In mathematics, a trigonometric number:ch. 5 is an irrational number produced by taking the sine or cosine of a rational multiple of a 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.
Expanding the left side and equating real parts gives an equation in and substituting gives a polynomial equation having as a solution, so by definition the latter is an algebraic number. Also is algebraic since it equals the algebraic number Finally, where again is a rational multiple of is algebraic as can be seen by equating the imaginary parts of the expansion of the de Moivre equation and dividing through by to obtain a polynomial equation in
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