The amplitude of a sinusoid is the distance from its axis to a high point or a low point.
When we read this, it follows that Tan and Cot don't have an amplitude. Nor do SEC or CSC. Now, I'm in an odd situation. (Note, I work in a high school, and function as an in house tutor and occasional sub). The trig classwork is using this term as in y=2tan(x) for example. The students would be asked to identify the 2 as the "amplitude".
Given my relationship to the teachers, I am there to support them, not criticize them or appear a threat. And I am at the point in my life where I don't need to be right or adhere to the past (Like what happened to the septagon? It worked well, and fit into the series sept, oct, non, dec.) The question is -
Am I doing damage by using non-standard terms with my students? In the same manner that I talk about 1/0 not equalling infinity, but rather, as the angle approaches say 90 degrees, Tan approaches infinity, do I need to awkwardly say "what your teacher calls 'amplitude'" / "The 'A' term in this equation" and not feed into the misnomer?
I can use the term intelligently with no sarcasm if needed -
When we look at TAN and COT, we see the parent functions have 2 points, tan(45)=1 and tan(-45)=-1. A translated function would see these points shift vertically, so the difference in Y value is consistent, twice the new amplitude. Similarly, when graphing SEC/CSC, I am ok to identify the space between the graph segments as twice the amplitude. All in the spirit of not going against what the teachers are choosing use as a colloquialism.
Edit - an example, for context.
This was from a worksheet. Which was from the section of chapter that specifically was beyond SIN & COS. To be clear, if it had all the trig functions, I'd put the amplitude for SIN and COS and N/A for the other 4. As of this moment, I don't know if the teachers created it, or if this was from a third party.