|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
rational sine and cosine
|
(Theorem)
|
|
|
Theorem. 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
 |
(1) |
Since the square sum of sine and cosine is always 1, we have
 |
(2) |
By removing the denominators we get the Diophantine equation $$a^2d^2+b^2c^2 = c^2d^2.$$ Since two of its terms are divisible by $c^2$ , also the third term $a^2d^2$ 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.
|
"rational sine and cosine" is owned by pahio.
|
|
(view preamble | get metadata)
Cross-references: sides, mean, relations, divisibility, corollary of Bézout's lemma, coprime, divisible, terms, Diophantine equation, denominators, sum, square, rational numbers, triangle, integers, right triangle, hypotenuse, catheti, proof, Pythagorean triplets, rational, cosine, sine, acute angles, theorem
This is version 4 of rational sine and cosine, born on 2008-03-15, modified 2008-12-12.
Object id is 10408, canonical name is RationalSineAndCosine.
Accessed 835 times total.
Classification:
| AMS MSC: | 11A67 (Number theory :: Elementary number theory :: Other representations) | | | 11D09 (Number theory :: Diophantine equations :: Quadratic and bilinear equations) | | | 26A09 (Real functions :: Functions of one variable :: Elementary functions) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
