Reference Angle

Author

Eric W. Weisstein
August 12, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/Trigonometry/ReferenceAngle.nb.

For more information, see Eric's MathWorld entry https://mathworld.wolfram.com/ReferenceAngle.html.

©2026 Wolfram Research, Inc. except for portions noted otherwise

Illustration

The reference angle is the acute angle between the terminal side and the x-axis.

Module[{angleColor, neutral, referenceColor, theta}, 
  theta = 2.25; angleColor = Red; referenceColor = Blue; neutral = LightDarkSwitched[Black, White]; 
   Graphics[{Directive[neutral, Dashing[{0.02, 0.02}]], Circle[{0, 0}, 1], 
     Directive[neutral, Dashing[{}], AbsoluteThickness[1.5]], Line[{{-1.3, 0}, {1.3, 0}}], 
     Line[{{0, -1.15}, {0, 1.15}}], {Directive[referenceColor, Opacity[0.22]], 
      Disk[{0, 0}, 0.58, {theta, Pi}]}, Directive[referenceColor, Opacity[1], 
      AbsoluteThickness[2.5]], Circle[{0, 0}, 0.58, {theta, Pi}], Directive[angleColor, Opacity[1], 
      AbsoluteThickness[2]], Circle[{0, 0}, 0.88, {0, theta}], AbsoluteThickness[2.5], 
     Line[{{0, 0}, 1.12*{Cos[theta], Sin[theta]}}], 
     Text[Style["θ", 18, angleColor, FontFamily -> "Times"], 0.7*{Cos[theta/2], Sin[theta/2]}], 
     Text[Style["α", 18, referenceColor, FontFamily -> "Times"], 
      0.78*{Cos[(theta + Pi)/2], Sin[(theta + Pi)/2]}], 
     Text[Style["x", 16, neutral, FontFamily -> "Times", FontSlant -> "Italic"], {1.38, 0}], 
     Text[Style["y", 16, neutral, FontFamily -> "Times", FontSlant -> "Italic"], {0, 1.25}]}]]