Template:Atan2/doc

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The {{atan2}} circular function template returns the four quadrant arc tangent (a circular angle in $\scriptstyle (-\pi ,\,+\pi ]\,$ radians as default, or a circular angle in $\scriptstyle [0,\,2\pi )\,$ radians) from $\scriptstyle y\,$ and $\scriptstyle x\,$, which are real numbers not both zero.

Usage

{{atan2|y=a real number|x=a real number|cut=an integer}}

or

{{atan2|a real number|a real number|an integer}}

Valid input

Two real numbers: $\scriptstyle y\,$ and $\scriptstyle x\,$, not both zero. Third argument (optional) as integer, where an odd integer (default 1) implies a circular angle in $\scriptstyle (-\pi ,\,+\pi ]\,$ radians and an even integer implies a circular angle in $\scriptstyle [0,\,2\pi )\,$ radians.

Examples

Examples with valid input

Code	Result
{{atan2|y=0|x=1|cut=1}}	0
{{atan2|y={{sin|pi/6}}|x={{cos|pi/6}}}}	0.5235987755983
{{atan2|y={{sin|pi/4}}|x={{cos|pi/4}}}}	0.78539816339745
{{atan2|y={{sin|pi/3}}|x={{cos|pi/3}}}}	1.0471975511966
{{atan2|y=1|x=0|cut=3}}	1.5707963267949
{{atan2|y={{sin|2*pi/3}}|x={{cos|2*pi/3}}}}	2.0943951023932
{{atan2|y={{sin|3*pi/4}}|x={{cos|3*pi/4}}}}	2.3561944901923
{{atan2|y={{sin|5*pi/6}}|x={{cos|5*pi/6}}}}	2.6179938779915
{{atan2|y=0|x=-1|cut=5}}	3.1415926535898
{{atan2|y={{sin|7*pi/6}}|x={{cos|7*pi/6}}}}	-2.6179938779915
{{atan2|y={{sin|5*pi/4}}|x={{cos|5*pi/4}}}}	-2.3561944901923
{{atan2|y={{sin|4*pi/3}}|x={{cos|4*pi/3}}}}	-2.0943951023932
{{atan2|y=-1|x=0|cut=-1}}	-1.5707963267949
{{atan2|y={{sin|5*pi/3}}|x={{cos|5*pi/3}}}}	-1.0471975511966
{{atan2|y={{sin|7*pi/4}}|x={{cos|7*pi/4}}}}	-0.78539816339745
{{atan2|y={{sin|11*pi/6}}|x={{cos|11*pi/6}}}}	-0.5235987755983
{{atan2|y={{sin|2*pi}}|x={{cos|2*pi}}}}	-2.4492935982947E-16

Code	Result
{{atan2|y=0|x=1|cut=0}}	0
{{atan2|y={{sin|pi/6}}|x={{cos|pi/6}}|cut=2}}	0.5235987755983
{{atan2|y={{sin|pi/4}}|x={{cos|pi/4}}|cut=2}}	0.78539816339745
{{atan2|y={{sin|pi/3}}|x={{cos|pi/3}}|cut=2}}	1.0471975511966
{{atan2|y=1|x=0|cut=2}}	1.5707963267949
{{atan2|y={{sin|2*pi/3}}|x={{cos|2*pi/3}}|cut=2}}	2.0943951023932
{{atan2|y={{sin|3*pi/4}}|x={{cos|3*pi/4}}|cut=2}}	2.3561944901923
{{atan2|y={{sin|5*pi/6}}|x={{cos|5*pi/6}}|cut=2}}	2.6179938779915
{{atan2|y=0|x=-1|cut=4}}	3.1415926535898
{{atan2|y={{sin|7*pi/6}}|x={{cos|7*pi/6}}|cut=2}}	3.6651914291881
{{atan2|y={{sin|5*pi/4}}|x={{cos|5*pi/4}}|cut=2}}	3.9269908169872
{{atan2|y={{sin|4*pi/3}}|x={{cos|4*pi/3}}|cut=2}}	4.1887902047864
{{atan2|y=-1|x=0|cut=-2}}	4.7123889803847
{{atan2|{{sin|5*pi/3}}|{{cos|5*pi/3}}|2}}	5.235987755983
{{atan2|{{sin|7*pi/4}}|{{cos|7*pi/4}}|2}}	5.4977871437821
{{atan2|{{sin|11*pi/6}}|{{cos|11*pi/6}}|2}}	5.7595865315813
{{atan2|y={{sin|2*pi}}|x={{cos|2*pi}}|cut=2}}	6.2831853071796

Examples with invalid input (returns an error message)

Code	Result	Comment
{{atan2}}
{{atan2|i}}
{{atan2|2}}
{{atan2|2|i}}
{{atan2|i|2}}
{{atan2|0|0}}
{{atan2|-1|0|2 pi}}
{{atan2|-1|0|6.28}}	4.7123889803847	(parity of cut value rounded towards zero, please use integers for clarity)
{{atan2|-1|0|6.7}}	4.7123889803847	(parity of cut value rounded towards zero, please use integers for clarity)
{{atan2|-1|0|-6.28}}	4.7123889803847	(parity of cut value rounded towards zero, please use integers for clarity)
{{atan2|-1|0|-6.7}}	4.7123889803847	(parity of cut value rounded towards zero, please use integers for clarity)

Code

<!-- Returns angle in (-pi, +pi] (default) or [0, 2*pi) -->{{ifnum| {{{y|{{{1|NAN}}}}}}{{{x|{{{2|NAN}}}}}}<!-- {{{cut|{{{3|1}}}}}} odd (for 1*pi, default) or even (for 2*pi) -->
| {{#switch: {{#expr: 3*{{sgn| {{{y|{{{1}}}}}} }} + {{sgn| {{{x|{{{2}}}}}} }} }} 
  | -4 = {{#expr: pi * {{#ifexpr: ({{{cut|{{{3|1}}}}}}) mod 2 = 0 | 1 | -1 }} + atan ( ({{{y|{{{1}}}}}}) / ({{{x|{{{2}}}}}}) ) }} <!--  y < 0 and x < 0 (third quadrant) -->
  | -3 = {{#expr: (pi / 2) * {{#ifexpr: ({{{cut|{{{3|1}}}}}}) mod 2 = 0 | 3 | -1 }} }} <!-- y < 0 and x = 0 -->
  | -2 = {{#expr: 2*pi * {{#ifexpr: ({{{cut|{{{3|1}}}}}}) mod 2 = 0 | 1 | 0 }} + atan ( ({{{y|{{{1}}}}}}) / ({{{x|{{{2}}}}}}) ) }} <!-- y < 0 and x > 0 (fourth quadrant) -->
  | -1 = {{#expr: pi }} <!--                                                              y = 0 and x < 0 -->
  |  0 = {{error| Atan2 error: Result is undefined when both arguments are zero }} <!--   y = 0 and x = 0 -->
  | +1 = 0 <!--                                                                           y = 0 and x > 0 -->
  | +2 = {{#expr: pi + atan ( ({{{y|{{{1}}}}}}) / ({{{x|{{{2}}}}}}) ) }} <!--             y > 0 and x < 0 (second quadrant) -->
  | +3 = {{#expr: pi / 2 }} <!--                                                          y > 0 and x = 0 -->
  | +4 = {{#expr: atan ( ({{{y|{{{1}}}}}}) / ({{{x|{{{2}}}}}}) ) }} <!--                  y > 0 and x > 0 (first quadrant) -->
  }}
| {{error| Atan2 error: Both arguments must be real numbers (not both zero) }}
}}

See also

• Trigonometric functions
  • Circular functions
  • Hyperbolic functions
• Inverse trigonometric functions
  • Inverse circular functions
  • Inverse hyperbolic functions

• {{sin}}
• {{cos}}
• {{tan}}

• {{csc}}
• {{sec}}
• {{cot}}

• {{arcsin}} or {{asin}}
• {{arccos}} or {{acos}}
• {{arctan}} or {{atan}} (see also {{atan2}})

• {{arccsc}} or {{acsc}}
• {{arcsec}} or {{asec}}
• {{arccot}} or {{acot}}