Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (occasionally also called arcus functions,[1][2][3][4][5] antitrigonometric functions[6] or cyclometric functions[7][8][9]) 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,[10] 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.

NotationEdit

Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin(x), arccos(x), arctan(x), etc.[6] (This convention is used throughout this article.) This notation arises from the following geometric relationships:[citation needed] when measuring in radians, an angle of θ radians will correspond to an arc whose length is , where r is the radius of the circle. Thus in the unit circle, "the arc whose cosine is x" is the same as "the angle whose cosine is x", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians.[11] In computer programming languages, the inverse trigonometric functions are often called by the abbreviated forms asin, acos, atan.[12]

The notations sin−1(x), cos−1(x), tan−1(x), etc., as introduced by John Herschel in 1813,[13][14] are often used as well in English-language sources,[6] much more than the also established sin[−1](x), cos[−1](x), tan[−1](x),—conventions consistent with the notation of an inverse function, that is useful e.g. to define the multivalued version of each inverse trigonometric function: e.g.  . However, this might appear to conflict logically with the common semantics for expressions such as sin2(x) (although only sin2 x, without parentheses, is the really common one), which refer to numeric power rather than function composition, and therefore may result in confusion between multiplicative inverse or reciprocal and compositional inverse.[15] The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name—for example, (cos(x))−1 = sec(x). Nevertheless, certain authors advise against using it for its ambiguity.[6][16] Another precarious convention used by a tiny number of authors is to use an uppercase first letter, along with a −1 superscript: Sin−1(x), Cos−1(x), Tan−1(x), etc.[17] Although its intention is to avoid confusion with the multiplicative inverse, which should be represented by sin−1(x), cos−1(x), etc., or, better, by sin−1 x, cos−1 x, etc., it in turn creates yet another major source of ambiguity: especially in light of the fact that many popular high-level programming languages (e.g. Wolfram's Mathematica, and University of Sydney's MAGMA) use those very same capitalised representations for the standard trig functions; whereas others (Python (ie SymPy and NumPy), Matlab, MAPLE etc) use lower-case.

Hence, since 2009, the ISO 80000-2 standard has specified solely the "arc" prefix for the inverse functions.

Basic conceptsEdit

Principal valuesEdit

Since none of the six trigonometric functions are one-to-one, they must be restricted in order to have inverse functions. Therefore, the result ranges of the inverse functions are proper (i.e. strict) subsets of the domains of the original functions.

For example, using function in the sense of multivalued functions, just as the square root function   could be defined from   the function   is defined so that   For a given real number   with   there are multiple (in fact, countably infinitely many) numbers   such that  ; for example,   but also     etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each   in the domain, the expression   will evaluate only to a single value, called its principal value. These properties apply to all the inverse trigonometric functions.

The principal inverses are listed in the following table.

Name Usual notation Definition Domain of   for real result Range of usual principal value
(radians)
Range of usual principal value
(degrees)
arcsine   x = sin(y)      
arccosine   x = cos(y)      
arctangent   x = tan(y) all real numbers    
arccotangent   x = cot(y) all real numbers    
arcsecant   x = sec(y)      
arccosecant   x = csc(y)      

Note: Some authors[citation needed] define the range of arcsecant to be  , because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range,   whereas with the range  , we would have to write   since tangent is nonnegative on   but nonpositive on   For a similar reason, the same authors define the range of arccosecant to be   or  

If   is allowed to be a complex number, then the range of   applies only to its real part.

The table below displays names and domains of the inverse trigonometric functions along with the range of their usual principal values in radians.

Name
Symbol Domain Image/Range Inverse
function
Domain Image of
principal values
sine                    
cosine                    
tangent                    
cotangent                    
secant                    
cosecant                    

The symbol   denotes the set of all real numbers and   denotes the set of all integers. The set of all integer multiples of   is denoted by

 

The symbol   denotes set subtraction so that, for instance,   is the set of points in   (that is, real numbers) that are not in the interval  

The Minkowski sum notation   and   that is used above to concisely write the domains of   is now explained.

Domain of cotangent   and cosecant  : The domains of   and   are the same. They are the set of all angles   at which   i.e. all real numbers that are not of the form   for some integer  

 

Domain of tangent   and secant  : The domains of   and   are the same. They are the set of all angles   at which  

 

Solutions to elementary trigonometric equationsEdit

Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of  :

  • Sine and cosecant begin their period at   (where   is an integer), finish it at   and then reverse themselves over   to  
  • Cosine and secant begin their period at   finish it at   and then reverse themselves over   to  
  • Tangent begins its period at   finishes it at   and then repeats it (forward) over   to  
  • Cotangent begins its period at   finishes it at   and then repeats it (forward) over   to  

This periodicity is reflected in the general inverses, where   is some integer.

The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions. It is assumed that the given values         and   all lie within appropriate ranges so that the relevant expressions below are well-defined. Note that "for some  " is just another way of saying "for some integer  "

The symbol   is logical equality. The expression "LHS   RHS" indicates that either (a) the left hand side (i.e. LHS) and right hand side (i.e. RHS) are both true, or else (b) the left hand side and right hand side are both false; there is no option (c) (e.g. it is not possible for the LHS statement to be true and also simultaneously for the RHS statement to false), because otherwise "LHS   RHS" would not have been written (see this footnote[note 1] for an example illustrating this concept).

Equation if and only if Solution Expanded form of solution where...
              for some                 or
 
for some  
              for some                 or
 
for some  
                for some                or
 
for some  
                for some                or
 
for some  
            for some  
            for some  

For example, if   then   for some   While if   then   for some   where   will be even if   and it will be odd if   The equations   and   have the same solutions as   and   respectively. In all equations above except for those just solved (i.e. except for  /  and  / ), the integer   in the solution's formula is uniquely determined by   (for fixed   and  ).

Detailed example and explanation of the "plus or minus" symbol  

The solutions to   and   involve the "plus or minus" symbol   whose meaning is now clarified. Only the solution to   will be discussed since the discussion for   is the same. We are given   between   and we know that there is an angle   in some interval that satisfies   We want to find this   The table above indicates that the solution is

 
which is a shorthand way of saying that (at least) one of the following statement is true:
  1.   for some integer  
    or
  2.   for some integer  

As mentioned above, if   (which by definition only happens when  ) then both statements (1) and (2) hold, although with different values for the integer  : if   is the integer from statement (1), meaning that   holds, then the integer   for statement (2) is   (because  ). However, if   then the integer   is unique and completely determined by   If   (which by definition only happens when  ) then   (because   and   so in both cases   is equal to  ) and so the statements (1) and (2) happen to be identical in this particular case (and so both hold). Having considered the cases   and   we now focus on the case where   and   So assume this from now on. The solution to   is still

 
which as before is shorthand for saying that one of statements (1) and (2) is true. However this time, because   and   statements (1) and (2) are different and furthermore, exactly one of the two equalities holds (not both). Additional information about   is needed to determine which one holds. For example, suppose that   and that all that is known about   is that   (and nothing more is known). Then
 
and moreover, in this particular case   (for both the   case and the   case) and so consequently,
 
This means that   could be either   or   Without additional information it is not possible to determine which of these values   has. An example of some additional information that could determine the value of   would be knowing that the angle is above the  -axis (in which case  ) or alternatively, knowing that it is below the  -axis (in which case  ).

Transforming equations

The equations above can be transformed by using the reflection and shift identities:[18]

Argument:              
                     
                     
                   
                   
                       
                       

These formulas imply, in particular, that the following hold:

 
where swapping   swapping   and swapping   gives the analogous equations for   respectively.

So for example, by using the equality   the equation   can be transformed into   which allows for the solution to the equation   (where  ) to be used; that solution being:   which becomes:

 
where using the fact that   and substituting   proves that another solution to   is:
 
The substitution   may be used express the right hand side of the above formula in terms of   instead of  

Equal identical trigonometric functionsEdit

The table below shows how two angles   and   must be related if their values under a given trigonometric function are equal or negatives of each other.

Equation if and only if Solution where... Also a solution to
                  for some    
                    for some    
                for some    
                  for some    
                      for some    
                  for some    
                  for some    
     

Relationships between trigonometric functions and inverse trigonometric functionsEdit

Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of length   then applying the Pythagorean theorem and definitions of the trigonometric ratios. Purely algebraic derivations are longer.[citation needed] It is worth noting that for arcsecant and arccosecant, the diagram assumes that   is positive, and thus the result has to be corrected through the use of absolute values and the signum (sgn) operation.

        Diagram
         
         
         
         
         
         

Relationships among the inverse trigonometric functionsEdit

 
The usual principal values of the arcsin(x) (red) and arccos(x) (blue) functions graphed on the cartesian plane.
 
The usual principal values of the arctan(x) and arccot(x) functions graphed on the cartesian plane.
 
Principal values of the arcsec(x) and arccsc(x) functions graphed on the cartesian plane.

Complementary angles:

 

Negative arguments:

 

Reciprocal arguments:

 

Useful identities if one only has a fragment of a sine table:

 

Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real).

A useful form that follows directly from the table above is

 .

It is obtained by recognizing that  .

From the half-angle formula,  , we get:

 

Arctangent addition formulaEdit

 

This is derived from the tangent addition formula

 

by letting

 

In calculusEdit

Derivatives of inverse trigonometric functionsEdit

The derivatives for complex values of z are as follows:

 

Only for real values of x:

 

For a sample derivation: if  , we get:

 

Expression as definite integralsEdit

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 seriesEdit

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 arctangentEdit

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 functionsEdit

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.

ExampleEdit

Using   (i.e. integration by parts), set

 

Then

 

which by the simple substitution   yields the final result:

 

Extension to complex planeEdit

 
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 formsEdit

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.

 

GeneralizationEdit

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