In mathematics, a trigonometric number is an irrational number

Irrational number

In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

 produced by taking the cosine of a rational

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

 multiple of a circle, or equivalently, the cosine in radian

Radian

Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

s of a rational multiple of π, or the cosine of a rational number of degrees.

Ivan Niven gave proofs regarding these numbers in Numbers: Rational and Irrational published in 1961. Li Zhou and Lubomir Markov recently improved and simplified Niven's proofs in the book.