1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Mathematical approximations in Physics for angles <20 degrees

  1. Aug 16, 2011 #1
    1. The problem statement, all variables and given/known data

    62. In physics, it is important to use mathematical approximations.
    (a) Demonstrate that for small angles (<20°)

    tan a ~ sin a ~ a = (Pia'/180 degrees)

    where a is in radians and a' is in degrees. (b) Use a calculator
    to find the largest angle for which tan a may be
    approximated by a with an error less than 10.0%.



    2. Relevant equations



    3. The attempt at a solution

    I'm unsure how I would go about this other than plugging in numbers less that 20 degrees, but I wouldn't know what is within ten percent error.
     
  2. jcsd
  3. Aug 16, 2011 #2

    PeterO

    User Avatar
    Homework Helper

    If you take sin 30 you get 0.5000. If you take tan 30 you get 0.5773

    That is a difference of 0.0773

    That represents a percentage error of (0.0773/.5000 * 100)% or 15.5% difference.
    That is bigger than 10%.

    If the angle is smaller , the error is smaller.
    NOTE: error doesn't mean a mistake, it means how far from the real answer would you calculated answer be if you used tan rather than sin - perhaps for some convenience reason.
     
  4. Aug 16, 2011 #3
    Oh, as you go from tan (a) to sin (a) to the actual (a) in radians it will be equal to the angle of (a) times Pi over 180?

    Edit: Thanks for your time and reply. It feels embarrassing that I should be strugglling with these questions in the first chapter as opposed to all the other questions I've been seeing.
     
  5. Aug 16, 2011 #4

    PeterO

    User Avatar
    Homework Helper

    the "angle of (a) times Pi over 180" is just there to convert degrees to radians
     
  6. Aug 16, 2011 #5
    Ah I see, thank you. It's been a few years since my Trig class.
     
  7. Jun 20, 2013 #6
    Follow Up Question.

    I understand that tan of a small angle is close to the sine of an angle, but what is the significance of it.

    In my book it states for waves that Tsin(theta) for small angles is approximately equal to Ttan(theta) and we should use that instead. My question is why? Why use tangent instead of sine?

    Thanks
     
  8. Jun 20, 2013 #7

    PeterO

    User Avatar
    Homework Helper

    depends where your angle is.

    If you are looking at an interference pattern on a screen for example.

    The source of the pattern (The actual double or single slit) is, say, 1 m from a screen.
    If the feature (max or min) you are working with it 2cm off centre, then the angle to that feature has a tangent of 2/100.
    If there is another feature 3cm off centre, then the angle to that feature has a tangent of 3/100.

    If you were after the sine of each of those angles you would have to measure the direct distance to each feature in turn. (tan is opposite/adjacent; sin is opposite/hypoteneuse)

    If the sin and tan value are basically the same, you are saved that problem.

    You would also be saved the problem if the subsequent formulae you were using had the tan function in them, but I think the formulae include sine.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Mathematical approximations in Physics for angles <20 degrees
  1. Physics 20 (Replies: 2)

Loading...