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ArcTan

ArcTan[z]
gives the arc tangent tan^(-1)(z) of the complex number z.
ArcTan[x, y]
gives the arc tangent of , taking into account which quadrant the point (x,y) is in.
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • All results are given in radians.
  • For real z, the results are always in the range -pi/2 to pi/2.
  • 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 z plane running from -i infty to -i and +i to +i infty.
  • If x or y is complex, then ArcTan[x, y] gives . When x^2+y^2=1, ArcTan[x, y] gives the number phi such that x=cosphi and y=sinphi.
EXAMPLESCLOSE ALL
Basic Examples   (5)
Results are in radians:
Divide by Degree to get results in degrees:
ArcTan[x, y] gives the angle of the point {x, y}:
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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Scope   (6)
Evaluate for complex arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
ArcTan threads element-wise over lists:
Simple exact values are generated automatically:
Parity transformations are automatically applied:
TraditionalForm formatting:
Generalizations & Extensions   (3)
Infinite arguments give symbolic results:
ArcTan can be applied to a power series:
ArcTan can deal with real-valued intervals:
Applications   (5)
Find angles of the right triangle with sides 3, 4 and hypotenuse 5:
They total to 90^ degrees:
Find integrals of rational functions in terms of ArcTan:
Addition theorem for tangent function:
Solve a differential equation:
Branch cuts of ArcTan run along the imaginary axis:
Properties & Relations   (4)
Use TrigToExp to express ArcTan using Log:
Use FullSimplify to simplify expressions with ArcTan:
ArcTan is a special case of some special functions:
Use Reduce to solve inequalities involving ArcTan:
Possible Issues   (1)
Because ArcTan is a multivalued function, tan^(-1)(tan(x))!=x
This differs from the original argument by a factor of pi:
Neat Examples   (1)
Expansion about the branch point x=ⅈ:
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