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sin theta ~ tan theta ~ theta. HELP watch

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    at the back of my (edexcel) physics paper

    it says

    sin theta ~(roughly equal to) tan theta ~ theta
    cos theta ~ 1

    i dont understand. :confused: :confused:
    can anyone explain it?

    thanx
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    If you use the series expansions you get
    sinx = x - (x^3)/3! + (x^5)/5! - ....
    cosx = 1 - (x^2)/2! + (x^4)/4! - ....
    For small x, x^3 terms and above can be neglected so sinx ~ x, cosx ~ 1 - (x^2)/2! More roughly if you neglect x^2 terms, cosx ~ 1. tanx = sinx/cosx ~ x
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    (Original post by scarlet ibis)
    at the back of my (edexcel) physics paper

    it says

    sin theta ~(roughly equal to) tan theta ~ theta
    cos theta ~ 1

    i dont understand. :confused: :confused:
    can anyone explain it?

    thanx
    These are the assumptions to make if theta is very small.
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    look at their graphs and draw the graph y = x

    or alternatively use the mclaurin expansions

    sinx = x - x^3/3! + x^5/5! .. etc

    you can see that when x is small, the terms after x will become negligible, and sin x = x

    same argument can be used to show that cos x = 1 for small x

    and tan x = sin x/cos x ... for small x, numerator = x for small x, denominator = 1 for small x, so tan x = x/1 = x for small x.
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    (Original post by Bezza)
    If you use the series expansions you get
    sinx = x - (x^3)/3! + (x^5)/5! - ....
    cosx = 1 - (x^2)/2! + (x^4)/4! - ....
    For small x, x^3 terms and above can be neglected so sinx ~ x, cosx ~ 1 - (x^2)/2! More roughly if you neglect x^2 terms, cosx ~ 1. Can't remember how you do tanx though

    Bloody hell yer quick
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    (Original post by fishpaste)
    Bloody hell yer quick
    Wasn't much quicker than you! And I forgot how to do tanx to start with
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    thanks. i think i understand, but i don't do maths, so its still be bit hard to grasp
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    (Original post by scarlet ibis)
    thanks. i think i understand, but i don't do maths, so its still be bit hard to grasp
    Indeed. In fact you don't learn where those expansions/approximations come from unless you do further maths! Never really used in the alevel. But they're useful in physics for things like SHM I guess.
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    Also remember that that's for theta in radians and not degrees.
 
 
 
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