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I asked my students the following question.

Q: Express $\cos(\pi+x)$ in terms of $\sin$ and $\cos$.

A: $-\cos(x)$.

Students: Yeah, but where is the $\sin$ part? If I got this in an exam then I'd think I was wrong...yada-yada-yada....

Their issue is with the use of the word "and". Logically speaking, it should be an "or", but that would confuse them even more! Therefore, I was wondering if anyone has either,

  • A better way of phrasing this, and similar, question(s).
  • A decent explanation of why they needn't look for the $\sin$ part.
  • A persuasive argument as to why I should bite the dust and just use "or".

I tried explaining to them that they could write it as "$-\cos(x)+0\sin(x)$", but this didn't seem to help.

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    $\begingroup$ ``using sine, cosine, or some combination of these functions?'' $\endgroup$
    – Steven Gubkin
    May 29, 2014 at 14:32
  • $\begingroup$ "As a polynomial function of sine and cosine?" $\endgroup$
    – Steven Gubkin
    May 29, 2014 at 14:33
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    $\begingroup$ As worded, the question has another problem: $\cos(\pi+x)$ is already expressed in terms of $\cos$. Isn't it? I think at the very least you need to say "in terms of $\cos(x)$". $\endgroup$
    – mweiss
    May 29, 2014 at 15:06
  • $\begingroup$ It's also $\sin(-\frac \pi2 + x)$. $\endgroup$
    – Joe Z.
    May 29, 2014 at 18:53
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    $\begingroup$ Use "or" and they will do their utmost to use only one... Englishis ambiguous ;-) $\endgroup$
    – vonbrand
    Sep 3, 2015 at 11:44

7 Answers 7

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Express $\cos(x + \pi)$ in terms of $\sin x$ or $\cos x$ (possibly both).

In my opinion, the "or" is logically more correct and the parenthetical recalls/stresses/clarifies this.

In general, I am in favor of redundancy and details for such things.

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  • $\begingroup$ I am in complete agreement with you and I really like this answer. Thanks! $\endgroup$
    – user1729
    May 29, 2014 at 14:39
  • $\begingroup$ You are welcome. The formulation of Steven Gubkin with "a combination there of" also seems in the same spirit and is perhaps smoother (language-wise). $\endgroup$
    – quid ♦
    May 29, 2014 at 14:44
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    $\begingroup$ No, I disagree. Most (almost all) of my students are not native English speakers, so fewer words is better. This gets the point across rather simply (and is the style I write with anyway). $\endgroup$
    – user1729
    May 29, 2014 at 14:49
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    $\begingroup$ @user1729 An alternative with fewer words, but perhaps less familiar to nonnative speakers, is to use: and/or $\endgroup$
    – Benjamin Dickman
    May 30, 2014 at 2:19
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Here's what happened:

  1. You asked a question using standard phrasing.
  2. The student found it surprising that the phrasing allowed the answer given.
  3. The student was worried about this and blamed the phrasing of the question.

Two things I think it's important that should happen:

  1. The student needs to understand what is meant by this standard terminology, and
  2. The student will also in the future have to get more used to mathematics allowing special cases that are covered by general cases without explicitly mentioning it, for example:
    • if we say something's a parallelogram, we allow it to be a square,
    • if we ask for complex roots of quadratics we allow real ones,
    • if we have a function of x and y we allow it to be 3y
    • if we express cos(x+a) in terms of cos(x) and sin(x) we allow it to be sin(x)
    • 0 is a number, 4 is complex, $1 \le 200$ and each set is a subset of itself.

The phrase "express cos(x+a) in terms of cos(x) and sin(x)" means, to mathematicians, find an equation of the form cos(x+a)=f(cos(x),sin(x)), and experience tells the mathematician that f will be a linear function. I feel strongly that students should be taught the meaning of standard mathematical terminology and that edge cases are valid.

Rephrasing the question does not help them with either of the two things that need to happen, and you were correct to point that 0 is a valid coefficient, something it is very important that they come to terms with.

Historically, the solution of quadratic equations was hampered by the fact that $x^2+x=6$, $x^2=x+6$, $x^2=6$ and $x^2=5x$ were seen as four completely separate types of quadratic equation. Only once you're prepared to accept negative and zero numbers, and write them all as $ax^2+bx+c$ is a general solution even expressible. It's very important to mathematical development to allow expressions to cover cases as generally as possible, and to resist the very understandable human trait to always treat edge cases separately.

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Better Phrasing

First of all, you should use $\sin(x)$ and $\cos(x)$ instead of $\sin$ and $\cos$ as mweiss explains in his comment.

Secondly, and and or are logical operators. They can naturally be interpreted as set operations $\cap$ and $\cup$. What you want is neither $\{\sin\}\cup\{\cos\}$ nor $\{\sin\}\cap\{\cos\}$, but rather something like $\mathrm{span}\{\sin,\cos\}$ where the underlying field is unclear, or the language of algebraic expressions using $\sin(x)$ and $\cos(x)$.

If it's clear to you, what you mean and your students will understand that, than use that explicitely for example:

Express $\cos(\pi+x)$ as a linear combination of $\sin(x)$ and $\cos(x)$ in the vector space of real valued functions over the field of real numbers.

If it's unclear to you or your students won't understand it, than the most explicit proposal is surely that of Steven Gubkin:

Express $\cos(\pi+x)$ as a (algebraic) combination $\sin(x)$ and $\cos(x)$.

Looking for the $\sin$-part

This is something, that should've been taught in high school. You can simply give several trivial transformations using $\sin(x)$:

$$-\cos(x)+0\sin(x)\quad,\quad -\cos(x)\cdot\sin(x):\sin(x)\quad,\quad -\cos(x)+\sin(x)-\sin(x)$$

So, if someone is looking for a $\sin$-part, he may do so, but he needs not necessarily find one. Besides, why bother?

Using or instead

You shouldn't bite the dust. or might be interpreted as either…or or as not allowing a combination of $\sin(x)$ and $\cos(x)$.

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Q: Simplify $\cos(\pi+x)$.

Back in the days I made it a point that the students should be able to deal with "soft" instructions and "high-level" instructions. One of my favorite 40 min tests is: "Discuss the function ...".

Clarification: the instruction "discuss the function" makes sense to the students at that point because by then they should have learned that it means working out

  • domain and value
  • symmetry
  • asymptotes
  • derivatives
  • zeroes of function and derivatives with multiplicity and effect on the function's graph
  • draw the graph in the "interesting" region with suitable elements discovered earlier
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    $\begingroup$ As "simplify $\cos(\pi + x)$" is ok with me, "discuss the function" feels ridiculous. The difference is that in the former one knows the approximate outcome, while the latter is essentially "guess the teacher's password". For example, in research or work it's very rare that you don't have some objective. $\endgroup$
    – dtldarek
    May 29, 2014 at 16:48
  • $\begingroup$ @dtldarek: I suppose it depends on how the students have been prepared to "discuss the function," so they know what's expected of them. The pros and cons of such a question might make for an interesting MESE question. $\endgroup$
    – J W
    May 29, 2014 at 19:45
  • $\begingroup$ For "Discuss the function...", did you provide a guideline/detailed example in class? If so, it is just a shortcut for a hard, detailed question. (I like the formulation though). $\endgroup$
    – Taladris
    Jun 2, 2014 at 4:57
  • $\begingroup$ @Taladris: yes. Mathematics is all about making better building blocks to make building more efficient. I'm making a point of doing the same with job instructions. $\endgroup$
    – Stefan Schmiedl
    Jun 2, 2014 at 8:15
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I think that there is an important point behind this question, that seems to have been missed by other answer (however interesting they are): in mathematics we use the word "and" sometimes as a logical connector, sometimes (just as in regular English) to mark an enumeration, as in "$\cos$ and $\sin$ are example of trigonometric functions".

It is important, and often neglected, to distinguish and allow both usages of the word. Once you make that clear, a lot of trouble (including yours) should disappear.

I learned this issue from reading some work by Zoe Mesnil, a French math didactic researcher (you kind find her on the internet, but she mostly writes in French I'm afraid).

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How about this:

Express $\sin(\pi+x)$ in terms of $\sin(x)$, $\cos(x)$ or a (linear) combination of these.

Depending on your audience and on how much information you want to convey, you may choose to include or omit the word “linear”.

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    $\begingroup$ The issue with the linear combination idea is that it rules out $\cos(x)\sin(x)$ and such things. It is giving them information, and I want the statement to be as general as possible. $\endgroup$
    – user1729
    May 30, 2014 at 8:10
  • $\begingroup$ I suspect that "(linear) combination" is meant to be read as "a combination, possibly linear." It's a usage of parentheses I often see in Dutch, and I suspect it may also occur in German. However, in English it could be read as "a combination that, by the way, is linear." $\endgroup$
    – J W
    May 30, 2014 at 8:46
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    $\begingroup$ I actually meant it as two alternatives of how to write the question, with or without that word. I just edited my post to make that clearer. $\endgroup$
    – MvG
    May 30, 2014 at 9:17
  • $\begingroup$ One way to make that clearer might be to use square brackets; they are more often used for words that can be included/omitted. $\endgroup$
    – Opal E
    Sep 3, 2015 at 19:57
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Use the phrasing "and/or".

As an example, this is the language in Michael Sullivan's Algebra & Trigonometry: "Express the sum or difference as a product of sines and/or cosines." (7th Edition, Section 7.6, p. 637-638).

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