MathLabs

Problem 4

Is there an infinite set of points in the plane such that no three points are collinear, and the distance between any two points is rational?
Step 2 of 4: Prove all relevant sines are rational
Previous stepNext step
sin⁡((k+1)θ)+sin⁡((k−1)θ)=2sin⁡(kθ)cos⁡θ\sin((k+1)\theta)+\sin((k-1)\theta)=2\sin(k\theta)\cos\theta
Detailed analysis

The recurrence and the rational initial values sin⁡(0)=0\sin(0)=0, sin⁡θ=4/5\sin\theta=4/5 show inductively that sin⁡(kθ)\sin(k\theta) is rational for every integer kk.