Inverse tangent function
For arctan in MATLAB®, see
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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.