Thursday, October 22, 2009

Fox 164

Submitted by:
http://geometri-problemleri.blogspot.com

6 comments:

  1. A*sinA/PI = a/b
    where A is the angle.

    The answer depends on the radius.
    If r=1 then I am not sure about the answer, but I don't think there are a and b integers.

    But say if r=sqrt(2)/PI, then it DE/|DB| can be a rational number.

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  2. thank you very much for publishing this question, great fox

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  3. This comment has been removed by the author.

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  4. 0 < angle(DCB)=DB <2PI

    in this distance there is infinite rational numbers can be shown x/y (x, y integers)
    let DB=x/y

    angle(DCB)=DB=x/y (radian)
    DE=rsin(x/y)
    DE/DB=rsin(x/y)*(y/x)

    r, y/x are rational numbers then sin(x/y) must be rational number where 0 < x/y < 2PI

    for example
    sin(PI/2)=1 but PI/2 is not a rational number
    sin(PI/6)=1/2 but PI/6 is not a rational num

    my math knowledge is not enough to solve it

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  5. DE/DB = sinX /(2pi*X/2pi) in radians.
    = sin(X) / X = f(x)
    f(x) i a monotonously decresing function between 0 and pi.
    f(0)=1 and f(pi/2)=2/pi, f(pi)=0.
    There are infinitely -many rational numbers between 1 and 2/pi()~0.6
    => there are infinitely-many rational values for DE/DB.
    -EOP

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  6. Thank you very much for this solution, Actually i meaned if arc length (DB)=X is a rational number then can sin(X) be a rational number? (0< x< 2pi)

    OR

    Let DE/DB=sin(x)/x be a rational number

    sin(x)-x*DE/DB=0 : Is solution of this equation a rational number?

    example
    sin(x)/x=0.5

    sin(x)-0.5*x=0 by solving this equation
    x=1,8954.....

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