Thursday, October 22, 2009

Fox 164

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

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

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.

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

3. This comment has been removed by the author.

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

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

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

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.....