"Amplitude" of Tan and Cot functions

Question

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.

If we take the definition of amplitude as "the distance from the middle of a periodic function to its local maximum" (but then again, what is the middle?), then tan and ctg have no such thing. — Rusty Core, Feb 26 '19 at 23:49
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
@RustyCore - agreed. My last para shows how I just roll with the impact that coefficient has on the shape of the curve. — JTP - Apologise to Monica, Feb 27 '19 at 00:20
@DanielHast - They are told the amplitude is the coefficient. As in "for y=2tan(x), '2' is the amplitude. I am not agreeing, just trying to minimize potential harm. — JTP - Apologise to Monica, Feb 27 '19 at 00:22
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 02:04
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 02:14
For the problems in which the word amplitude is used, the coefficient is sufficient. I realize the potential pitfalls, as Adam mentions. Worst case, I can admit I don’t know how to apply the word to a given circumstance, but so far, I can predict what the teachers expect. — JTP - Apologise to Monica, Feb 27 '19 at 02:25
Correct. The former. I know better, and would be happy to stick to the definition I posted. I understand that Tan has no amplitude to speak of. — JTP - Apologise to Monica, Feb 27 '19 at 03:18
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 03:25
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 08:05
Do the students already know about parabolas? How is the $a$ in $y=ax^2$ called? — Jasper, Feb 27 '19 at 11:42
@DaveLRenfro - In light of how I worded the question, your comment can be expanded into an answer. It starts with "Yes, reinforcing incorrect definitions is damaging. The ACT or Math SAT subject test has trig questions, and such word play can cause students to lose points and your trust. Further, for non-SIN/COS, we have the phrase 'vertical stretch' which can be legitimately applied to your scenario. Students are used to this phrase and its meaning." <<< You are welcome to lift any/all of this, it's how I read your comment. (and thanks) — JTP - Apologise to Monica, Feb 27 '19 at 12:40
@JoeTaxpayer: I'll try to write something up within a day or two. I just made a few comments in various Stack Exchanges, but now I need to get working on something I got late yesterday that's due about 24 hours from now (which will take me 6 to 12 hours, the uncertainty being why I want to get started now). — Dave L Renfro, Feb 27 '19 at 16:42
"you don't have to be confrontational—just asking for clarification" — asking for clarification or exact definition means being confrontational to those who lack the required knowledge. — Rusty Core, Feb 27 '19 at 17:45
@DaveLRenfro I like your suggestion of referring to it as a "stretch" or "dilation" factor, but I worry that both of those words suggest an increase in size. How about calling it the "scaling" factor (so $a=2$ is a scaling by a factor of 2, or a vertical "stretch", while $a=1/2$ is a scaling by a factor of $1/2$, or a vertical "compression"). The amplitude of a sinusoidal function can then be described as the magnitude (or, better yet, "size") of the vertical scaling factor. — Xander Henderson, Feb 27 '19 at 19:05
@Jasper, ‘a’ is vertical stretch. Some students call it slope. I respond that if we draw a point from the vertex to x+1, f(x+1), ‘a’ is the slope of that secant line. And I tell them it’s a non standard way to talk about ‘a’. — JTP - Apologise to Monica, Feb 28 '19 at 00:00
But that's also wrong. The slope of a parabola is not constant :/ — Jasper, Feb 28 '19 at 06:48
@Jasper - indeed, but, since the students told me that's just what they call it, (a) I have no teachers to contradict, and (b) I can help them codify the phenomenon they are articulating. The fact that the point x+1 from the vertex will identify the function's 'A' value. It's a variation of my original issue, but not identical, and one I'm happy with my approach. — JTP - Apologise to Monica, Feb 28 '19 at 10:25
"As of this moment, I don't know if the teachers created it, or if this was from a third party." - this is the clear reason for teachers to create their own tests and be responsible for them. Outsourcing everything means that no one is responsible, everyone points to "the system". — Rusty Core, Feb 28 '19 at 17:37
I think the natural thing to say is the reciprocal trig. functions as well as tangent and cotangent have infinite amplitude. Or, simply say those worksheets were intended for just sine and cosine. However, it is true that sums of sine and cosine of the same angular frequency combine to a single sinusoid with some phase angle and that does also have a well-defined amplitude. Certainly calling the distance which is closest for the reciprocal function the "amplitude" is a wrong practice that should not be continued. That's just weird. — James S. Cook, Mar 04 '19 at 01:23

score 5 · Accepted Answer · edited Mar 11 '19 at 21:23

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.

score 0 · Answer 2 · answered Feb 27 '19 at 14:32

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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.

answered Feb 27 '19 at 14:32

Chris Cunningham

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For a more realistic example, I call values of $x$ where $f''(x) = 0$ by the name "Hergert numbers." I am clear with students that other people may not learn this term, as it is mostly a regional phenomenon. The question is: if one of my students' tutors contradicts me and says "actually, there is no term for those," is that tutor being productive, or counterproductive? – Chris Cunningham Feb 27 '19 at 15:08
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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
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@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
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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
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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 $\sin+\cos$. – Feb 27 '19 at 18:37
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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
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Chris - still I appreciate being asked to look at issues from all sides. I still have a decision to make, and from all these comments, it seems not so straightforward. – JTP - Apologise to Monica Feb 28 '19 at 01:51

score 0 · Answer 3 · answered Mar 02 '19 at 15:54

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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.

answered Mar 02 '19 at 15:54

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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 02 '19 at 16:55
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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 03 '19 at 02:53
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@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 03 '19 at 14:12

"Amplitude" of Tan and Cot functions

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