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Ordering Products, Powers, and Parameters of Trigonometric Functions

Date: 10/31/2010 at 21:11:03
From: Jim
Subject: Order of operations

I am tying to clarify the order of operations as it applies to
trigonometric functions.

I want to know what the correct order of operations is for an expression
like sin2x. When do we know that the multiplication is implied?

When my textbook says sin2x, I know that it means sin(2x). But often the
parentheses are missing. Is it then correct to assume that the
multiplication is always implied to be in parentheses? If that's true,
wouldn't that imply that sinxcosy would have to be read as sin(xcosy)?
(I've seen that written in my textbook too, but I know I am supposed to
interpret that as (sinx)(cosy).)

I know that grouping symbols should always be included to avoid
ambiguity; but when there are none, what is the correct way to interpret
these expressions?

Date: 10/31/2010 at 23:20:32
From: Doctor Peterson
Subject: Re: Order of operations

Hi, Jim.

I've pondered this a number of times, and my own conclusion is that there
are no actual rules.

What we are looking at here is a language that has developed not by
deliberate design, but organically -- with agreement among its users --
just as any natural language evolves. Linguists can study a language to
figure out its rules, but there are essentially reverse-engineering
something that exists without explicit rules. You can see some of that
development in the order of operations here:

  History of the Order of Operations 

The language of trigonometry is especially hard to explain logically. I
just looked through Cajori's _History of Mathematical Notations_, and the
variety of early forms is remarkable. In particular, he shows how much
inconsistency there was in writing 

Some would write it as 

   sin x^2
Others would render it as

   (sin x)^2
And still others would denote it as

   sin^2 x

That last way is the usual one today, despite long-acknowledged
difficulties. Specifically, it could be taken to mean sin(sin(x)) -- a
repeated application of the sine function. But Cajori points out

      "As functions of the last type [sin(sin x)] do not ordinarily
      present themselves, the danger of misinterpretation is very much
      less than in case of log^2 x, where log x * log x and log(log x) are
      of frequent occurrence in analysis."

What he's saying is that this abbreviated notation is used because it is
sufficient to distinguish commonly used forms from one another: experience
tells us that someone wouldn't mean sin(sin(x)).

I think that basic rule of common sense lies behind the laxness in the use
of other forms, such as the second and third examples you mention:

   sin 2x         means      sin(2x)

   sin x cos y    means      sin(x)cos(y)

These mean what they do simply because we know enough NOT to expect that

   sin 2x         would mean      sin(2)*x
   sin x cos y    would mean      sin(x*cos(y))

There is perhaps also a bit of typographical consideration: the spacing
generally suggests that 2x belongs as a unit, as do sinx and cosy in the
second example.

So it may be possible to examine all usages linguistically and come up
with some rules -- for example, "multiplication precedes trig functions
except where another trig function would be a factor." But what we really
do in reading these expressions is to use common sense based on
mathematical experience.

Does that help at all?

- Doctor Peterson, The Math Forum 

Date: 11/01/2010 at 00:50:07
From: Jim
Subject: Thank you (Order of operations)

This helps a lot. I appreciate knowing that this is not as cut and dried
as I thought it must be.

Thanks for the very clear explanation -- I really appreciate it!


Date: 11/01/2010 at 11:29:36
From: Doctor Peterson
Subject: Re: Thank you (Order of operations)

Hi, Jim.

I just checked for what others might have said about this, and found the
following conversation, which was not archived. 


      Dear Doctor Math,
      Are there rules (relating to order of operations maybe) that set
      guidelines for using parentheses with trig functions?

      For example, we are told to write the derivative of ...
         sin(4x^2 - 2) 

      ... as

         8x cos(4x^2)
      This leaves no ambiguity as to whether ...
         cos(4x^2 - 2)      is multiplied by      8x
      ... or just 
         4x^2 - 2           is multiplied by      8x

      Positioning the 8x term in front helps avoid confusion about where
      one expression ends and another one begins.
      But mathematically, and conventionally, what would this mean?
         cos(4x^2 - 2)(8x)
      Personally, I would take it to mean
         [cos(4x^2 - 2)] (8x)
      If I wanted to convey "take the cosine of the whole quantity," I
      would write 
      I understand it's best to use parentheses and brackets to erase any
      doubt, but would it be wrong to write cos(a) (b) when you mean

      When no parentheses are used -- for example, with sin 2x -- I know
      we usually take this to mean sin(2x). So that's something else to
      think about!


      Hi Bev,
      Good question! You've caught us mathematicians with our pants down.
      I think it's mostly out of laziness that we even write things like
         "sin x"
      Almost every other function other than the trigonometric (and
      possibly logarithmic) ones requires that there be parentheses around
      the parameter(s).

      Once you admit the legality of a form like "sin 2x," you've opened a
      can of worms. Even something like this ...
         f(x) = sin x + cos x
      ... could then mean this, right?
         f(x) = sin(x + cos(x))
      This is going to sound awful, but I have seen this sloppy usage so
      often that I just "know" what's intended. We mathematicians simply
      do a lot of "assumed grouping." For example, no professional
      mathematician would ever write a product this way:
      She'd always write it like this:
      In the same way, I'd never write ...
         sin x 2
      ... or even
         (sin x)2
      I'd move the 2 in front, to make it this ...
         2 sin x 
      ... or this:
         2 sin(x)
      Likewise, interpreting this ...
         sin 2x
      ... as "the sine of 2, multiplied by x" would be very unusual, since
      first off, we'd almost surely put the "x" in front of the "sin 2";
      and second, unless there's a "degree" symbol after the "2," it would
      be incredibly unlikely that you'd be taking the sine of 2 radians.

      Most computer languages require parentheses for the trig functions
      (well, any functions), because they have to be precise.

      If you do put parentheses after sin, cos, et cetera, then there's no
      doubt that the contents of those parentheses represent exactly the
      parameter to the function.

      So for your example of "cos(4x^2 - 2)(8x)", every mathematician in
      the world would interpret this as the cosine of 4x^2 - 2, with that
      result then multiplied by 8x.

      I'm sorry there's not a clean, clear answer, but that's the way it
      is: an ugly skeleton in the mathematician's closet.
      - Doctor Tom, The Math Forum 


So, Dr. Tom more or less agrees with me, but says it in ways that
complement my explanations.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Basic Algebra
High School Exponents
High School Trigonometry

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