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arcsin, arccos, arctan, arccsc, arcsec, arccot -- the inverse trigonometric functions

Introduction

arcsin(x) represents the inverse of the sine function.

arccos(x) represents the inverse of the cosine function.

arctan(x) represents the inverse of the tangent function.

arccsc(x) represents the inverse of the cosecant function.

arcsec(x) represents the inverse of the secant function.

arccot(x) represents the inverse of the cotangent function.

Call(s)

arcsin(x)
arccos(x)
arctan(x)
arccsc(x)
arcsec(x)
arccot(x)

Parameters

x - an arithmetical expression

Returns

an arithmetical expression.

Overloadable:

x

Side Effects

When called with a floating point argument, the functions are sensitive to the environment variable DIGITS which determines the numerical working precision.

Related Functions

sin, cos, tan, csc, sec, cot

Details

Example 1

We demonstrate some calls with exact and symbolic input data:

>> arcsin(1), arccos(1/sqrt(2)), arctan(5 + I), arccsc(1/3), 
   arcsec(I), arccot(1)
          PI  PI                            PI                 PI
          --, --, arctan(5 + I), arcsin(3), -- + I arcsinh(1), --
          2   4                             2                  4
>> arcsin(-x), arccos(x + 1), arctan(1/x)
                                                   / 1 \
                  -arcsin(x), arccos(x + 1), arctan| - |
                                                   \ x /

Floating point values are computed for floating point arguments:

>> arcsin(0.1234), arccos(5.6 + 7.8*I), arccot(1.0/10^20)
          0.1237153458, 0.9506879769 - 2.956002937 I, 1.570796327

Example 2

Some special values are implemented:

>> arcsin(1/sqrt(2)), arccos((5^(1/2) - 1)/4), arctan(3^(1/2) - 2)
                              PI  2 PI    PI
                              --, ----, - --
                              4    5      12

Such simplifications occur for arguments that are trigonometric images of rational multiples of PI:

>> sin(9/10*PI), arcsin(sin(9/10*PI))
                               1/2
                              5           PI
                              ---- - 1/4, --
                               4          10
>> cos(PI/8)/sin(PI/8), arctan(cos(PI/8)/sin(PI/8))
                              1/2     1/2
                            (2    + 2)     3 PI
                            -------------, ----
                                  1/2 1/2   8
                            (2 - 2   )

Example 3

Arguments that are rational multiples of I are rewritten in terms of hyperbolic functions:

>> arcsin(5*I), arccos(5/4*I), arctan(-3*I)
                           PI
             I arcsinh(5), -- - I arcsinh(5/4), -I arctanh(3)
                           2

For other complex arguments unevaluated function calls without simplifications are returned:

>> arcsin(1/2^(1/2) + I), arccos(1 -3*I)
                          /  1/2     \
                          | 2        |
                    arcsin| ---- + I |, arccos(1 - 3 I)
                          \  2       /

Example 4

The values jump when crossing a branch cut:

>> arcsin(2.0 + I/10^10), arcsin(2.0 - I/10^10)
         1.570796327 + 1.316957897 I, 1.570796327 - 1.316957897 I

On the branch cut, the values of arcsin coincide with the limit ``from below'' for real arguments x>1. The values coincide with the limit ``from above'' for real x<-1:

>> arcsin(1.2), arcsin(1.2 - I/10^10), arcsin(1.2 + I/10^10)
      1.570796327 - 0.6223625037 I, 1.570796327 - 0.6223625037 I,
      
         1.570796327 + 0.6223625037 I
>> arcsin(-1.2), arcsin(-1.2 + I/10^10), arcsin(-1.2 - I/10^10)
      - 1.570796327 + 0.6223625037 I,
      
         - 1.570796327 + 0.6223625037 I,
      
         - 1.570796327 - 0.6223625037 I

Example 5

The inverse trigonometric functions can be rewritten in terms of the logarithm function with complex arguments:

>> rewrite(arcsin(x), ln), rewrite(arctan(x), ln)
                         2 1/2
      - I ln(I x + (1 - x )   ), 1/2 I ln(1 - I x) -
      
         1/2 I ln(I x + 1)

Example 6

Various system functions such as diff, float, limit, or series handle expressions involving the inverse trigonometric functions:

>> diff(arcsin(x^2), x), float(arccos(3)*arctan(5 + I))
                   2 x
               -----------, - 0.06540673615 + 2.433548516 I
                     4 1/2
               (1 - x )
>> limit(arcsin(x^2)/arctan(x^2), x = 0)
                                     1
>> series(arctan(sin(x)) - arcsin(tan(x)), x = 0, 10)
                                  7      9
                          3   83 x    4 x       10
                       - x  - ----- - ---- + O(x  )
                               120    189

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