Inverse cotangent function
This functionality does not run in MATLAB.
arccot(x) represents the inverse of the cotangent function.
The angle returned by this function is measured in radians, not in degrees. E.g., the result π represents an angle of 180o.
arccot is defined for complex arguments.
Floating-point values are returned for floating-point arguments. Floating-point intervals are returned for interval arguments. Unevaluated function calls are returned for most exact arguments.
If the argument is a rational multiple of I, the result is expressed in terms of hyperbolic functions. See Example 2.
The inverse cotangent function is multivalued. The MuPAD® arccot function returns the value on the main branch. The branch cut is the interval [- i, i] on the imaginary axis. Thus, arcsin returns values, such that y = arccot(x) satisfies for any finite complex x.
The cot function returns explicit values for arguments that are certain rational multiples of π. For these values, arccot returns an appropriate rational multiple of π on the main branch. See Example 3.
The values jump when the arguments cross a branch cut. See Example 4.
Note: MuPAD defines arccot as arccot(x) = arctan(1/x), although arccot may return an unevaluated function call and does not rewrite itself in terms of arctan. As a consequence of this definition, the real line crosses the branch cut and arccot has a jump discontinuity at the origin!
The float attributes are kernel functions. Thus, floating-point evaluation is fast.
When called with a floating-point argument, arccot is sensitive to the environment variable DIGITS which determines the numerical working precision.
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)
arcsin(-x), arccos(x + 1), arctan(1/x)
Floating-point values are computed for floating-point arguments:
arcsin(0.1234), arccos(5.6 + 7.8*I), arccot(1.0/10^20)
On input of floating-point intervals, these functions compute floating-point intervals containing the image sets:
Note that certain types of input lead to severe overestimation, sometimes returning the whole image set of the function in question:
Arguments that are rational multiples of I are rewritten in terms of hyperbolic functions:
arcsin(5*I), arccos(5/4*I), arctan(-3*I)
For other complex arguments unevaluated function calls without simplifications are returned:
arcsin(1/2^(1/2) + I), arccos(1 -3*I)
Some special values are implemented:
arcsin(1/sqrt(2)), arccos((5^(1/2) - 1)/4), arctan(3^(1/2) - 2)
Such simplifications occur for arguments that are trigonometric images of rational multiples of π:
The values jump when crossing a branch cut:
arcsin(2.0 + I/10^10), arcsin(2.0 - I/10^10)
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)
arcsin(-1.2), arcsin(-1.2 + I/10^10), arcsin(-1.2 - I/10^10)
The inverse trigonometric functions can be rewritten in terms of the logarithm function with complex arguments:
rewrite(arcsin(x), ln), rewrite(arctan(x), ln)
diff(arcsin(x^2), x), float(arccos(3)*arctan(5 + I))
limit(arcsin(x^2)/arctan(x^2), x = 0)
series(arctan(sin(x)) - arcsin(tan(x)), x = 0, 10)
series(arccos(2 + x), x, 3)