (along with 1 − cos values), in 3.75-degree intervals from
0 to 90 degrees.
The Hindu works were later translated and expanded by
the Arabs, who by the 10th century (in the work of Abu'l-Wefa) were using all
six trigonometric functions, and had sine tables in 0.25-degree increments, to 8
decimal places of accuracy, as well as tables of tangent values.
Our
modern word sine comes, via sinus ("bay" or "fold") in Latin, from a
mistranslation of the Sanskrit jiva (or jya). jiva (originally called
ardha-jiva, "half-chord", in the 6th century Aryabhatiya) was transliterated by
the Arabs as jiba (جب), but was confused for another word, jaib (جب) ("bay"), by
European translators such as Robert of Chester and Gherardo of Cremona in Toledo
in the 12th century, probably because jiba (جب) and jaib (جب) are written the
same in Arabic (many vowels are excluded from words written in the Arabic
alphabet).
All of these earlier works on trigonometry treated it mainly
as an adjunct to astronomy; perhaps the first treatment as a subject in its own
right was by the De triangulis omnimodus (1464) of Regiomontanus (14361476), as
well as his later Tabulae directionum (which included the tangent function,
unnamed).
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The Opus palatinum de triangulis of Rheticus, a student of
Copernicus, was the first to define trigonometric functions directly in terms of
right triangles instead of circles, with tables for all six trigonometric
functions; this work was finished by Rheticus' student Valentin Otho in
1596.
The Introductio in analysin infinitorum (1748) of Euler was
primarily responsible for establishing the analytic treatment of trigonometric
functions, defining them as infinite series and presenting "Euler's formula" eix
= cos(x) + i sin(x). Euler used the near-modern abbreviations sin., cos., tang.,
cot., sec., and cosec..
Right triangle definitions
A right triangle always includes a 90° (π/2 radians) angle, here labeled C. Angles A and B may vary. Trigonometric functions specify the relationships between side lengths and interior angles of a right triangle. |
In order to define the trigonometric functions for the angle A, start with an
arbitrary right triangle that contains the angle A:
We use the following
names for the sides of the triangle:
* The hypotenuse is the side opposite
the right angle, or defined as the longest side of a right-angled triangle, in
this case h.
* The opposite side is the side opposite to the angle we are
interested in, in this case a.
* The adjacent side is the side that is in
contact with the angle we are interested in and the right angle, hence its name.
In this case the adjacent side is b.
All triangles are taken to exist in the
Euclidean plane so that the inside angles of each triangle sum to π radians (or
180°). Then,
1) The sine of an angle is the ratio of the length of the
opposite side to the length of the hypotenuse. In our case
Note
that this ratio does not depend on the particular right triangle chosen, as long
as it contains the angle A, since all those triangles are similar.
The set
of zeroes of sine is
2) The cosine of an angle is the ratio of the length of the adjacent side to
the length of the hypotenuse. In our case
The set of zeroes of
cosine is
3) The tangent of an angle is the ratio of the length of the opposite side
to the length of the adjacent side. In our case
The set of zeroes of
tangent is
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The remaining three functions are best defined using the above three
functions.
4) The cosecant csc(A) is the multiplicative inverse of
sin(A), i.e. the ratio of the length of the hypotenuse to the length of the
opposite side:
5) The secant sec(A) is the multiplicative inverse of cos(A),
i.e. the ratio of the length of the hypotenuse to the length of the adjacent
side:
6) The cotangent cot(A) is the multiplicative inverse of
tan(A), i.e. the ratio of the length of the adjacent side to the length of the
opposite side:
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Mnemonics
There are a number
of mnemonics for the above definitions, for example SOHCAHTOA (sounds like "soak
a toe-a" or "sock-a toe-a" depending upon which side of the Atlantic you hail
from. Can also be read as "soccer tour"). It means:
* SOH ... sin =
opposite/hypotenuse
* CAH ... cos = adjacent/hypotenuse
* TOA ... tan =
opposite/adjacent.
Many other such words and phrases have been contrived.
For more see: trigonometry mnemonics.
Slope
definitions
Equivalent to the right-triangle definitions, the
trigonometric functions can be defined in terms of the rise, run, and slope of a
line segment relative to some horizontal line. The slope is commonly taught as
"rise over run" or rise/run. The three main trigonometric functions are commonly
taught in the order sine, cosine, tangent. With a unit circle, this gives rise
to the following matchings:
1. Sine is first, rise is first. Sine takes
an angle and tells the rise.
2. Cosine is second, run is second. Cosine takes
an angle and tells the run.
3. Tangent is the slope formula that combines the
rise and run. Tangent takes an angle and tells the slope.
This shows the
main use of tangent and arctangent, which is converting between the two ways of
telling how slanted a line is: angles and slopes.
While the radius of the
circle makes no difference for the slope (the slope doesn't depend on the length
of the slanted line), it does affect rise and run. To adjust and find the actual
rise and run, just multiply the sine and cosine by the radius. For instance, if
the circle has radius 5, the run at an angle of 1 is 5 cos(1).
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Unit-circle definitions
The Unit Circle |
The six trigonometric functions can also be defined in terms of the unit
circle, the circle of radius one centered at the origin. The unit circle
definition provides little in the way of practical calculation; indeed it relies
on right triangles for most angles. The unit circle definition does, however,
permit the definition of the trig functions for all positive and negative
arguments, not just for angles between 0 and π/2 radians. It also provides a
single visual picture that encapsulates at once all the important triangles used
so far. The equation for the unit circle is:
In the picture, some
common angles, measured in radians, are given. Measurements in the counter
clockwise direction are positive angles and measurements in the clockwise
direction are negative angles. Let a line making an angle of θ with the positive
half of the x-axis intersect the unit circle. The x- and y-coordinates of this
point of intersection are equal to cos θ and sin θ, respectively. The triangle
in the graphic enforces the formula; the radius is equal
to the hypotenuse and has length 1, so we have sin θ = y/1 and cos θ = x/1. The unit circle can be thought of as a way of looking at an infinite number of triangles by varying the lengths of their legs but keeping the lengths of their hypotenuses equal to 1.
The f(x) = sin(x) and f(x) = cos(x) functions graphed on the cartesian plane |
For angles greater than 2π or less than −2π, simply continue to rotate around the circle. In this way, sine and cosine become periodic functions w
ith period 2π:
for any angle θ and any integer k.
The smallest positive period of a periodic function is called the primitive
period of the function. The primitive period of the sine, primitive period of
the tangent or cotangent is only a half-circle, i.e. π radians or 180
degrees. Above, only sine and cosine were defined directly by the unit
circle, but the other four trig functions can be defined by:
All of the trigonometric functions can be constructed geometrically in terms of a unit circle centered at O. |
Alternatively, all of the basic trigonometric functions can be defined in
terms of a unit circle centered at O (shown at right), and similar such
geometric definitions were used historically. In particular, for a chord AB of
the circle, where θ is half of the subtended angle, sin(θ) is AC (half of the
chord), a definition introduced in India (see below). cos(θ) is the horizontal
distance OC, and versin(θ) = 1 − cos(θ) is CD. tan(θ) is the length of the
segment AE of the tangent line through A, hence the word tangent for this
function. cot(θ) is another tangent segment, AF. sec(θ) = OE and csc(θ) = OF are
segments of secant lines (intersecting the circle at two points), and can also
be viewed as projections of OA along the tangent at A to the horizontal and
vertical axes, respectively. DE is exsec(θ) = sec(θ) − 1 (the portion of the
secant outside, or ex, the circle). From these constructions, it is easy to see
that the secant and tangent functions diverge as θ approaches π/2 (90 degrees)
and that the cosecant and cotangent diverge as θ approaches zero. (Many similar
constructions are possible, and the basic trigonometric identities can also be
proven graphically.)
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Series definitions
The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin. |
Please note: Here, and generally in calculus, all angles are measured in
radians. (See also below).
Using only geometry and properties of limits, it
can be shown that the derivative of sine is cosine and the derivative of cosine
is the opposite of sine. One can then use the theory of Taylor series to show
that the following identities hold for all real numbers x:
These identities are
often taken as the definitions of the sine and cosine function. They are often
used as the starting point in a rigorous treatment of trigonometric functions
and their applications (e.g. in Fourier series), since the theory of infinite
series can be developed from the foundations of the real number system,
independent of any geometric considerations. The differentiability and
continuity of these functions are then established from the series definitions
alone.
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Other series can be found:
where
is the nth Euler
number, and
is the nth up/down number.
Relationship to exponential
function
It can be shown from the series definitions that the
sine and cosine functions are the imaginary and real parts, respectively, of the
complex exponential function when its argument is purely imaginary:
This
relationship was first noted by Euler and the identity is called Euler's
formula. In this way, trigonometric functions become essential in the geometric
interpretation of complex analysis. For example, with the above identity, if one
considers the unit circle in the complex plane, defined by eix, and as above, we
can parametrize this circle in terms of cosines and sines, the relationship
between the complex exponential and the trigonometric functions becomes more
apparent.
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Furthermore, this allows for the definition of the
trigonometric functions for complex arguments z:
where i^{2} =
−1. Also, for purely real x,
It is also shown that
exponential processes are intimately linked to periodic
behavior.
Definitions via differential equations
Both the sine and cosine functions satisfy the differential equation
i.e. each is the
additive inverse of its own second derivative. Within the 2-dimensional vector
space V consisting of all solutions of this equation, the sine function is the
unique solution satisfying the initial conditions y(0) = 0 and y′(0) = 1, and
the cosine function is the unique solution satisfying the initial conditions
y(0) = 1 and y′(0) = 0. Since the sine and cosine functions are linearly
independent, together they form a basis of V. This method of defining the sine
and cosine functions is essentially equivalent to using Euler's formula. (See
linear differential equation.) It turns out that this differential equation can
be used not only to define the sine and cosine functions but also to prove the
trigonometric identities for the sine and cosine functions. See the
trigonometric identity article for this technique.
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The tangent function is the unique solution of the nonlinear differential
equation
satisfying the initial condition y(0) = 0. There is a very
interesting visual proof that the tangent function satisfies this differential
equation.
The significance of radians
Radians
constitute a special argument to the sine and cosine functions. In particular,
only those sines and cosines which map radians to ratios satisfy the
differential equations which classically describe them. If an argument to sine
or cosine in radians is scaled by frequency,
then the derivatives will
scale by amplitude.
Here, k is a constant that represents a mapping between units. If
x is in degrees, then
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This means that the second derivative of a sine in degrees satisfies not the
differential equation
but
similarly for cosine.
This means that these sines and
cosines are different functions, and that the fourth derivative of sine will be
sine again only if the argument is in radians.
Other
definitions
Theorem: There exists exactly one pair of real functions s, c with the
following properties:
For any
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Computation
The computation of trigonometric functions is a complicated subject, which
can today be avoided by most people because of the widespread availability of
computers and scientific calculators that provide built-in trigonometric
functions for any angle. In this section, however, we describe more details of
their computation in three important contexts: the historical use of
trigonometric tables, the modern techniques used by computers, and a few
"important" angles where simple exact values are easily found. (Below, it
suffices to consider a small range of angles, say 0 to π/2, since all other
angles can be reduced to this range by the periodicity and symmetries of the
trigonometric functions.)
Prior to computers, people typically evaluated
trigonometric functions by interpolating from a detailed table of their values,
calculated to many significant figures. Such tables have been available for as
long as trigonometric functions have been described (see History, above), and
were typically generated by repeated application of the half-angle and
angle-addition identities starting from a known value (such as sin(π/2)=1). See
also: Generating trigonometric tables.
Modern computers use a variety of
techniques (Kantabutra, 1996). One common method, especially on higher-end
processors with floating point units, is to combine a polynomial approximation
(such as a Taylor series or a rational function) with a table lookup they
first look up the closest angle in a small table, and then use the polynomial to
compute the correction. On simpler devices that lack hardware multipliers, there
is an algorithm called CORDIC (as well as related techniques) that is more
efficient, since it uses only shifts and additions. All of these methods are
commonly implemented in hardware for performance reasons.
Finally, for
some simple angles, the values can be easily computed by hand using the
Pythagorean theorem, as in the following examples. In fact, the sine, cosine and
tangent of any integer multiple of π/60 radians (three degrees) can be found
exactly by hand.
Consider a right triangle where the two other angles are
equal, and therefore are both π/4 radians (45 degrees). Then the length of side
b and the length of side a are equal; we can choose a = b = 1. The values of
sine, cosine and tangent of an angle of π/4 radians (45 degrees) can then be
found using the Pythagorean theorem:
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Therefore:
To determine
the trigonometric functions for angles of π/3 radians (60 degrees) and π/6
radians (30 degrees), we start with an equilateral triangle of side length 1.
All its angles are π/3 radians (60 degrees). By dividing it into two, we obtain
a right triangle with π/6 radians (30 degrees) and π/3 radians (60 degrees)
angles. For this triangle, the shortest side = 1/2, the next largest side
=(√3)/2 and the hypotenuse = 1. This yields:
See also: Exact
trigonometric constants
Inverse functions
The trigonometric functions are periodic, so we must restrict their domains
before we are able to define a unique inverse. In the following, the functions
on the left are defined by the equation on the right; these are not proved
identities. The principal inverses are usually defined as:
For inverse trigonometric functions, the notations sin−1 and cos−1 are often
used for arcsin and arccos, etc. When this notation is used, the inverse
functions are sometimes confused with the multiplicative inverses of the
functions. Our notation avoids such confusion.
The following series
definition may be obtained:
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These functions may also be defined by proving that
they are antiderivatives of other functions.
Inverse trigonometric
functions can be generalized to complex arguments using the complex
logarithm.
Note: arcsec can also mean arcsecond.
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Identities
See also trigonometric identity.
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Properties and applications
The trigonometric functions, as the name suggests, are of crucial importance
in trigonometry, mainly because of the following two results:
Law
of sines
The law of sines for an arbitrary triangle
states:
It can be proven by dividing the triangle into two right ones
and using the above definition of sine. The common number (sinA)/a occurring in
the theorem is the reciprocal of the diameter of the circle through the three
points A, B and C. The law of sines is useful for computing the lengths of the
unknown sides in a triangle if two angles and one side are known. This is a
common situation occurring in triangulation, a technique to determine unknown
distances by measuring two angles and an accessible enclosed
distance.
Law of cosines
The law of cosines
(also known as the cosine formula) is an extension of the Pythagorean
theorem:
Again, this theorem can be proven by dividing the triangle into
two right ones. The law of cosines is useful to determine the unknown data of a
triangle if two sides and an angle are known.
If the angle is not
contained between the two sides, the triangle may not be unique. Be aware of
this ambiguous case of the Cosine law.
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Law of tangents
There is also a law of
tangents:
Functions based on sine and cosine can make appealing pictures.
Functions based on sine and cosine can make appealing pictures. |
The trigonometric functions are also important outside of the study of
triangles. They are periodic functions with characteristic wave patterns as
graphs, useful for modelling recurring phenomena such as sound or light waves.
Every signal can be written as a (typically infinite) sum of sine and cosine
functions of different frequencies; this is the basic idea of Fourier analysis,
where trigonometric series are used to solve a variety of boundary-value
problems in partial differential equations.
The image on the right
displays a two-dimensional graph based on such a summation of sines and cosines,
illustrating the fact that arbitrarily complicated closed curves can be
described by a Fourier series. Its equation is:
where
F(n) is the nth Fibonacci number.
For a compilation of many relations
between the trigonometric functions, see trigonometric identities.
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References
* Carl B. Boyer, A History of Mathematics, 2nd ed. (Wiley, New York,
1991).
* Eli Maor, Trigonometric Delights (Princeton Univ. Press, 1998).
*
"Trigonometric functions", MacTutor History of Mathematics Archive.
* Tristan
Needham, Visual Complex Analysis, (Oxford University Press, 2000), ISBN
0198534469 Book website
* Vitit Kantabutra, "On hardware for computing
exponential and trigonometric functions," IEEE Trans. Computers 45 (3), 328-339
(1996).