trigonometric number

In 1961, mathematician Ivan Niven published a proof that the cosine of a rational number given in degrees between 0° and 90° is irrational with one notable exception at 60°.

Such a number is called a trigonometric number.

His proof involves intermediate trigonometric identities and arguments based on rationality that should be accessible to a university student in first year.

The outline of the proof goes like this :

Take the identity

2 cos (n + 1)θ = {2 cos nθ}{2 cos θ} - 2 cos (n - 1)θ

And then show that

2 cos nθ = (2 cos θ)ⁿ + c_n-1(2 cos θ)^n-1 + ... + c₁(2 cos θ) + c₀

This part of the proof is a standard proof by induction.

And Niven then argues that 2 cos θ must then be the root of the polynomial equation with integer coefficients

xⁿ + c_n-1x^n-1 + c_n-2x^n-2 + ... + c₁x + c₀ - 2 = 0

But 2 cos θ has a maximum of 2 and a minimum of -2, and since 0° < θ < 90° then cos θ is between 1 and 0. So 0 < 2 cos θ < 2. The only integer between 0 and 2 is 1, and the only value of θ satisfying this is 60°

Niven then goes on to prove similar properties for the sin and tan functions based on this.

Niven tied together the concepts of irrationality, rationality and trigonometry in one proof in the same manner that Euler tied together the concepts of e, π, zero and complex numbers.