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In mathematics, the trigonometric functions are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other applications. 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 define them equally well geometrically or by other means and then derive these relations. A few other functions were common historically (and appeared in the earliest tables), but are now seldom used, such as the versine (1 − cos θ) and the exsecant (sec θ − 1). Many more relations between these functions are listed in the article about trigonometric identities.
The earliest use of sine appears in the Sulba Sutras written in ancient India from the 8th century BC to the 6th century BC. Trigonometric functions were later studied by Hipparchus of Nicaea (180-125 BC), Aryabhata (476–550), Varahamihira, Brahmagupta, Muḥammad ibn Mūsā al-Ḵwārizmī, Abu'l-Wafa, Omar Khayyam, Bhaskara II, Nasir al-Din Tusi, Regiomontanus (1464), Ghiyath al-Kashi (14th century), Ulugh Beg's (14th century), Madhava (c. 1400), Rheticus , Rheticus' student Valentin Otho. Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, defining them as infinite series and presenting "Euler's formula."
The notion that there should be some standard correspondence between the length of the sides of a triangle and the angles of the triangle comes as soon as one recognises that similar triangles maintain the same ratios between their sides. That is, for any set of similar triangles the ratio of the hypotenuse (for example) and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides. It is just these ratios that the trig. functions express.
Right triangle definitions
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; here we call it h.
- The opposite side is the side opposite to the angle we are interested in; we call it 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. We call the adjacent side 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°); therefore, for a right triangle the two non-right angles are between zero and π/2 radians. The reader should note that the following definitions, strictly speaking, only define the trigonometric functions for angles in this range. We extend them to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions.
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
- . Intuitively, because when the angle becomes very small, the opposite side also becomes very small.
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
- When the angle gets close to (a right angle), the adjacent side becomes very small, so .
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
the same set as for the sine function since
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:
- . The secant and cosecant have no zeroes; all their values are either or .
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:
- . The set of zeroes is the same as for .
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 the use of American English or British English, or even "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.
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:
- Sine is first, rise is first. Sine takes an angle and tells the rise.
- Cosine is second, run is second. Cosine takes an angle and tells the run.
- 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: converting between the two ways of telling the slant of a line, i.e., angles and slopes. (Note that the arctangent or "inverse tangent" is not to be confused with the cotangent, which is 1 divided by the tangent.)
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).
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 trigonometric 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.
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 with 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, cosine, secant, or cosecant is a full circle, i.e. 2π radians or 360 degrees; the 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 trigonometric functions can be defined by:
To the right is an image that displays a noticeably different graph of the trigonometric function f(θ)= tan(θ) graphed on the cartesian plane. Note that its x-intercepts correspond to that of sin(θ) while its undefined values correspond to the x-intercepts of the cos(θ). Observe that the function's results change slowly around angles of kπ, but change rapidly at angles close to (k/2)π. The graph of the tangent function also has a vertical asymptote at θ = kπ/2. This is the case because the function approaches infinity as θ approaches k/π from the left and minus infinity as it approaches k/π from the right.
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 above). 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.)
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 negative of sine. (Here, and generally in calculus, all angles are measured in radians; see also the significance of radians below.) 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.
Other series can be found:
When this is expressed in a form in which the denominators are the corresponding factorials, and the numerators, called the "tangent numbers", have a combinatorial interpretation: they enumerate alternating permutations of finite sets of odd cardinality.
When this is expressed in a form in which the denominators are the corresponding factorials, the numerators, called the "secant numbers", have a combinatorial interpretation: they enumerate alternating permutations of finite sets of even cardinality.
From a theorem in complex analysis, there is a unique analytic extension of this real function to the complex numbers. They have the same Taylor series, and so the trigonometric functions are defined on the complex numbers using the Taylor series above.
Relationship to exponential function and complex numbers
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.
Furthermore, this allows for the definition of the trigonometric functions for complex arguments z:
where i2 = −1. Also, for purely real x,
It is also known that exponential processes are intimately linked to periodic behavior.
Definitions via differential equations
Both the sine and cosine functions satisfy the differential equation
That is to say, 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.
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; see Needham's Visual Complex Analysis.
The significance of radians
Radians specify an angle by measuring the length around the path of the circle and 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
This means that the second derivative of a sine in degrees satisfies not the differential equation
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.
- Main article: List of trigonometric identities.
Many identities exist which interrelate the trigonometric functions. Among the most frequently used is the Pythagorean identity, which states that for any angle, the square of the sine plus the square of the cosine is always 1. This is easy to see by studying a right triangle of hypotenuse 1 and applying the Pythagorean theorem. In symbolic form, the Pythagorean identity reads,
which is more commonly written in the following way:
Other key relationships are the sum and difference formulas, which give the sine and cosine of the sum and difference of two angles in terms of sines and cosines of the angles themselves. These can be derived geometrically, using arguments which go back to Ptolemy; one can also produce them algebraically using Euler's formula.
When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulas.
Definitions using functional equations
In mathematical analysis, one can define the trigonometric functions using functional equations based on properties like the sum and difference formulas. Taking as given these formulas and the Pythagorean identity, for example, one can prove that only two real functions satisfy those conditions. Symbolically, we say that there exists exactly one pair of real functions s and c such that for all real numbers x and y, the following equations hold:
with the added condition that
Other derivations, starting from other functional equations, are also possible, and such derivations can be extended to the complex numbers. As an example, this derivation can be used to define trigonometry in Galois fields.
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).
Modern computers use a variety of techniques. 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:
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:
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. The notation using the "arc-" prefix avoids such confusion, though "arcsec" may occasionally be confused with "arcsecond".
Just like the sine and cosine, the inverse trigonometric functions can also be defined in terms of infinite series. For example,
These functions may also be defined by proving that they are antiderivatives of other functions. The arcsine, for example, can be written as the following integral:
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:
The law of sines can be proved by dividing the triangle into two right ones and using the above definition of sine. The common number (sin A)/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
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 or if all three sides are known.
Other useful properties
There is also a law of tangents:
The sine and the cosine functions also appear when describing a simple harmonic motion, another important concept in physics. In this context the sine and cosine functions are used to describe one dimension projections of the uniform circular motion, the mass in a string movement, and a small angle approximation of the mass on a pendulum movement.
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. For example the square wave, can be written as the Fourier series
- Abramowitz; Weisstein.
- Needham, p. ix.