ArcTan

ArcTan[z]

gives the arc tangent of the complex number .

ArcTan[x,y]

gives the arc tangent of , taking into account which quadrant the point is in.

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • All results are given in radians.
  • For real , the results are always in the range to .
  • For certain special arguments, ArcTan automatically evaluates to exact values.
  • ArcTan can be evaluated to arbitrary numerical precision.
  • ArcTan automatically threads over lists.
  • ArcTan[z] has branch cut discontinuities in the complex plane running from to and to .
  • If or is complex, then ArcTan[x,y] gives . When , ArcTan[x,y] gives the number such that and .

Background & Context

  • ArcTan is the inverse tangent function. For a real number x, ArcTan[x] represents the radian angle measure such that . The two-argument form ArcTan[x,y] represents the arc tangent of y/x, taking into account the quadrant in which the point lies. It therefore gives the angular position (expressed in radians) of the point measured from the positive axis. ArcTan is consequently useful when converting from Cartesian to polar coordinate systems and for finding the phase in phasor notation x+ⅈ y=TemplateBox[{z}, Abs]ⅇ^(ⅈ phi).
  • ArcTan automatically threads over lists. For certain special exact arguments, ArcTan automatically evaluates to exact values. When given exact numeric expressions as arguments, ArcTan may be evaluated to arbitrary numeric precision. Operations useful for manipulation of symbolic expressions involving ArcTan include FunctionExpand, TrigToExp, TrigExpand, Simplify, and FullSimplify.
  • ArcTan is defined for complex argument via . ArcTan[z] has branch cut discontinuities in the complex plane.
  • Related mathematical functions include Arg, Tan, ArcCot, ArcTanh, and Gudermannian.

Examples

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Basic Examples  (5)

Results are in radians:

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Divide by Degree to get results in degrees:

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ArcTan[x,y] gives the angle of the point {x,y}:

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Plot over a subset of the reals:

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Series expansion:

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Scope  (35)

Generalizations & Extensions  (1)

Applications  (5)

Properties & Relations  (4)

Possible Issues  (1)

Neat Examples  (1)

Introduced in 1988
(1.0)