rational sine and cosine

Theorem.  The only acute anglesMathworldPlanetmathPlanetmath, whose sine and cosine are rational, are those determined by the Pythagorean triplets(a,b,c).

Proof. 1o¯. When the catheti a, b and the hypotenuseMathworldPlanetmath c of a right triangleMathworldPlanetmath are integers, i.e. they form a Pythagorean triplet, then the sine ac and the cosine bc of one of the acute angles of the triangle are rational numbersPlanetmathPlanetmath.

2o¯. Let the sine and the cosine of an acute angle ω be rational numbers


where the integers a, b, c, d satisfy

gcd(a,c)=gcd(b,d)= 1. (1)

Since the square sum of sine and cosine is always 1, we have

a2c2+b2d2= 1. (2)

By removing the denominators we get the Diophantine equationMathworldPlanetmath


Since two of its terms are divisible by c2, also the third term a2d2 is divisible by c2.  But because by (1), the integers a2 and c2 are coprimeMathworldPlanetmath, we must have  c2d2 (see the corollary of Bézout’s lemma).  Similarly, we also must have  d2c2.  The last divisibility relations mean that  c2=d2,  whence (2) may be written


and accordingly the sides a,b,c of a corresponding right triangle are integers.

Title rational sine and cosine
Canonical name RationalSineAndCosine
Date of creation 2013-03-22 17:54:50
Last modified on 2013-03-22 17:54:50
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Theorem
Classification msc 26A09
Classification msc 11D09
Classification msc 11A67
Related topic RationalPointsOnTwoDimensionalSphere
Related topic GreatestCommonDivisor
Related topic GeometricProofOfPythagoreanTriplet
Related topic RationalBriggsianLogarithmsOfIntegers
Related topic AlgebraicSinesAndCosines