# rational sine and cosine

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

Proof. ${1}^{\underset{\xaf}{o}}$. 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^{}.

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

$$\mathrm{sin}\omega =\frac{a}{c},\mathrm{cos}\omega =\frac{b}{d},$$ |

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

$\mathrm{gcd}(a,c)=\mathrm{gcd}(b,d)=\mathrm{\hspace{0.33em}1}.$ | (1) |

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

$\frac{{a}^{2}}{{c}^{2}}}+{\displaystyle \frac{{b}^{2}}{{d}^{2}}}=\mathrm{\hspace{0.33em}1}.$ | (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 |