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Math Help - Canceling out and trig identities

  1. #1
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    Canceling out and trig identities

    I just have a quick question:

    Example:

    (2sinxcosx/sinx) - (cos2x/cosx)

    So this is one side of a trig identity. I was just wondering...

    in (2sinxcosx/sinx) can you cancel out the sinx and leave it to be (2cosx)
    but for the (cos2x/cosx), you cannot cancel out the cos2x and cosx, right?

    Or can you NOT cancel out the (2sinxcosx/sinx) and put it as (2cosx) because of the subtraction sign? I don't understand when you can and cannot cancel. Ok for multiplication and division you can cancel, but for addition and subtraction you can only cancel the ones on top of eachother, not across right? or is that wrong?

    Main idea:

    When can you cancel and when can you not? Provide examples in relation to this please and thank you!
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  2. #2
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    Quote Originally Posted by skeske1234 View Post
    I just have a quick question:

    Example:

    (2sinxcosx/sinx) - (cos2x/cosx)

    So this is one side of a trig identity. I was just wondering...

    in (2sinxcosx/sinx) can you cancel out the sinx and leave it to be (2cosx)
    but for the (cos2x/cosx), you cannot cancel out the cos2x and cosx, right?

    Or can you NOT cancel out the (2sinxcosx/sinx) and put it as (2cosx) because of the subtraction sign? I don't understand when you can and cannot cancel. Ok for multiplication and division you can cancel, but for addition and subtraction you can only cancel the ones on top of eachother, not across right? or is that wrong?

    Main idea:

    When can you cancel and when can you not? Provide examples in relation to this please and thank you!
    Hello again skeske1234,

    Have we been helpful, yet.

    To your first question, you may cancel across factors like this:

    \frac{2 \sin x \cos x}{\sin x}=\frac{2}{1}\cdot \frac{{\rlap{////}\sin x}}{{\rlap{////}\sin x}} \cdot \frac{\cos x}{1}=2 \cos x

    As far as \frac{\cos 2x}{\cos x} goes, consider:

    \frac{\cos(2 \cdot 30)}{\cos(30)}

    If you tried to cancel out the 30, what do you have? \cos 2?? Does that make sense?
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  3. #3
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    Quote Originally Posted by masters View Post
    Hello again skeske1234,

    Have we been helpful, yet.

    To your first question, you may cancel across factors like this:

    \frac{2 \sin x \cos x}{\sin x}=\frac{2}{1}\cdot \frac{{\rlap{////}\sin x}}{{\rlap{////}\sin x}} \cdot \frac{\cos x}{1}=2 \cos x

    As far as \frac{\cos 2x}{\cos x} goes, consider:

    \frac{\cos(2 \cdot 30)}{\cos(30)}

    If you tried to cancel out the 30, what do you have? \cos 2?? Does that make sense?
    If you cancelled out the 30, you would have cos2/cos.. so no it does not make sense??? Right or wrong?But if you cancelled out the cos30, you would have 2. Can you do either one of them? What about my questions relating to adding and subtraction?

    If you had
    (18a/9a) - (5b/10b)
    can you cancel out the 18a/9a? and then subtract it from the 5b/10b?
    when you do:
    18a/9a do you get 2 or do you get 2a? and then subtract that from 1/2b or 1/2?

    Or is canceling out when subtracting and adding not possible. HELP! Please be straight forward...
    Last edited by skeske1234; April 10th 2009 at 10:45 AM.
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  4. #4
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    Get the word "Cancel" out of your head.

    Use the Multiplicative Identity Property of Unity (1).

    Looking at \frac{\sin(2x)}{\sin(x)}, tell me where you see FACTOR that is exactly one (1). There is only one factor in each of the numerator and the denominator. You can't do it!

    The one that was not confusing? \frac{2\sin(x)cos(x)}{sin(x)}

    As was demonstrated, there was a FACTOR of the expression that is exactly one (1).

    \frac{2\sin(x)cos(x)}{sin(x)}\;=\;\frac{\sin(x)}{\  sin(x)} \cdot 2 \cdot \cos(x)\;=\;(1) \cdot 2 \cdot \cos(x)\;=\;2 \cdot \cos(x)

    This is one of the reasons you studied properties and factoring and definitions. It all comes together right here and all this wonderful framework is destroyed by the silly idea of "cancelling". Just say "no" to cancelling.
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  5. #5
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    Quote Originally Posted by TKHunny View Post
    Get the word "Cancel" out of your head.

    Use the Multiplicative Identity Property of Unity (1).

    [snip]

    This is one of the reasons you studied properties and factoring and definitions. It all comes together right here and all this wonderful framework is destroyed by the silly idea of "cancelling". Just say "no" to cancelling.
    The word 'cancel' when used in a mathematical context is euphemistically implying the multiplicative identity. It has widespread usage and acceptance in this area. The use of the word 'cancel' or any other 'shortspeak' should not in any way replace the foundation properties of mathematics. It just makes them a little more palatable for some. It's sort of like "cross multiplying", which I also know you don't like. But it's a technique, a method of solving proportions. It works for those 'non-math' oriented folks who only need their math as a means to some end (which probably does not include a lot of math). Just an opinion.


    References:


    cancel - definition of cancel by the Free Online Dictionary, Thesaurus and Encyclopedia.

    can·cel (knsl)
    v. can·celed also can·celled, can·cel·ing also can·cel·ling, can·cels also can·cels

    5. Mathematics
    a. To remove (a common factor) from the numerator and denominator of a fractional expression.
    b. To remove (a common factor or term) from both sides of an equation or inequality.

    6. Printing To omit or delete.

    cancelling definition | Dictionary.com

    can⋅cel [kan-suhl] Show IPA verb, -celed, -cel⋅ing or (especially British) -celled, -cel⋅ling, noun –verb (used with object)

    6.Mathematics. to eliminate by striking out a factor common to both the denominator and numerator of a fraction, equivalent terms on opposite sides of an equation, etc.
    7.to cross out (words, letters, etc.) by drawing a line over the item.
    8.Printing. to omit.

    –verb (used without object)
    9.to counterbalance or compensate for one another; become neutralized (often fol. by out): The pros and cons cancel out.
    10.Mathematics. (of factors common to both the denominator and numerator of a fraction, certain terms on opposite sides of an equation, etc.) to be equivalent; to allow cancellation.

    cancel - Definition from the Merriam-Webster Online Dictionary

    can·cel

    2 a: to mark or strike out for deletion
    b: omit , delete

    3 a: to remove (a common divisor) from numerator and denominator b: to remove (equivalents) on opposite sides of an equation or account
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  6. #6
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    Educators' Dictionary *


    fool·ish
    a.
    1. Perpetuating ideas, techniques, or processes that are intended to help students but clearly have confused the student at hand - no matter how authoritatively you may be able to justify your actions.

    can·cel
    v. can·celed also can·celled, can·cel·ing also can·cel·ling, can·cels also can·cels
    1. Mathematics - A technique used to simplfy primary consumption of multiplicative properties. When the educator observes that the student is confused by the concept of "cancelling", abandon the concept immediately and teach the fundamental priciples.

    Note to Student: Get the word "Cancel" out of your head. It is confusing you, and is, therefore, of no value.

    * So far, I have only two entries in my "Educators' Dictionary". Publish date is unclear.
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  7. #7
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    Just one more reference, then I'll hush. We can revive this in a Chat Room thread if you like. I would like to hear opinions from the other site members. I believe we could have some good clean healthy discussions about this and other similar bastardizations of mathematical propertities, terms and shortcuts.

    Six references to the word 'cancel' on this page alone:
    Complex Fractions

    Eleven references here:
    Complex Fractions: More Examples

    Twenty-three here:
    Multiplying Rational Expressions

    It's out there, TKH. It's all over the place. Teachers have been using it for years and are continuing to do so.

    By the way. Are you familiar with the referenced web site?
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  8. #8
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    Whew! I thought you were going to quote me on some post where I was not paying my usual attention.

    I have had a chat with Elizabeth (aka stapel, "Purple Math Lady") on many occasions. I think we get along rather well.
    • When I see "Cancel" confusing a student, I tell them to forget the word exists.
    • When I see a student failing to generalize "FOIL" to trinomials, I tell them to forget "FOIL" ever existed.
    • When I see a student forgetting what is happening, I tell them to forget the pratice of "cross multiplication".
    • When I see a student simply substituting to find a limit, without any consideration of continuity, I tell them never to substitute like that again.
    • There are others, but these are the most common.
    I've been an Ivory Towerist on these points since I met the concepts. "Cancel" and "Cross Multiply" go all the way back to 2nd Grade!!

    In any case, I am not so locked in my tower as to reject the methods if it is determined that the student actually is benefitted by them. On the other hand, the student may not know there is some other way to think about it, having been required to perform a certain way by a classroom environment. My background is not so much in the classroom. I tend to get students who were confused by their classroom experience. Thus, this makes me a natrual heretic when it comes to certain prevailing methods. It's just a pedagogical choice, not really a Crusade. (Well, for "FOIL", it is kind of a Crusade.)

    Note: When I took up this cause, I was close to amazed at what a massive proportion of students thanked me openly and asked "Why couldn't my teacher say that?" I was pretty high on myself for a little while. It dawned on me that if I were in the calssroom, that teacher they were talking about would be outside getting the same remarks from the other half the class.

    P.S. I used to live in Lunchburg, VA. Great area!
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  9. #9
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    Quote Originally Posted by TKHunny View Post
    P.S. I used to live in Lunchburg, VA. Great area!
    TK,

    You make a good argument. I believe we're probably on the same page about this.

    By the way, I taught math at Natural Bridge High School (long before consolidation) for a few years right after graduation. That was my first job. Used to go to Lynchburg a lot. All I can remember from back then were all the railroad tracks.

    If one of our friendly mods is watching, maybe they will move this conversation to the Chat Room.
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