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Limits of trigonometric functions

  1. Feb 3, 2008 #1
    Why do some problems return the wrong answer while others do not on the ti-89.

    For example:

    [tex]\[ \lim_{x \to 0} \frac{\cos\theta \tan\theta}{\theta}\][/tex]

    Shows up wrong (shows up as pi over 180).

    But

    [tex]\[ \lim_{x \to 0} \frac{\sin x(1 - \cos x)}{2x^2}\][/tex]

    does not?
     
    Last edited: Feb 3, 2008
  2. jcsd
  3. Feb 3, 2008 #2
    Not sure, but you should be able to do these easy by hand.
     
  4. Feb 3, 2008 #3

    mathman

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    What is the something? 3.14159...?
     
  5. Feb 3, 2008 #4
    No. Pi/180. But that isn't one. My question is, why do certain trigonometric equations show up as the textbook answers, but not others.

    For example, like I said, the second one returns correctly, the first one does not. I understand it should be in radian mode now, but why does degree mode give the right answer 50-75% of the time in my experience?
     
  6. Feb 3, 2008 #5

    Gib Z

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    I am very confused :( These problems have nothing to do with angles!! It shouldn't matter what mode you shove these into your calculator.
     
  7. Feb 3, 2008 #6
    Right, Gib Z. That's exactly what I was thinking. Just thought it would be helpful for the forum if someone had a quick explanation.
     
  8. Feb 3, 2008 #7
    Nothing happens to that limit as x changes, maybe that's why your calculator comes up with something different.

    [tex]\[ \lim_{\theta \to 0} \frac{\cos\theta \tan\theta}{\theta}\][/tex]
     
  9. Feb 3, 2008 #8

    lurflurf

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    Not wrong

    [tex]\[ \lim_{x \to 0} \frac{\cos\theta \tan\theta}{\theta}= \lim_{x \to 0} \frac{\sin\theta }{\theta}=\frac{\pi}{2 \arcsin 1}\][/tex]
    in degrees pi/180 deg^-1
    in grad pi/200 grad^-1
    in rad 1 rad^-1
    in mil pi/3200 mil^-1
    in clock pi/6 hours^-1
    in rotations pi/.5 rot^-1

    Angle measure units matter
    Rad make calculus things look nice
    Why use the calculator at all save that for later
     
  10. Feb 4, 2008 #9

    HallsofIvy

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    Make sure your calculator is in "radian" mode rather than "degree" mode!
     
  11. Feb 4, 2008 #10
    Yeah, thanks guys. And I think I see what you're saying lurflurf. If you convert it from degrees to randians its 1 anyway. (What's the deg^-1?)
     
  12. Feb 5, 2008 #11

    lurflurf

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    It is from unit analysis

    (10 feet)/(5 seconds)=2 feet seconds^-1

    if
    units(x)=degrees
    units(sin(x))=1 (ie no units)
    then
    units(sin(x)/x)=1/degrees=deg^-1
    angle measure units are not entirely well defined
    but tracking them can prevent errors especially when radians are not being used

    if anyone like -1 better than 2
    pi/arccos(-1)=pi/(2 arcsin(1))
    =limit x->0 sin(x)/x
    for that matter may expressions are possible
     
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