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Trigonometric function

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   All of the trigonometric functions of an angle θ can be constructed
   geometrically in terms of a unit circle centered at O.
   Enlarge
   All of the trigonometric functions of an angle θ can be constructed
   geometrically in terms of a unit circle centered at O.

   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.

   The study of trigonometric functions dates back to Babylonian times,
   and a considerable amount of fundamental work was done by Persian and
   Greek mathematicians.

   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.
   Function Abbreviation Relation
   Sine sin \sin \theta = \cos \left(\frac{\pi}{2} - \theta \right) \,
   Cosine cos \cos \theta = \sin \left(\frac{\pi}{2} - \theta \right)\,
   Tangent tan \tan \theta = \frac{1}{\cot \theta} = \frac{\sin
   \theta}{\cos \theta} = \cot \left(\frac{\pi}{2} - \theta \right) \,
   Cotangent cot \cot \theta = \frac{1}{\tan \theta} = \frac{\cos
   \theta}{\sin \theta} = \tan \left(\frac{\pi}{2} - \theta \right) \,
   Secant sec \sec \theta = \frac{1}{\cos \theta} = \csc
   \left(\frac{\pi}{2} - \theta \right) \,
   Cosecant csc
   (or cosec) \csc \theta =\frac{1}{\sin \theta} = \sec
   \left(\frac{\pi}{2} - \theta \right) \,

History

   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),
   Ptolemy, Aryabhata (476–550), Varahamihira, Brahmagupta, Muḥammad ibn
   Mūsā al-Ḵwārizmī, Abu'l-Wafa, Omar Khayyam, Bhaskara II, Nasir al-Din
   Tusi, Ghiyath al-Kashi (14th century), Ulugh Beg (14th century),
   Regiomontanus (1464), Rheticus, and Rheticus' student Valentin Otho.
   Madhava (c. 1400) made early strides in the analysis of trigonometric
   functions in terms of infinite series. Leonhard Euler's Introductio in
   analysin infinitorum (1748) was mostly responsible for establishing the
   analytic treatment of trigonometric functions in Europe, also defining
   them as infinite series and presenting " Euler's formula", as well as
   the near-modern abbreviations sin., cos., tang., cot., sec., and cosec.

   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 similar triangle 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

   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.
   Enlarge
   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 °); 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

          \sin A = \frac {\textrm{opposite}} {\textrm{hypotenuse}} = \frac
          {a} {h}.

   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

          \left\{n\pi\big| n\isin\mathbb{Z}\right\}.

   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

          \cos A = \frac {\textrm{adjacent}} {\textrm{hypotenuse}} = \frac
          {b} {h}.

   The set of zeroes of cosine is

          \left\{\frac{\pi}{2}+n\pi\bigg| n\isin\mathbb{Z}\right\}.

   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

          \tan A = \frac {\textrm{opposite}} {\textrm{adjacent}} = \frac
          {a} {b}.

   The set of zeroes of tangent is

          \left\{n\pi\big| n\isin\mathbb{Z}\right\}.

   The same set of the sine function since

          \tan A = \frac {\sin A}{\cos A}.

   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:

          \csc A = \frac {\textrm{hypotenuse}} {\textrm{opposite}} = \frac
          {h} {a} .

   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:

          \sec A = \frac {\textrm{hypotenuse}} {\textrm{adjacent}} = \frac
          {h} {b} .

   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:

          \cot A = \frac {\textrm{adjacent}} {\textrm{opposite}} = \frac
          {b} {a} .

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
   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
   mnemonic.

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: 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).

Unit-circle definitions

   The unit circle
   Enlarge
   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 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:

          x^2 + y^2 = 1 \,

   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.
   Enlarge
   The f(x) = sin(x) and f(x) = cos(x) functions graphed on the cartesian
   plane.
   A sin(x) animation which shows the connection between circle and sine.
   Enlarge
   A sin(x) animation which shows the connection between circle and sine.

   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:

          \sin\theta = \sin\left(\theta + 2\pi k \right)
          \cos\theta = \cos\left(\theta + 2\pi k \right)

   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:

          \tan\theta = \frac{\sin\theta}{\cos\theta} \quad \sec\theta =
          \frac{1}{\cos\theta}

          \csc\theta = \frac{1}{\sin\theta} \quad \cot\theta =
          \frac{\cos\theta}{\sin\theta}

   The f(x) = tan(x) function graphed on the cartesian plane.
   Enlarge
   The f(x) = tan(x) function graphed on the cartesian plane.

   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.
   All of the trigonometric functions can be constructed geometrically in
   terms of a unit circle centered at O.
   Enlarge
   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
   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.)

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.
   Enlarge
   The sine function (blue) is closely approximated by its Taylor
   polynomial of degree 7 (pink) for a full cycle centered on the origin.

   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:

          \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} +
          \cdots = \sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}

          \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} +
          \cdots = \sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!}

   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:

      \tan x \, {} = \sum_{n=0}^\infty \frac{U_{2n+1} x^{2n+1}}{(2n+1)!}
                {} = \sum_{n=1}^\infty \frac{(-1)^{n-1} 2^{2n} (2^{2n}-1) B_{2n}
                x^{2n-1}}{(2n)!}
                {} = x + \frac{x^3}{3} + \frac{2 x^5}{15} + \frac{17 x^7}{315} +
                \cdots, \qquad \mbox{for } |x| < \frac {\pi} {2}

   where

          U_n \, is the nth up/down number,
          B_n \, is the nth Bernoulli number, and
          E_n \, (below) is the nth Euler number.

   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.

  \csc x \, {} = \sum_{n=0}^\infty \frac{(-1)^{n+1} 2 (2^{2n-1}-1) B_{2n}
            x^{2n-1}}{(2n)!}
            {} = \frac {1} {x} + \frac {x} {6} + \frac {7 x^3} {360} + \frac {31
            x^5} {15120} + \cdots, \qquad \mbox{for } 0 < |x| < \pi

\sec x \, {} = \sum_{n=0}^\infty \frac{U_{2n} x^{2n}}{(2n)!} =
          \sum_{n=0}^\infty \frac{(-1)^n E_n x^{2n}}{(2n)!}
          {} = 1 + \frac {x^2} {2} + \frac {5 x^4} {24} + \frac {61 x^6} {720} +
          \cdots, \qquad \mbox{for } |x| < \frac {\pi} {2}

   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.

  \cot x \, {} = \sum_{n=0}^\infty \frac{(-1)^n 2^{2n} B_{2n}
            x^{2n-1}}{(2n)!}
            {} = \frac {1} {x} - \frac {x}{3} - \frac {x^3} {45} - \frac {2 x^5}
            {945} - \cdots, \qquad \mbox{for } 0 < |x| < \pi

   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:

          e^{i \theta} = \cos\theta + i\sin\theta \,.

   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 e^ix, 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:

          \sin z \, = \,
          \sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)!}z^{2n+1} \, = \,
          {e^{\imath z} - e^{-\imath z} \over 2\imath} = -\imath \sinh
          \left( \imath z\right)

          \cos z \, = \, \sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n)!}z^{2n}
          \, = \, {e^{\imath z} + e^{-\imath z} \over 2} = \cosh
          \left(\imath z\right)

   where i^2 = −1. Also, for purely real x,

          \cos x \, = \, \mbox{Re } (e^{\imath x})

          \sin x \, = \, \mbox{Im } (e^{\imath x})

   It is also known that exponential processes are intimately linked to
   periodic behaviour.

Definitions via differential equations

   Both the sine and cosine functions satisfy the differential equation

          y\,''=-y.

   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

          y\,'=1+y^2

   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,

          f(x) = \sin(kx); k \ne 0, k \ne 1 \,

   then the derivatives will scale by amplitude.

          f'(x) = k\cos(kx) \, .

   Here, k is a constant that represents a mapping between units. If x is
   in degrees, then

          k = \frac{\pi}{180^\circ}.

   This means that the second derivative of a sine in degrees satisfies
   not the differential equation

          y'' = -y \, ,

   but

          y'' = -k^2y \, ;

   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.

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,

          \left(\sin x\right)^2 + \left(\cos x\right)^2 = 1,

   which is more commonly written in the following way:

          sin^2x + cos^2x = 1.

   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.

          \sin \left(x+y\right)=\sin x \cos y + \cos x \sin y
          \cos \left(x+y\right)=\cos x \cos y - \sin x \sin y

          \sin \left(x-y\right)=\sin x \cos y - \cos x \sin y
          \cos \left(x-y\right)=\cos x \cos y + \sin x \sin y

   When the two angles are equal, the sum formulas reduce to simpler
   equations known as the double-angle formulas.

   For integrals and derivatives of trigonometric functions, see the
   relevant sections of table of derivatives, table of integrals, and list
   of integrals of trigonometric functions.

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:

          s(x)^2 + c(x)^2 = 1,\,
          s(x+y) = s(x)c(y) + c(x)s(y),\,
          c(x+y) = c(x)c(y) - s(x)s(y),\,

   with the added condition that

          0 < xc(x) < s(x) < x\ \mathrm{for}\ 0 < x < 1.

   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.

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).

   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:

          c = \sqrt { a^2+b^2 } = \sqrt2.

   Therefore:

          \sin \left(\pi / 4 \right) = \sin \left(45^\circ\right) = \cos
          \left(\pi / 4 \right) = \cos \left(45^\circ\right) = {1 \over
          \sqrt2},
          \tan \left(\pi / 4 \right) = \tan \left(45^\circ\right) =
          {\sqrt2 \over \sqrt2} = 1.

   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:

          \sin \left(\pi / 6 \right) = \sin \left(30^\circ\right) = \cos
          \left(\pi / 3 \right) = \cos \left(60^\circ\right) = {1 \over
          2},
          \cos \left(\pi / 6 \right) = \cos \left(30^\circ\right) = \sin
          \left(\pi / 3 \right) = \sin \left(60^\circ\right) = {\sqrt3
          \over 2},
          \tan \left(\pi / 6 \right) = \tan \left(30^\circ\right) = \cot
          \left(\pi / 3 \right) = \cot \left(60^\circ\right) = {1 \over
          \sqrt3}.

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:

          \begin{matrix} \mbox{for} & -\frac{\pi}{2} \le y \le
          \frac{\pi}{2}, & y = \arcsin(x) & \mbox{if and only if} & x =
          \sin(y) \\ \\ \mbox{for} & 0 \le y \le \pi, & y = \arccos(x) &
          \mbox{if and only if} & x = \cos(y) \\ \\ \mbox{for} &
          -\frac{\pi}{2} < y < \frac{\pi}{2}, & y = \arctan(x) & \mbox{if
          and only if} & x = \tan(y) \\ \\ \mbox{for} & -\frac{\pi}{2} \le
          y \le \frac{\pi}{2}, y \ne 0, & y = \arccsc(x) & \mbox{if and
          only if} & x = \csc(y) \\ \\ \mbox{for} & 0 \le y \le \pi, y \ne
          \frac{\pi}{2}, & y = \arcsec(x) & \mbox{if and only if} & x =
          \sec(y) \\ \\ \mbox{for} & -\frac{\pi}{2} < y < \frac{\pi}{2}, y
          \ne 0, & y = \arccot(x) & \mbox{if and only if} & x = \cot(y)
          \end{matrix}

   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,

          \arcsin z = z + \left( \frac {1} {2} \right) \frac {z^3} {3} +
          \left( \frac {1 \cdot 3} {2 \cdot 4} \right) \frac {z^5} {5} +
          \left( \frac{1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6 } \right)
          \frac{z^7} {7} + \cdots

   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:

          \arcsin\left(x\right) = \int_0^x \frac 1 {\sqrt{1 -
          z^2}}\,\mathrm{d}z, \quad |x| < 1

   Analogous formulas for the other functions can be found at Inverse
   trigonometric function. Using the complex logarithm, one can generalize
   all these functions to complex arguments:

          \arcsin (z) = -i \log \left( i z + \sqrt{1 - z^2} \right)
          \arccos (z) = -i \log \left( z + \sqrt{z^2 - 1}\right)
          \arctan (z) = \frac{i}{2} \log\left(\frac{1-iz}{1+iz}\right)

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:

          \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}

   also known as:

          \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

   A Lissajous curve, a figure formed with a trigonometry-based function.
   Enlarge
   A Lissajous curve, a figure formed with a trigonometry-based function.

   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:

          c^2=a^2+b^2-2ab\cos C \,

   also known as:

          \cos C=\frac{a^2+b^2-c^2}{2ab}

   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.

Other useful properties

   There is also a law of tangents:

          \frac{a+b}{a-b} =
          \frac{\tan[\frac{1}{2}(A+B)]}{\tan[\frac{1}{2}(A-B)]}

Periodic functions

   Animation of the additive synthesis of a square wave with an increasing
   number of harmonics
   Enlarge
   Animation of the additive synthesis of a square wave with an increasing
   number of harmonics

   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
   projection 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

          x_{\mathrm{square}}(t) = \frac{4}{\pi} \sum_{k=1}^\infty
          {\sin{\left ( (2k-1)t \right )}\over(2k-1)}.

   Retrieved from " http://en.wikipedia.org/wiki/Trigonometric_function"
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   with only minor checks and changes (see www.wikipedia.org for details
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