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}
+ ... + c_{1}(2 cos θ) + c_{0}
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

x^{n} + c_{n-1}x^{n-1} + c_{n-2}x^{n-2}
+ ... + c_{1}x + c_{0} - 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.