Existen varias notaciones para las funciones trigonométricas inversas. La convención más común es nombrar funciones trigonométricas inversas usando un prefijo de arco : arcsin ( x ) , arccos ( x ) , arctan ( x ) , etc. [10] [6] (Esta convención se usa a lo largo de este artículo). La notación surge de las siguientes relaciones geométricas: [ cita requerida ] cuando se mide en radianes, un ángulo de θ radianes corresponderá a un arco cuya longitud es rθ , donde r es el radio del círculo. Así, en el círculo unitario , "el arco cuyo coseno es x " es lo mismo que "el ángulo cuyo coseno es x ", porque la longitud del arco del círculo en radios es la misma que la medida del ángulo en radianes. [12] En los lenguajes de programación de computadoras, las funciones trigonométricas inversas a menudo se denominan con las formas abreviadas asin, acos, atan. [13]
Las notaciones sin −1 ( x ) , cos −1 ( x ) , tan −1 ( x ) , etc., como las introdujo John Herschel en 1813, [14] [15] también se utilizan a menudo en fuentes en inglés. [6]: convenciones consistentes con la notación de una función inversa . Esto puede parecer que entra en conflicto lógicamente con la semántica común para expresiones como sin 2 ( x ) , que se refieren al poder numérico en lugar de a la composición de funciones y, por lo tanto, puede dar lugar a una confusión entre el inverso multiplicativo o el recíproco y el inverso compositivo . [16] La confusión se mitiga un poco por el hecho de que cada una de las funciones trigonométricas recíprocas tiene su propio nombre, por ejemplo, (cos ( x )) −1 = sec ( x ) . Sin embargo, algunos autores desaconsejan su uso por su ambigüedad. [6] [17] Otra convención usada por algunos autores es usar una primera letra mayúscula , junto con un superíndice −1 : Sin −1 ( x ) , Cos −1 ( x ) , Tan −1 ( x ) , etc. . [18] Esto potencialmente evita la confusión con el inverso multiplicativo, que debería estar representado por sin −1 ( x ) , cos −1 ( x ) , etc.
Desde 2009, la norma ISO 80000-2 ha especificado únicamente el prefijo "arco" para las funciones inversas.
Conceptos básicos
Valores principales
Dado que ninguna de las seis funciones trigonométricas es uno a uno , deben restringirse para tener funciones inversas. Por lo tanto, los rangos de las funciones inversas son subconjuntos propios de los dominios de las funciones originales.
Por ejemplo, usando función en el sentido de funciones multivalor , así como la función raíz cuadrada y = √ x podría definirse a partir de y 2 = x , la función y = arcsin ( x ) se define de modo que sin ( y ) = x . Para un número real dado x , con −1 ≤ x ≤ 1 , hay múltiples (de hecho, numerables infinitos) números y tales que sin ( y ) = x ; por ejemplo, sin (0) = 0 , pero también sin (π) = 0 , sin (2π) = 0 , etc. Cuando solo se desea un valor, la función puede restringirse a su rama principal . Con esta restricción, para cada x en el dominio, la expresión arcsin ( x ) evaluará solo a un valor único, llamado su valor principal . Estas propiedades se aplican a todas las funciones trigonométricas inversas.
Las principales inversas se enumeran en la siguiente tabla.
(Nota: algunos autores [ cita requerida ] definen el rango de arcosecante como (0 ≤ y < π/2o π ≤ y < 3 π/2), porque la función tangente no es negativa en este dominio. Esto hace que algunos cálculos sean más consistentes. Por ejemplo, usando este rango, tan (arcsec ( x )) = √ x 2 - 1 , mientras que con el rango (0 ≤ y < π/2 o π/2< y ≤ π ), tendríamos que escribir tan (arcsec ( x )) = ± √ x 2 - 1 , ya que la tangente no es negativa en 0 ≤ y < π/2, pero no positivo en π/2< y ≤ π . Por una razón similar, los mismos autores definen el rango de arcososecante como - π < y ≤ - π/2o 0 < y ≤ π/2.)
Si se permite que x sea un número complejo , entonces el rango de y se aplica solo a su parte real.
Soluciones generales
Cada una de las funciones trigonométricas es periódica en la parte real de su argumento, recorriendo todos sus valores dos veces en cada intervalo de 2 π :
El seno y la cosecante comienzan su período en 2 π k - π/2(donde k es un número entero), termínelo en 2 π k + π/2, y luego se invierten sobre 2 π k + π/2hasta 2 π k + 3 π/2.
El coseno y la secante comienzan su período en 2 π k , lo terminan en 2 π k + π y luego se invierten sobre 2 π k + π a 2 π k + 2 π .
La tangente comienza su período en 2 π k - π/2, lo termina en 2 π k + π/2, y luego lo repite (hacia adelante) sobre 2 π k + π/2hasta 2 π k + 3 π/2.
La cotangente comienza su período en 2 π k , lo termina en 2 π k + π y luego lo repite (hacia adelante) sobre 2 π k + π a 2 π k + 2 π .
Esta periodicidad se refleja en las inversas generales, donde k es un número entero.
La tabla siguiente muestra cómo funciones trigonométricas inversas pueden ser utilizados para resolver igualdades que implican las seis funciones trigonométricas estándar, donde se supone que r , s , x , y y todos se encuentran dentro del rango apropiado.
El símbolo ⇔ es igualdad lógica . La expresión "LHS ⇔ RHS" indica que o bien (a) el lado izquierdo (es decir, LHS) y derecho de la parte (es decir, RHS) son tanto verdad, o bien (b) el lado izquierdo y lado derecho son ambos false; no hay ninguna opción (c) (por ejemplo, es no posible que la declaración LHS es cierto, y también al mismo tiempo por la declaración RHS a falso), porque de lo contrario "LHS ⇔ RHS" no habría sido escrito (ver esta nota [nota 1 ] para ver un ejemplo que ilustra este concepto).
Condición
Solución
dónde...
pecado θ = y
⇔
θ = (-1) k arcosin ( y ) + π k
para algunos k ∈ ℤ
⇔
θ = arcosen ( y ) + 2 π k o θ = - arcosen ( y ) + 2 π k + π
para algunos k ∈ ℤ
csc θ = r
⇔
θ = (-1) k arccsc ( r ) + π k
para algunos k ∈ ℤ
⇔
θ = arccsc ( y ) + 2 π k o θ = - arccsc ( y ) + 2 π k + π
para algunos k ∈ ℤ
cos θ = x
⇔
θ = ± arcos ( x ) + 2 π k
para algunos k ∈ ℤ
⇔
θ = arccos ( x ) + 2 π k o θ = - arccos ( x ) + 2 π k + 2 π
para algunos k ∈ ℤ
seg θ = r
⇔
θ = ± segundos de arco ( r ) + 2 π k
para algunos k ∈ ℤ
⇔
θ = segundos de arco ( x ) + 2 π k o θ = - segundos de arco ( x ) + 2 π k + 2 π
para algunos k ∈ ℤ
tan θ = s
⇔
θ = arctan ( s ) + π k
para algunos k ∈ ℤ
cuna θ = r
⇔
θ = arco ( r ) + π k
para algunos k ∈ ℤ
Funciones trigonométricas idénticas iguales
En la siguiente tabla, mostramos cómo dos ángulos θ y φ deben estar relacionados, si sus valores bajo una función trigonométrica dada son iguales o negativos entre sí.
Igualdad
Solución
dónde...
También una solución para
pecado θ
=
pecado φ
⇔
θ =
(-1) k
φ
+
π k
para algunos k ∈ ℤ
csc θ = csc φ
porque θ
=
porque φ
⇔
θ =
±
φ
+
2
π k
para algunos k ∈ ℤ
seg θ = seg φ
bronceado θ
=
bronceado φ
⇔
θ =
φ
+
π k
para algunos k ∈ ℤ
cuna θ = cuna φ
- pecado θ
=
pecado φ
⇔
θ =
(-1) k +1
φ
+
π k
para algunos k ∈ ℤ
csc θ = - csc φ
- cos θ
=
porque φ
⇔
θ =
±
φ
+
2
π k
+ π
para algunos k ∈ ℤ
seg θ = - seg φ
- bronceado θ
=
bronceado φ
⇔
θ =
-
φ
+
π k
para algunos k ∈ ℤ
cuna θ = - cuna φ
| pecado θ |
=
| pecado φ |
⇔
θ =
±
φ
+
π k
para algunos k ∈ ℤ
| tan θ | = | tan φ |
⇕
| csc θ | = | csc φ |
| cos θ |
=
| cos φ |
| sec θ | = | sec φ |
| cuna θ | = | cuna φ |
Relaciones entre funciones trigonométricas y funciones trigonométricas inversas
Las funciones trigonométricas de las funciones trigonométricas inversas se tabulan a continuación. Una forma rápida de derivarlos es considerando la geometría de un triángulo rectángulo, con un lado de longitud 1 y otro lado de longitud x , luego aplicando el teorema de Pitágoras y las definiciones de las razones trigonométricas. Las derivaciones puramente algebraicas son más largas. [ cita requerida ] Vale la pena señalar que para arcsecant y arcososecante, el diagrama asume que x es positivo y, por lo tanto, el resultado debe corregirse mediante el uso de valores absolutos y la operación signum (sgn).
Diagrama
Relaciones entre las funciones trigonométricas inversas
Los valores principales habituales de las funciones arcsin ( x ) (rojo) y arccos ( x ) (azul) se grafican en el plano cartesiano.
Los valores principales habituales de las funciones arctan ( x ) y arccot ( x ) se grafican en el plano cartesiano.
Los valores principales de las funciones arcsec ( x ) y arccsc ( x ) se grafican en el plano cartesiano.
Ángulos complementarios:
Argumentos negativos:
Argumentos recíprocos:
Identidades útiles si solo se tiene un fragmento de una tabla sinusoidal:
Siempre que se use aquí la raíz cuadrada de un número complejo, elegimos la raíz con la parte real positiva (o la parte imaginaria positiva si el cuadrado es real negativo).
Un formulario útil que se deriva directamente de la tabla anterior es
.
Se obtiene reconociendo que .
De la fórmula de medio ángulo ,, obtenemos:
Fórmula de adición arcangente
Esto se deriva de la fórmula de adición de tangente
Dejando
En cálculo
Derivadas de funciones trigonométricas inversas
Las derivadas para valores complejos de z son las siguientes:
Only for real values of x:
For a sample derivation: if , we get:
Expression as definite integrals
Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral:
When x equals 1, the integrals with limited domains are improper integrals, but still well-defined.
Infinite series
Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using power series, as follows. For arcsine, the series can be derived by expanding its derivative, , as a binomial series, and integrating term by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding its derivative in a geometric series, and applying the integral definition above (see Leibniz series).
Series for the other inverse trigonometric functions can be given in terms of these according to the relationships given above. For example, , , and so on. Another series is given by:[19]
Leonhard Euler found a series for the arctangent that converges more quickly than its Taylor series:
[20]
(The term in the sum for n = 0 is the empty product, so is 1.)
Alternatively, this can be expressed as
Another series for the arctangent function is given by
where is the imaginary unit.[21]
Continued fractions for arctangent
Two alternatives to the power series for arctangent are these generalized continued fractions:
The second of these is valid in the cut complex plane. There are two cuts, from −i to the point at infinity, going down the imaginary axis, and from i to the point at infinity, going up the same axis. It works best for real numbers running from −1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the first) are just (nz)2, with each perfect square appearing once. The first was developed by Leonhard Euler; the second by Carl Friedrich Gauss utilizing the Gaussian hypergeometric series.
Indefinite integrals of inverse trigonometric functions
For real and complex values of z:
For real x ≥ 1:
For all real x not between -1 and 1:
The absolute value is necessary to compensate for both negative and positive values of the arcsecant and arccosecant functions. The signum function is also necessary due to the absolute values in the derivatives of the two functions, which create two different solutions for positive and negative values of x. These can be further simplified using the logarithmic definitions of the inverse hyperbolic functions:
The absolute value in the argument of the arcosh function creates a negative half of its graph, making it identical to the signum logarithmic function shown above.
All of these antiderivatives can be derived using integration by parts and the simple derivative forms shown above.
Example
Using (i.e. integration by parts), set
Then
which by the simple substitution yields the final result:
Extensión a plano complejo
A Riemann surface for the argument of the relation tan z = x. The orange sheet in the middle is the principal sheet representing arctan x. The blue sheet above and green sheet below are displaced by 2π and −2π respectively.
Since the inverse trigonometric functions are analytic functions, they can be extended from the real line to the complex plane. This results in functions with multiple sheets and branch points. One possible way of defining the extension is:
where the part of the imaginary axis which does not lie strictly between the branch points (−i and +i) is the branch cut between the principal sheet and other sheets. The path of the integral must not cross a branch cut. For z not on a branch cut, a straight line path from 0 to z is such a path. For z on a branch cut, the path must approach from Re[x]>0 for the upper branch cut and from Re[x]<0 for the lower branch cut.
The arcsine function may then be defined as:
where (the square-root function has its cut along the negative real axis and) the part of the real axis which does not lie strictly between −1 and +1 is the branch cut between the principal sheet of arcsin and other sheets;
which has the same cut as arcsin;
which has the same cut as arctan;
where the part of the real axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and other sheets;
which has the same cut as arcsec.
Logarithmic forms
These functions may also be expressed using complex logarithms. This extends their domains to the complex plane in a natural fashion. The following identities for principal values of the functions hold everywhere that they are defined, even on their branch cuts.
Generalization
Because all of the inverse trigonometric functions output an angle of a right triangle, they can be generalized by using Euler's formula to form a right triangle in the complex plane. Algebraically, this gives us:
or
where is the adjacent side, is the opposite side, and is the hypotenuse. From here, we can solve for .
or
Simply taking the imaginary part works for any real-valued and , but if or is complex-valued, we have to use the final equation so that the real part of the result isn't excluded. Since the length of the hypotenuse doesn't change the angle, ignoring the real part of also removes from the equation. In the final equation, we see that the angle of the triangle in the complex plane can be found by inputting the lengths of each side. By setting one of the three sides equal to 1 and one of the remaining sides equal to our input , we obtain a formula for one of the inverse trig functions, for a total of six equations. Because the inverse trig functions require only one input, we must put the final side of the triangle in terms of the other two using the Pythagorean Theorem relation
The table below shows the values of a, b, and c for each of the inverse trig functions and the equivalent expressions for that result from plugging the values into the equations above and simplifying.
In this sense, all of the inverse trig functions can be thought of as specific cases of the complex-valued log function. Since this definition works for any complex-valued , this definition allows for hyperbolic angles as outputs and can be used to further define the inverse hyperbolic functions. Elementary proofs of the relations may also proceed via expansion to exponential forms of the trigonometric functions.
Example proof
Using the exponential definition of sine, one obtains
Let
Solving for
(the positive branch is chosen)
Color wheel graphs of inverse trigonometric functions in the complex plane
Aplicaciones
Application: finding the angle of a right triangle
A right triangle.
Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine and cosine, it follows that
Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the Pythagorean Theorem: where is the length of the hypotenuse. Arctangent comes in handy in this situation, as the length of the hypotenuse is not needed.
For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle θ with the horizontal, where θ may be computed as follows:
In computer science and engineering
Two-argument variant of arctangent
The two-argument atan2 function computes the arctangent of y / x given y and x, but with a range of (−π, π]. In other words, atan2(y, x) is the angle between the positive x-axis of a plane and the point (x, y) on it, with positive sign for counter-clockwise angles (upper half-plane, y > 0), and negative sign for clockwise angles (lower half-plane, y < 0). It was first introduced in many computer programming languages, but it is now also common in other fields of science and engineering.
In terms of the standard arctan function, that is with range of (− π/2, π/2), it can be expressed as follows:
It also equals the principal value of the argument of the complex number x + iy.
This function may also be defined using the tangent half-angle formulae as follows:
provided that either x > 0 or y ≠ 0. However this fails if given x ≤ 0 and y = 0 so the expression is unsuitable for computational use.
The above argument order (y, x) seems to be the most common, and in particular is used in ISO standards such as the C programming language, but a few authors may use the opposite convention (x, y) so some caution is warranted. These variations are detailed at atan2.
Arctangent function with location parameter
In many applications[22] the solution of the equation is to come as close as possible to a given value . The adequate solution is produced by the parameter modified arctangent function
The function rounds to the nearest integer.
Numerical accuracy
For angles near 0 and π, arccosine is ill-conditioned and will thus calculate the angle with reduced accuracy in a computer implementation (due to the limited number of digits).[23] Similarly, arcsine is inaccurate for angles near −π/2 and π/2.
Ver también
Inverse exsecant
Inverse versine
Inverse hyperbolic functions
List of integrals of inverse trigonometric functions
List of trigonometric identities
Trigonometric function
Trigonometric functions of matrices
Notas
^To clarify, suppose that it is written "LHS ⇔ RHS" where LHS (which abbreviates "Left Hand Side") and RHS are both statements that can individually be either be true or false. For example, if θ and s are some given and fixed numbers and if the following is written:
tan θ = s ⇔ θ = arctan(s) + π k for some k ∈ ℤ
then LHS is the statement "tan θ = s". Depending on what specific values θ and s have, this LHS statement can either be true or false. For instance, LHS is true if θ = 0 and s = 0 (because in this case tan θ = tan 0 = 0 = s) but LHS is false if θ = 0 and s = 2 (because in this case tan θ = tan 0 = 0 which is not equal to s = 2); more generally, LHS is false if θ = 0 and s ≠ 0. Similarly, RHS is the statement "θ = arctan(s) + π k for some k ∈ ℤ". The RHS statement can also either true or false (as before, whether the RHS statement is true or false depends on what specific values θ and s have). The logical equality symbol ⇔ means that (a) if the LHS statement is true then the RHS statement is also necessarily true, and moreover (b) if the LHS statement is false then the RHS statement is also necessarily false. Similarly, ⇔also means that (c) if the RHS statement is true then the LHS statement is also necessarily true, and moreover (d) if the RHS statement is false then the LHS statement is also necessarily false.
Referencias
^Taczanowski, Stefan (1978-10-01). "On the optimization of some geometric parameters in 14 MeV neutron activation analysis". Nuclear Instruments and Methods. ScienceDirect. 155 (3): 543–546. Bibcode:1978NucIM.155..543T. doi:10.1016/0029-554X(78)90541-4.
^Hazewinkel, Michiel (1994) [1987]. Encyclopaedia of Mathematics (unabridged reprint ed.). Kluwer Academic Publishers / Springer Science & Business Media. ISBN 978-155608010-4.
^Ebner, Dieter (2005-07-25). Preparatory Course in Mathematics (PDF) (6 ed.). Department of Physics, University of Konstanz. Archived (PDF) from the original on 2017-07-26. Retrieved 2017-07-26.
^Mejlbro, Leif (2010-11-11). Stability, Riemann Surfaces, Conformal Mappings - Complex Functions Theory (PDF) (1 ed.). Ventus Publishing ApS / Bookboon. ISBN 978-87-7681-702-2. Archived from the original (PDF) on 2017-07-26. Retrieved 2017-07-26.
^Durán, Mario (2012). Mathematical methods for wave propagation in science and engineering. 1: Fundamentals (1 ed.). Ediciones UC. p. 88. ISBN 978-956141314-6.
^ a b c dHall, Arthur Graham; Frink, Fred Goodrich (January 1909). "Chapter II. The Acute Angle [14] Inverse trigonometric functions". Written at Ann Arbor, Michigan, USA. Trigonometry. Part I: Plane Trigonometry. New York, USA: Henry Holt and Company / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. p. 15. Retrieved 2017-08-12. […] α = arcsin m: It is frequently read "arc-sine m" or "anti-sine m," since two mutually inverse functions are said each to be the anti-function of the other. […] A similar symbolic relation holds for the other trigonometric functions. […] This notation is universally used in Europe and is fast gaining ground in this country. A less desirable symbol, α = sin-1m, is still found in English and American texts. The notation α = inv sin m is perhaps better still on account of its general applicability. […]
^Klein, Christian Felix (1924) [1902]. Elementarmathematik vom höheren Standpunkt aus: Arithmetik, Algebra, Analysis (in German). 1 (3rd ed.). Berlin: J. Springer.
^Klein, Christian Felix (2004) [1932]. Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis. Translated by Hedrick, E. R.; Noble, C. A. (Translation of 3rd German ed.). Dover Publications, Inc. / The Macmillan Company. ISBN 978-0-48643480-3. Retrieved 2017-08-13.
^Dörrie, Heinrich (1965). Triumph der Mathematik. Translated by Antin, David. Dover Publications. p. 69. ISBN 978-0-486-61348-2.
^ a b"Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-29.
^Weisstein, Eric W. "Inverse Trigonometric Functions". mathworld.wolfram.com. Retrieved 2020-08-29.
^Beach, Frederick Converse; Rines, George Edwin, eds. (1912). "Inverse trigonometric functions". The Americana: a universal reference library. 21.
^John D. Cook (2021-02-11). "Trig functions across programming languages". Retrieved 2021-03-10.
^Cajori, Florian (1919). A History of Mathematics (2 ed.). New York, NY: The Macmillan Company. p. 272.
^Herschel, John Frederick William (1813). "On a remarkable Application of Cotes's Theorem". Philosophical Transactions. Royal Society, London. 103 (1): 8. doi:10.1098/rstl.1813.0005.
^Korn, Grandino Arthur; Korn, Theresa M. (2000) [1961]. "21.2.-4. Inverse Trigonometric Functions". Mathematical handbook for scientists and engineers: Definitions, theorems, and formulars for reference and review (3 ed.). Mineola, New York, USA: Dover Publications, Inc. p. 811. ISBN 978-0-486-41147-7.
^Bhatti, Sanaullah; Nawab-ud-Din; Ahmed, Bashir; Yousuf, S. M.; Taheem, Allah Bukhsh (1999). "Differentiation of Trigonometric, Logarithmic and Exponential Functions". In Ellahi, Mohammad Maqbool; Dar, Karamat Hussain; Hussain, Faheem (eds.). Calculus and Analytic Geometry (1 ed.). Lahore: Punjab Textbook Board. p. 140.
^Borwein, Jonathan; Bailey, David; Gingersohn, Roland (2004). Experimentation in Mathematics: Computational Paths to Discovery (1 ed.). Wellesley, MA, USA: A. K. Peters. p. 51. ISBN 978-1-56881-136-9.
^Hwang Chien-Lih (2005), "An elementary derivation of Euler's series for the arctangent function", The Mathematical Gazette, 89 (516): 469–470, doi:10.1017/S0025557200178404
^S. M. Abrarov and B. M. Quine (2018), "A formula for pi involving nested radicals", The Ramanujan Journal, 46: 657–665, arXiv:1610.07713, doi:10.1007/s11139-018-9996-8
^when a time varying angle crossing should be mapped by a smooth line instead of a saw toothed one (robotics, astromomy, angular movement in general)[citation needed]
^Gade, Kenneth (2010). "A non-singular horizontal position representation" (PDF). The Journal of Navigation. Cambridge University Press. 63 (3): 395–417. Bibcode:2010JNav...63..395G. doi:10.1017/S0373463309990415.