Trigonometric number

In mathematics, a trigonometric number:ch. 5 is an irrational number produced by taking the sine or cosine of a rational multiple of a full circle, or equivalently, the sine or cosine of an angle which in radians is a rational multiple of π, or the sine or cosine of a rational number of degrees. In mathematics, a trigonometric number:ch. 5 is an irrational number produced by taking the sine or cosine of a rational multiple of a full circle, or equivalently, the sine or cosine of an angle which in radians is a rational multiple of π, or the sine or cosine of a rational number of degrees. A real number different from 0, 1, –1 is a trigonometric number if and only if it is the real part of a root of unity. Thus every trigonometric number is half the sum of two complex conjugate roots of unity. This implies that a trigonometric number is an algebraic number, and twice a trigonometric number is an algebraic integer. Ivan Niven gave proofs of theorems regarding these numbers.:ch. 3 Li Zhou and Lubomir Markov recently improved and simplified Niven's proofs. Any trigonometric number can be expressed in terms of radicals. Those that can be expressed in terms of square roots are well characterized (see below). To express the other ones in terms of radicals, one requires nth roots of non-real complex numbers, with n > 2. An elementary proof that every trigonometric number is an algebraic number is as follows.:pp. 29-30. One starts with the statement of de Moivre's formula for the case of θ = 2 π k / n {displaystyle heta =2pi k/n} for coprime k and n: Expanding the left side and equating real parts gives an equation in cos ⁡ θ {displaystyle cos heta } and sin 2 ⁡ θ ; {displaystyle sin ^{2} heta ;} substituting sin 2 ⁡ θ = 1 − cos 2 ⁡ θ {displaystyle sin ^{2} heta =1-cos ^{2} heta } gives a polynomial equation having cos ⁡ θ {displaystyle cos heta } as a solution, so by definition the latter is an algebraic number. Also sin ⁡ θ {displaystyle sin heta } is algebraic since it equals the algebraic number cos ⁡ ( θ − π / 2 ) . {displaystyle cos( heta -pi /2).} Finally, tan ⁡ θ , {displaystyle an heta ,} where again θ {displaystyle heta } is a rational multiple of π , {displaystyle pi ,} is algebraic as being the ratio of two algebraic numbers. In a more elementary way, this can also be seen by equating the imaginary parts of the two sides of the expansion of the de Moivre equation to each other and dividing through by cos n ⁡ θ {displaystyle cos ^{n} heta } to obtain a polynomial equation in tan ⁡ θ . {displaystyle an heta .}