MathLabs

Problem 5

Determine, with proof, whether one can find 19751975 points on the circumference of a circle with unit radius such that the distance between any two of them is a rational number.
Step 2 of 5: Good angles are closed under sums and differences
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sin⁡(α±β)=sin⁡αcos⁡β±cos⁡αsin⁡β\sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta
Detailed analysis

The addition and subtraction formulas show that the sum or difference of two good angles is again good, because it is built from rational sums and products.