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Thread: 2 sin(x/2) cos(x/2) = sin(x)?

  1. #1
    Newbie B9766's Avatar
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    2 sin(x/2) cos(x/2) = sin(x)?

    Hi,

    I'm having trouble with a proof in a course assignment. We aren't using calculus yet. The lesson covers the sine of 1 degree and describes the equations sin(x+y), cos(x+y), sin(x-y), cos(x-y), sin(2x) and cos(2x).

    In one particular problem there's a step I don't understand:

    Equation 1:  2sin(\frac{x}{2})cos(\frac{x}{2}) + 2sin(\frac{y}{2})cos(\frac{y}{2}) = sin(x)+sin(y)

    I know from the equation for sin(2x) that:

    Equation 2: sin(2x) = 2sin(x)cos(x)

    but it looks as if they're dividing all the angles by 2 in this second equation. Is this just a given that will be covered in calculus later or is there some way other than calculus to derive:

    Equation 3:  2sin(\frac{x}{2})cos(\frac{x}{2}) = sin(x)

    I've drawn a circle with r = 1 and tried to reconcile the concept that, for any point on the circle, half the angle of elevation produces a rise that when multiplied by its overage, and then doubled, is equal to the total elevation.

    Any help would, as always, be much appreciated.
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  2. #2
    MHF Contributor
    skeeter's Avatar
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    Re: 2 sin(x/2) cos(x/2) = sin(x)?

    no calculus required, just simple algebraic substitution ...

    $2\sin(u)\cos(u) = \sin(2u)$

    now let $u = \dfrac{x}{2}$

    don't make it more difficult than it isn't ...
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  3. #3
    Newbie B9766's Avatar
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    Re: 2 sin(x/2) cos(x/2) = sin(x)?

    Thanks skeeter. I guess I was getting all bound up in the rule sin(2x) does not = 2sin(x).

    I appreciate the help.
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