How this calculator works
Adding or removing complete revolutions changes how far an angle turns without changing its terminal ray. The principal result lies from 0 inclusive to 360 degrees exclusive. Separate strictly positive and strictly negative examples are also shown. Multiples-of-pi mode accepts a coefficient such as 3/2 without typing pi. Two angles are treated as coterminal within 10^-10 radians; very large floating-point inputs may lose small angular details.
Formula
Coterminal angles have the form theta + 2k pi radians, where k is an integer. Their difference divided by 2 pi is a whole number.
Worked example
-45 degrees and 315 degrees differ by one revolution and are coterminal. Also 9pi/2 radians reduces to pi/2 radians after subtracting two complete turns.
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Frequently asked questions
Are there infinitely many coterminal angles?
Yes. Every integer k adds a different whole-turn count to the same terminal direction.
What happens for a multiple of 360 degrees?
The principal representative is 0 degrees. The strictly positive and negative examples are 360 degrees and -360 degrees.
Can I add half a turn?
A half turn changes the terminal ray. Use a whole number in the turns field to keep the angle coterminal.