Niven's theorem

Niven's theorem states that if an angle is a rational multiple of π and its sine is rational, its sine is 0, 1/2, −1/2, 1, or −1.

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Roots of unity: Complex numbers whose positive integer powers equal 1. Exponentiating a rational multiple of π produces a root of unity whose algebraic properties constrain the sine.
Unit circle: The circle of radius one centered at the origin in the coordinate plane. It identifies sine with a coordinate and makes the theorem’s five possible values geometrically visible.
Exact trigonometric values: Closed-form values of trigonometric functions at specified angles. Niven’s theorem identifies the only rational sine values possible when the angle is a rational multiple of π.
Hermite–Lindemann theorem: A transcendence theorem stating that the exponential of every nonzero algebraic number is transcendental. It offers a broader transcendence framework, whereas Niven’s theorem gives a direct finite list under a specific angle condition.
Chebyshev polynomials: Polynomial families defined by identities such as Tₙ(cos θ) = cos(nθ). Their recurrence relations can turn rational-angle trigonometric identities into polynomial constraints.
Sine: A trigonometric function giving the vertical coordinate of a point on the unit circle. The theorem restricts precisely the rational values this function can take at rational multiples of π.
Sine of a rational multiple of π: A sine value whose angle is a rational multiple of π. This is the precise class of values restricted by the theorem.
Lindemann–Weierstrass theorem: A theorem asserting the linear independence of exponentials of distinct algebraic numbers over the algebraic numbers. Its powerful general result differs from Niven’s elementary restriction on rational sine values.
Algebraic integer: A complex number that is a root of a monic polynomial with integer coefficients. The proof exploits an algebraic integer built from the angle, then uses its rationality to restrict its value.
Rational multiple of π: An angle equal to π times a rational number. This angle condition is the theorem’s other essential hypothesis.