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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.

Notation edit

 
For a circle of radius 1, arcsin and arccos are the lengths of actual arcs determined by the quantities in question.

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 (for example) to define the multivalued version of each inverse trigonometric function:   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 use), which refer to numeric power rather than function composition, and therefore may result in confusion between notation for the reciprocal (multiplicative inverse) and inverse function.[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, since it is ambiguous.[6][16] Another precarious convention used by a small 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 it is intended to avoid confusion with the reciprocal, 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 since many popular high-level programming languages (e.g. Mathematica, and MAGMA) use those very same capitalised representations for the standard trig functions, whereas others (Python, SymPy, 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 concepts edit

 
The points labelled 1, Sec(θ), Csc(θ) represent the length of the line segment from the origin to that point. Sin(θ), Tan(θ), and 1 are the heights to the line starting from the x-axis, while Cos(θ), 1, and Cot(θ) are lengths along the x-axis starting from the origin.

Principal values edit

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  

Domains edit

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 equations edit

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 be 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
              for some  
              for some  
                for some  
                for some  
            for some  
            for some  

where the first four solutions can be written in expanded form as:

Equation if and only if Solution
     
          or
 
for some  
     
          or
 
for some  
     
         or
 
for some  
     
         or
 
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 ± edit

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  ).

Equal identical trigonometric functions edit

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 (for some  ) Also a solution to
       
       
       
       
       
       
       

The vertical double arrow   in the last row indicates that   and   satisfy   if and only if they satisfy  

Set of all solutions to elementary trigonometric equations

Thus given a single solution   to an elementary trigonometric equation (  is such an equation, for instance, and because   always holds,   is always a solution), the set of all solutions to it are:

If   solves then Set of all solutions (in terms of  )
  then              
  then              
  then            
  then            
  then        
  then