Inverse cotangent function
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arccot(x) represents the inverse of the cotangent
The angle returned by this function is measured in radians, not in degrees. For example, 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
the result is expressed in terms of hyperbolic functions. See Example 2.
The inverse cotangent function is multivalued. The MuPAD®
returns the value on the main branch. The branch cut is the interval [- i, i] on
the imaginary axis. Thus,
arccot returns values,
such that y = arccot(x) satisfies for
any finite complex x.
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.
The float attributes are kernel functions. Thus, floating-point evaluation is fast.
When called with a floating-point argument,
sensitive to the environment variable
DIGITS which determines
the numerical working precision.
arccot with the following exact and
symbolic input arguments:
arccot(1), arccot(1/sqrt(2)), arccot(5 + I), arccot(1/3), arccot(0), arccot(I/2)
arccot(-x), arccot(x + 1), arccot(1/x)
Floating-point values are computed for floating-point arguments:
arccot(0.1234), arccot(5.6 + 7.8*I), arccot(1.0/10^20)
Floating-point intervals are computed for interval arguments:
Arguments that are rational multiples of
rewritten in terms of hyperbolic functions:
arccot(5*I), arccot(5/4*I), arccot(-3*I)
For other complex arguments unevaluated function calls without simplifications are returned:
arccot(1/2^(1/2) + I), arccot(1 - 3*I)
Some special values are implemented:
arccot(1), arccot((5 - 2*5^(1/2))^(1/2)), arccot(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:
arccot(0.5*I + 10^(-10)), arccot(0.5*I - 10^(-10))
On the branch cut, the values of
with the limit "from the right" for imaginary arguments
= c*i where
0 < c < 1:
limit(arccot(0.5*I - 1/n), n = infinity); limit(arccot(0.5*I + 1/n), n = infinity); arccot(0.5*I)
The values coincide with the limit "from the left"
for imaginary arguments
x = c*i where
< c < 0:
limit(arccot(-0.5*I - 1/n), n = infinity); limit(arccot(-0.5*I + 1/n), n = infinity); arccot(-0.5*I)
The inverse cotangent function can be rewritten in terms of the logarithm function with complex arguments:
diff(arccot(x^2), x), float(arccos(3)*arccot(5 + I))
limit(arccot(1 - sin(x)/x), x = 0)
taylor(arccot(1/x), x = 0)
series(arccot(x), x = 0)
Arithmetical expression or floating-point interval.