rational sine and cosine
Proof. . When the catheti , and the hypotenuse of a right triangle are integers, i.e. they form a Pythagorean triplet, then the sine and the cosine of one of the acute angles of the triangle are rational numbers.
. Let the sine and the cosine of an acute angle be rational numbers
where the integers , , , satisfy
Since the square sum of sine and cosine is always 1, we have
Since two of its terms are divisible by , also the third term is divisible by . But because by (1), the integers and are coprime, we must have (see the corollary of Bézout’s lemma). Similarly, we also must have . The last divisibility relations mean that , whence (2) may be written
and accordingly the sides of a corresponding right triangle are integers.
|Title||rational sine and cosine|
|Date of creation||2013-03-22 17:54:50|
|Last modified on||2013-03-22 17:54:50|
|Last modified by||pahio (2872)|