"other ways besides the geometrical definitions above, using tools from calculus and infinite series. With these definitions the trigonometric functions can be defined for complex numbers. The complex exponential function is particularly useful. See Euler's and De Moivre's formulas. A common use of mnemonics is to remember facts and relationships in trigonometry. For example, the ""sine"", ""cosine"", and ""tangent"" ratios in a right triangle can be remembered by representing them and their corresponding sides as strings of letters. For instance, a mnemonic is SOH-CAH-TOA: One way to remember the letters is to sound them out phonetically (i.e., ""SOH-CAH-TOA"", which is"
"Trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle. They relate the angles of a triangle to the lengths of its sides. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, and tangent. In the context of the standard unit circle (a circle with radius 1 unit), where a triangle is formed by a ray starting at the origin and making some angle with the -axis, the sine of the angle"
"Trigonometric functions of matrices The trigonometric functions (especially sine and cosine) for real or complex square matrices occur in solutions of second-order systems of differential equations. They are defined by the same Taylor series that hold for the trigonometric functions of real and complex numbers: with being the th power of the matrix , and being the identity matrix of appropriate dimensions. Equivalently, they can be defined using the matrix exponential along with the matrix equivalent of Euler's formula, , yielding For example, taking to be a standard Pauli matrix, one has as well as, for the cardinal sine function,"
"antiderivatives of other functions. The arcsine, for example, can be written as the following integral: Analogous formulas for the other functions can be found at inverse trigonometric functions. Using the complex logarithm, one can generalize all these functions to complex arguments: In an inner product space, the angle between two non-zero vectors is defined to be The trigonometric functions, as the name suggests, are of crucial importance in trigonometry, mainly because of the following two results. The law of sines states that for an arbitrary triangle with sides , , and and angles opposite those sides , and : where"
"prefix ""co-"" (in ""cosine"", ""cotangent"", ""cosecant"") is found in Edmund Gunter's ""Canon triangulorum"" (1620), which defines the ""cosinus"" as an abbreviation for the ""sinus complementi"" (sine of the complementary angle) and proceeds to define the ""cotangens"" similarly. Trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle. They relate the angles of a triangle to the lengths of its sides. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, and tangent. In"
"sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than . However, when defined with the unit circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2. In fact, all six standard trigonometric functions – sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like versine and exsecant – can be defined geometrically in terms of a unit circle, as shown at right. Using the unit circle, the values of any trigonometric function for many angles other than those labeled can be"
"arguments to the whole number line and to the complex numbers. Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles (often right triangles). In this use, trigonometric functions are used, for instance, in navigation, engineering, and physics. A common use in elementary physics is resolving a vector into Cartesian coordinates. The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations through the year. In modern usage, there"
"be deduced for other angles of an integer number of degrees, for example, where , and and are the above algebraic expressions for, respectively, and , and the principal cube root (that is, the cube root with the largest real part) is to be taken. Trigonometric functions are analytic functions. 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. One can then use the theory of Taylor series to show that the following identities hold for all real numbers . Here,"
"the length of the opposite side, so called because it is the tangent of the complementary or co-angle: Equivalent to the right-triangle definitions, the trigonometric functions can also be defined in terms of the ""rise"", ""run"", and ""slope"" of a line segment relative to horizontal. The slope is commonly taught as ""rise over run"" or . The three main trigonometric functions are commonly taught in the order sine, cosine and tangent. With a line segment length of 1 (as in a unit circle), the following mnemonic devices show the correspondence of definitions: This shows the main use of tangent and"
"the expression will evaluate only to a single value, called its principal value. ""k"" is some integer: Or in one equation: Arcsin satisfies: and For the sine function: The derivative is: The antiderivative is: ""C"" denotes the constant of integration. It is possible to express any trigonometric function in terms of any other (up to a plus or minus sign, or using the sign function). Sine in terms of the other common trigonometric functions: Note that for all equations which use plus/minus (±), the result is positive for angles in the first quadrant. The basic relationship between the sine and"
"the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities. The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to ""co-"". With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are"
"Trigonometric polynomial In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(""nx"") and cos(""nx"") with ""n"" taking on the values of one or more natural numbers. The coefficients may be taken as real numbers, for real-valued functions. For complex coefficients, there is no difference between such a function and a finite Fourier series. Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions. They are used also in the discrete Fourier transform. The term ""trigonometric polynomial"" for the real-valued case can be"
"architectures this method may offer no advantage. The CORDIC algorithm is commonly used in scientific calculators. Sine In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse). More generally, the definition of sine (and other trigonometric functions) can be extended to any real value in terms of the length of a certain line"
"geometry and trigonometry courses, the notation sin, cos, etc., are often used in place of arcsin, arccos, etc. However, the arcsin, arccos, etc., notation is standard in higher mathematics where trigonometric functions are commonly raised to powers, as this avoids confusion between multiplicative inverse and compositional inverse. The law of sines, or sine rule, states that the ratio of the length of a side to the sine of its corresponding opposite angle is constant, that is This ratio is equal to the diameter of the circumscribed circle of the given triangle. Another interpretation of this theorem is that every triangle"
"and , the following equation holds: with the added condition that This may also be used for extending sine and cosine to the complex numbers. Other functional equations are also possible for defining trigonometric functions. 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. This section, however, describes 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"
"picture). These considerations are particularly important for defining the inverses of trigonometric functions. For example, the sine function is not one-to-one, since for every real (and more generally for every integer ). However, the sine is one-to-one on the interval , and the corresponding partial inverse is called the arcsine. This is considered the principal branch of the inverse sine, so the principal value of the inverse sine is always between − and . The following table describes the principal branch of each inverse trigonometric function: If , a left inverse for (or ""retraction"" of ) is a function such"
"of the inner product between line segments taken as vectors). The three main laws in trigonometry – Pythagoras's theorem, the sine law and the cosine law – are given in rational (square-equivalent) form, and are augmented by two further laws – the triple quad formula (relating the quadrances of three collinear points) and the triple spread formula (relating the spreads of three concurrent lines) –, giving the five main laws of the subject. Rational trigonometry is otherwise broadly based on Cartesian analytic geometry, with ""a point"" defined as an ordered pair of rational numbers and ""a line"" as a general"
"formula_15 = formula_16 = formula_17. Also, using the angle addition and subtraction formulae for both the sine and cosine one obtains: Pairwise addition of the above four formulae yields: Setting formula_23 and formula_24 and substituting yields: Dividing the sum of sines by the sum of cosines one arrives at: Applying the formulae derived above to the rhombus figure on the right, it is readily shown that In the unit circle, application of the above shows that formula_28. According to similar triangles, In various applications of trigonometry, it is useful to rewrite the trigonometric functions (such as sine and cosine) in"
"gives the -component (the opposite to the angle or the rise) of the triangle, the cosine gives the -component (the adjacent of the angle or the run), and the tangent function gives the slope (-component divided by the -component). For angles less than a right angle, trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and their values can be found in the lengths of various line segments around a unit circle. Modern definitions express trigonometric functions as infinite series or as solutions of certain differential equations, allowing the extension of the"
"Versine The versine or versed sine is a trigonometric function already appearing in some of the earliest trigonometric tables. The versine of an angle equals 1 minus its cosine. There are several related functions, most notably the coversine and haversine. The latter, half a versine, is of particular importance in the haversine formula of navigation. The versine or versed sine is a trigonometric function already appearing in some of the earliest trigonometric tables. It is written as versin(""θ""), sinver(""θ""), vers(""θ""), ver(""θ"") or siv(""θ""). In Latin, it is known as the ""sinus versus"" (flipped sine), ""versinus"", ""versus"" or the ""sagitta"" (arrow)."
"Sine In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse). More generally, the definition of sine (and other trigonometric functions) can be extended to any real value in terms of the length of a certain line segment in a unit circle. More modern definitions express the sine as an infinite series or"
"Inverse trigonometric functions In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry. There are several notations used for the inverse trigonometric functions. The most common convention is to name inverse trigonometric functions using an arc- prefix, e.g., , , ,"
"""Euler's formula"", as well as near-modern abbreviations (""sin."", ""cos."", ""tang."", ""cot."", ""sec."", and ""cosec.""). A few functions were common historically, but are now seldom used, such as the chord (), the versine () (which appeared in the earliest tables), the coversine (), the haversine (), the exsecant (), and the excosecant (). See List of trigonometric identities for more relations between these functions. The word ""sine"" derives from Latin ""sinus"", meaning ""bend; bay"", and more specifically ""the hanging fold of the upper part of a toga"", ""the bosom of a garment"", which was chosen as the translation of what was"
"The analog of the Pythagorean trigonometric identity holds: If is a diagonal matrix, and are also diagonal matrices with and , that is, they can be calculated by simply taking the sines or cosines of the matrice's diagonal components. The analogs of the trigonometric addition formulas are true if and only if : The tangent, as well as inverse trigonometric functions, hyperbolic and inverse hyperbolic functions have also been defined for matrices: and so on. Trigonometric functions of matrices The trigonometric functions (especially sine and cosine) for real or complex square matrices occur in solutions of second-order systems of differential"
"the cosine can also be expressed as the Pythagorean trigonometric identity: where sin(""x"") means (sin(""x"")). The table below displays many of the key properties of the sine function (sign, monotonicity, convexity), arranged by the quadrant of the argument. For arguments outside those in the table, one may compute the corresponding information by using the periodicity formula_15 of the sine function. The following table gives basic information at the boundary of the quadrants. Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine, and that the derivative of cosine is the negative of"
"defined as the trigonometric functions and today are pervasive in both pure and applied mathematics: fundamental methods of analysis such as the Fourier transform, for example, or the wave equation, use trigonometric functions to understand cyclical phenomena across many applications in fields as diverse as physics, mechanical and electrical engineering, music and acoustics, astronomy, ecology, and biology. Trigonometry is also the foundation of surveying. Trigonometry is most simply associated with planar right-angle triangles (each of which is a two-dimensional triangle with one angle equal to 90 degrees). The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle (on a"
"of the rail tracks and it is the basis of the Hallade method for rail surveying. The term ""sagitta"" (often abbreviated ""sag"") is used similarly in optics, for describing the surfaces of lenses and mirrors. Versine The versine or versed sine is a trigonometric function already appearing in some of the earliest trigonometric tables. The versine of an angle equals 1 minus its cosine. There are several related functions, most notably the coversine and haversine. The latter, half a versine, is of particular importance in the haversine formula of navigation. The versine or versed sine is a trigonometric function already"
"Jyā, koti-jyā and utkrama-jyā Jyā, koti-jyā and utkrama-jyā are three trigonometric functions introduced by Indian mathematicians and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta. These are functions of arcs of circles and not functions of angles. Jyā and koti-jyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of ""sine"" and ""cosine"" have been traced back to the Sanskrit words jyā and koti-jyā. Let 'arc AB' denote an arc whose two extremities are A and B of a circle with center O."
"connection with Euclid's demonstration of the Pythagorean theorem. Pythagorean trigonometric identity The Pythagorean trigonometric identity is a trigonometric identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions. The identity is given by the formula: (Note that means . This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity. Any similar triangles have the property that if we select the same angle in all of them, the ratio of the two sides defining the angle is the same"
"Trigonometric integral In mathematics, the trigonometric integrals are a family of integrals involving trigonometric functions. A number of the basic trigonometric integrals are discussed at the list of integrals of trigonometric functions. The different sine integral definitions are Note that the integrand is the function, and also the zeroth . Since is an even entire function (holomorphic over the entire complex plane), is entire, odd, and the integral in its definition can be taken along any path connecting the endpoints. By definition, is the antiderivative of which is zero for , and is the antiderivative of which is zero for"
"Law of sines In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of a triangle (any shape) to the sines of its angles. According to the law, where , and are the lengths of the sides of a triangle, and , and are the opposite angles (see the figure to the right), while is the diameter of the triangle's circumcircle. When the last part of the equation is not used, the law is sometimes stated using the reciprocals; The law of sines can be used to compute"
"Differentiation of trigonometric functions The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. Common trigonometric functions include sin(""x""), cos(""x"") and tan(""x""). For example, the derivative of ""f""(""x"") = sin(""x"") is represented as ""f"" ′(""a"") = cos(""a""). ""f"" ′(""a"") is the rate of change of sin(""x"") at a particular point ""a"". All derivatives of circular trigonometric functions can be found using those of sin(""x"") and cos(""x""). The quotient rule is then implemented to differentiate the resulting expression. Finding the derivatives of the inverse"
"angle across to make a right angle with one side, the right angle triangle's hypotenuse is length one and the sides are of length and . From this the trigonometric function tangent of 60° equals , and the sine of 60° and the cosine of 30° both equal . The square root of 3 also appears in algebraic expressions for various other trigonometric constants, including the sines of 3°, 12°, 15°, 21°, 24°, 33°, 39°, 48°, 51°, 57°, 66°, 69°, 75°, 78°, 84°, and 87°. It is the distance between parallel sides of a regular hexagon with sides of length"