# “Amplitude” of Tan and Cot functions

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.

• Have the students been given a definition of "amplitude", and does it match the definition you gave at the start of your question? – Daniel Hast Feb 26 '19 at 23:51
• It sounds like there are two separate problems here: nonstandard use of the word "amplitude" in a way that disagrees with widespread standard usage, and a lack of clear, precise definitions. If the students aren't being given precise definitions of all terms they're expected to work with—for instance, if some terms are only "defined" by example—then they don't have a solid basis for reasoning, which is a much more serious issue than nonstandard use of one bit of terminology. – Daniel Hast Feb 27 '19 at 2:04
• To further @DanielHast 's last comment: Without a clear definition, how would a student decide on the amplitude of $\sin(x)+\cos(x)$? Should they provide the coefficient, 1, or the usual amplitude, $\sqrt{2}$? – Adam Feb 27 '19 at 2:14
• Are you able to speak to the teacher to ask for clarification on their definition of "amplitude"? The examples of $\tan(x)$ and $\sin(x) + \cos(x)$ should suffice to illustrate the issue. You don't have to be confrontational—just asking for clarification on their definition should bring the issue to light. – Daniel Hast Feb 27 '19 at 3:25
• I agree with @Daniel Hast in that you should ask the teacher for clarification in a non-confrontational way. I would call $2$ the stretch factor (or the dilation, if you want to use a more mathy word), and for functions with an amplitude it can be used to determine the new amplitude, but it can also be used for something without an amplitude (such as $f(x) = x^2).$ It might also be helpful to point out (I'm virtually certain, but you may want to look for a specific example) that on the ACT test, where a few trig questions occur, knowing that tangent has no amplitude can definitely arise. – Dave L Renfro Feb 27 '19 at 8:05

I used to teach my PreCalc students an early unit on the Function Toolbox.

Take $$f(x) = x^2$$. Something happens to it when you add a term: $$f(x) = x^2 + a$$. Also if you change it to $$f(x) = (x - b)^2$$. And how ‘bout $$f(x) = cx^2$$.

Now change the function to $$f(x) = x^3$$. And $$f(x) = \sqrt{x}$$. And $$f(x) = \sin(x)$$. And $$f(x) = \frac1x$$.

My point is that the constant that is traditionally called amplitude in trig class actually does something more general, and to a ton of functions, even to those with asymptotes.

I think I fall on your side when calling it “amplitude,” but would love students to know what $$2f(x)$$ does to $$f(x)$$, regardless of the function.

This looks to me like a balance between two concerns:

• Tradition: Students should learn correct and standard terminology.
• Professionalism: Everyone involved should show a united front to the students.

Suppose the teacher, Mr. Flample, wants to call the coefficient of a cosecant function the "flamplitude." You have two choices of what to say to the student:

• "Which number does your teacher call the 'flamplitude'?"
• "Which number do we call the 'flamplitude'?"

This is not a subtle distinction, nor is it a difficult choice.

Choosing the first option is actively giving the student a reason to disengage from the class. That's bad for the student. Don't help students develop excuses to not learn mathematics.

• Have I misinterpreted Dave's comment above? I'm looking to understand the result of the path I take. If that result is an incorrect SAT answer, or a student confused by the teacher in the next year's class who stays with the strict definition, have I done right by my student? – JTP - Apologise to Monica Feb 27 '19 at 15:49
• @ChrisCunningham "Flamplitude" and "Hergert Numbers" do not have existing mathematical uses which contradict the definitions you are giving them. A more realistic comparison would be if they used the word "extremum" to mean any place where the derivative is zero. This contradicts the usual definition, and could cause real problems down the line. – Steven Gubkin Feb 27 '19 at 17:09
• Wait, what? I thought you would say that the first question is the correct one. I would definitely ask the first question, stressing out that it is the particular teacher's word choice, not a common one. Then I would suggest looking in a textbook (if the students have one, because often there are no textbooks whatsoever) or online, at least in two different sources. "Everyone involved should show a united front to the students" - this is the same kind of BS that the police use when not prosecuting its own for offenses for which any regular Joe would serve time. – Rusty Core Feb 27 '19 at 17:49
• I disagree that this is about tradition. If the teacher had a useful and logically coherent definition of something, and decided to call it amplitude, but it differed from the standard definition, then the only concern would be tradition. But if there is a useful and logically coherent definition that has been proposed by this teacher, we haven't seen it. As pointed out in a comment by Adam, there is no obvious way to extend this teacher's definition to an example like $\sin+\cos$. – Ben Crowell Feb 27 '19 at 18:37
• I like Steven's and Ben's point is a good one, and I now believe that my answer is wrong. I'll leave it up here because I think the discussion is useful. Please don't +1 the answer. Thanks friends! – Chris Cunningham Feb 27 '19 at 21:28

The other teach is wrong. But since you are in a support role, I would just ignore it and roll onward.

Surely there are a bazillion other things to work on, to help the kids with. It's not like 50% of your time is on this wrong amplitude.

If it still bugs you the wrongness, you could say something like "I don't know amplitude so well--have Mrs. WrongAmp help on that one. Just circle it and move on, kid. I'll help with next problem." That way you don't endorse the wrongness or tie yourself in knots promulgating it. But you still move along and do something useful. And don't make students lose faith in WrongAmp lady.

• Yes, this a tiny fraction of my time. But, 6 different classes doing trig over next 3 weeks, so I’m likely to hit this issue a number of times “if” the others use same material. If not, much easier. – JTP - Apologise to Monica Mar 2 '19 at 16:55
• While I agree that discrediting the other teacher is probably not the way to go, discrediting yourself to some extent (by saying you "don't know amplitude so well", when in fact you do!) does not sound like a fair compromise. – orion2112 Mar 3 '19 at 2:53
• @orion2112 - thanks, agreed. As someone with a 33 year gap between being a HS student and a tutor, I'm humble enough to admit my limits, but not willing to claim ignorance on a topic I've mastered. – JTP - Apologise to Monica Mar 3 '19 at 14:12