Inverse tangent function
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For arctan in MATLAB®, see
arctan(x) represents the inverse of the tangent
arctan(y, x) is an alias for
The angle returned by this function is measured in radians, not in degrees. For example, the result π represents an angle of 180o.
arctan 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 tangent function is multivalued. The MuPAD®
returns the value on the main branch. The branch cuts are the intervals and on
the imaginary axis. Thus,
arctan returns values,
such that y = arctan(x) satisfies for
any finite complex x.
tan function returns explicit values
for arguments that are certain rational multiples of π.
For these values,
arctan 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.
arctan with the following exact and
symbolic input arguments:
arctan(-5), arctan(1/sqrt(2)), arctan(5 + I), arctan(1/3), arctan(0), arctan(1)
arctan(-x), arctan(x + 1), arctan(1/x)
Floating-point values are computed for floating-point arguments:
arctan(0.1234), arctan(5.6 + 7.8*I), arctan(1.0/10^20)
Floating-point intervals are computed for interval arguments:
Arguments that are rational multiples of
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:
arctan(1), arctan((5 - 2*5^(1/2))^(1/2)), 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:
arctan(2.0*I + 10^(-10)), arctan(2.0*I - 10^(-10))
On the branch cut, the values of
with the limit “from the right” for imaginary arguments
= c*i where
c > 1:
limit(arctan(2.0*I - 1/n), n = infinity); limit(arctan(2.0*I + 1/n), n = infinity); arctan(2.0*I)
The values coincide with the limit “from the left”
for imaginary arguments
x = c*i where
limit(arctan(-2.0*I - 1/n), n = infinity); limit(arctan(-2.0*I + 1/n), n = infinity); arctan(-2.0*I)
The inverse tangent function can be rewritten in terms of the logarithm function with complex arguments:
diff(arctan(x), x), float(arccos(3)*arctan(5 + I))
limit(arctan(sin(x)/tan(x)), x = 0)
taylor(arctan(x), x = 0)
When you call
arctan with two arguments, MuPAD calls
and computes the polar angle of a complex number:
Arithmetical expressions representing real numbers
Arithmetical expression or floating-point interval.