# rational sine and cosine

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

Proof. $1^{\underline{o}}$. When the catheti $a$, $b$ and the hypotenuse $c$ of a right triangle are integers, i.e. they form a Pythagorean triplet, then the sine $\frac{a}{c}$ and the cosine $\frac{b}{c}$ of one of the acute angles of the triangle are rational numbers.

$2^{\underline{o}}$. Let the sine and the cosine of an acute angle $\omega$ be rational numbers

 $\sin\omega\;=\;\frac{a}{c},\quad\cos\omega=\frac{b}{d},$

where the integers $a$, $b$, $c$, $d$ satisfy

 $\displaystyle\gcd(a,\,c)\;=\;\gcd(b,\,d)\;=\;1.$ (1)

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

 $\displaystyle\frac{a^{2}}{c^{2}}\!+\!\frac{b^{2}}{d^{2}}\;=\;1.$ (2)

By removing the denominators we get the Diophantine equation

 $a^{2}d^{2}\!+\!b^{2}c^{2}\;=\;c^{2}d^{2}.$

Since two of its terms are divisible by $c^{2}$, also the third term $a^{2}d^{2}$ is divisible by $c^{2}$.  But because by (1), the integers $a^{2}$ and $c^{2}$ are coprime, we must have  $c^{2}\mid d^{2}$ (see the corollary of Bézout’s lemma).  Similarly, we also must have  $d^{2}\mid c^{2}$.  The last divisibility relations mean that  $c^{2}=d^{2}$,  whence (2) may be written

 $a^{2}+b^{2}\;=\;c^{2},$

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