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rational sine and cosine∗
pahio†
2013-03-22 0:36:44
Theorem. The only acute angles, 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 hypotenuse c of a right triangle are
integers, i.e. they form a Pythagorean triplet, then the sine ac and the cosine cb
of one of the acute angles of the triangle are rational numbers.
2o . Let the sine and the cosine of an acute angle ω be rational numbers
sin ω =
a
,
c
cos ω =
b
,
d
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
a2 b2
+
= 1.
c2 d 2
(2)
By removing the denominators we get the Diophantine equation
a2 d2 +b2 c2 = c2 d2 .
Since two of its terms are divisible by c2 , also the third term a2 d2 is divisible
by c2 . But because by (1), the integers a2 and c2 are coprime, we must have
c2 | d2 (see the corollary of Bézout’s lemma). Similarly, we also must have
d2 | c2 . The last divisibility relations mean that c2 = d2 , whence (2) may be
written
a2 + b2 = c2 ,
and accordingly the sides a, b, c of a corresponding right triangle are integers.
∗ hRationalSineAndCosinei
created: h2013-03-2i by: hpahioi version: h40408i Privacy
setting: h1i hTheoremi h26A09i h11D09i h11A67i
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