Talk:Slowly varying function

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Sine, Cosine

Is sine or cosine slowly varying?

--130.149.114.63 (talk) 15:53, 29 October 2017 (UTC)[reply]

From the definition given, they can only be slowly varying if sin(ax)/sin(x) and cos(ax)/cos(x) converge as x tends to infinity for all a. It's fairly clear this isn't the case, just from the formulae for sin and cos of a sum. E.g

sin(ax)/sin(x) = [sin((a-1)x+x)]/sin(x)

              = [sin((a-1)x)cos(x)+cos((a-1)x)sin(x)]/sin(x)
              = sin((a-1)x)sec(x)+cos((a-1)x)

Usually this will have singularities whenever sec(x) does, preventing it from converging, so the needed limit doesn't exist.

86.30.22.26 (talk) 22:06, 9 March 2018 (UTC)[reply]