Algebraic trigonometric values at rational multipliers of π

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1 ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume, Number, June 04 Available online at Algebraic trigonometric values at rational multipliers of π Pinthira Tangsupphathawat Abstract. The problem of finding all algebraic values of α [, ] when arccos α, arcsin α and arctan α are rational multiples of π is solved. The values of such α of degree less than five are explicitly determined.. Introduction Rational and higher degree algebraic values of the cosine function have been of much interest for quite some time, cf. [4,,, 0]. As early as 9, Lehmer [4] proved that if k/n, n >, is an irreducible fraction, then cos(πk/n) and sin(πk/n) are algebraic integers of degree ϕ(n)/, where ϕ(n) is the Euler s totient function. Lehmer s proof makes use of cyclotomic polynomials. As a consequence, we have [7, Theorem 6.6, pp. 0 09]: let θ = rπ be a rational multiple of π. Then cos θ, sin θ and tan θ are irrational numbers apart from the cases where tan θ is undefined, and the exceptions cos θ = 0, ±/, ±; sin θ = 0, ±/, ±; tan θ = 0, ±. Recently, Varona [0] proved that if r Q [0, ] then arccos( r) is a rational multiple of π if and only if r {0, /4, /, /4, }. His proof is elementary and is similar to the proof of [, Theorem 4, p. ]. In Section, we push further the result of Varona, using elementary trigonometric identities, to find all possible nonnegative rational and some quadratic (i.e., algebraic of degree ) values of the cosine function at rational multiples of π. In Section, we adopt the approach of Lehmer in [4], i.e., using cyclotomic polynomials, to determine all other algebraic values. All algebraic values of degree less than 5 are explicitly worked out. In the last section, we consider the same problem for the sine and tangent functions. Received December, Mathematics Subject Classification. J7, B0. Key words and phrases. Trigonometric functions, algebraic values, multiples of π. 9

2 0 PINTHIRA TANGSUPPHATHAWAT. Nonnegative rational and some quadratic cosine values From the well-known trigonometric identities cos θ = ( cos θ) and cos(n + )θ = ( cos θ)( cos nθ) cos(n )θ (n N), using an idea from [7, Theorem 6.6, pp. 0 09], it follows that there exists a monic f n (x) Z[x] of degree n such that Clearly, cos nθ = f n ( cos θ). (.) f (x) = x, f (x) = x, f n+ (x) = xf n (x) f n (x) (n N). (.) The polynomials f n (x) are closely related to the Chebyshev polynomials of the first kind ([, pp. 6 6]) defined by T 0 (x) =, T (x) = x, T n+ (x) = xt n (x) T n (x) (n ), i.e., f n (x) = T n (x). Taking θ = kπ/n, k N, in (.), we get f n ( cos(kπ/n)) = 0. (.) Since f n Z[x] is monic of degree n, we have thus proved Theorem.. If k, n N, then cos(kπ/n) is an algebraic integer of degree n. In [0], the author s main idea is to take a proof from THE BOOK for the case π arccos(/ n) (n N, n odd, n ) and to do a simple variation of the proof to analyze the case π arccos( r) (r Q) that is much more general. We now proceed analogously using Theorem. to give a simple proof of the result in [0]. Corollary.. Let r Q with 0 r. Then, the number π arccos( r) is rational if and only if r {0, /4, /, /4, }. Proof. Assume that π arccos( r) = k/n is a rational number. Theorem. shows then that r = cos(kπ/n) is an algebraic integer. Thus, 4r is also an algebraic integer. Since r Q [0, ], we deduce that 4r is a rational integer in [0, 4], which implies r {0, /4, /, /4, }. The converse is trivial. An alternative simple derivation of Corollary. is to start from [7, Theorem 6.6] which asserts that for r Q, the values cos(rπ) Q only for cos(rπ) {0, ±, ±/}. By using cos (x) = ( + cos(x))/, we get that cos (rπ) Q if and only if cos(rπ) {0, ±, ±/}. Consequently, cos (rπ) can only take the rational values ( + 0)/, ( ± )/ and ( ± /)/, so the possible rational values for cos (rπ) correspond to the cases cos(rπ) { 0, ±, ±, ±, ± }.

3 ALGEBRAIC TRIGONOMETRIC VALUES Recall from [9, Theorem 6.6] that a quadratic integer in Q( d), where d is a square-free rational integer, takes one of the following two forms: and. all numbers of the form a + b d, where a and b are rational integers,. if d (mod4), but not otherwise, all numbers of the form (a+b d)/, where a and b are odd. This result is proved by analyzing solutions a + b d c (.4) with a, d, c, d N {0}, c, d 0, gcd(a, b, c) =, of a quadratic equation x + sx + t = 0 with s, t Z. Extending the proof of Corollary., we obtain the following generalization. Theorem.. Let α = a + b d [0, ] c with a, b, c, d N {0}, c, d 0, gcd(a, b, c) = and d square free. Then, the number π arccos(α) is a rational number if and only if α takes one of the following values 0 π, π, 0, π 4, π 6, 5+ 4 π 5. (.5) Proof. The sufficiency part is clear from (.5). We proceed now to prove the necessity part. If π arccos(α) = k/n is a rational number, then, by Theorem., ( ) ( kπ a + b ) d cos = α = [0, ] n c is an algebraic integer. From the shapes of quadratic integers, given in (.4), we must have c = or 4, the latter possibility occuring only when d mod 4 and a, b odd. If c =, then α = a + b d [0, ]. Since a, b are nonnegative, the only possible values of α are 0,,,,. If c = 4, d mod 4, a and b odd, then α = (a + b d)/ [0, ] implying that the only possible value of α is ( + 5)/. To deal with the case where a, b are integers, we need more information about cos(kπ/n). We carry this out in the next section using Lehmer s results in [4].

4 PINTHIRA TANGSUPPHATHAWAT. General algebraic cosine values Let n N, n > and ζ n = e πi/n. The nth cyclotomic polynomial (see, e.g., [6, pp. ]) is defined by Φ n (x) = (x ζn). k (.) k n gcd(k,n)= Clearly, deg Φ n (x) = ϕ(n), where ϕ is the Euler totient function. It is well known that Φ n (x) Z[x]. Since the roots of Φ n (x) occur in reciprocal pairs, there is a monic polynomial ψ n (x) Z[x] of degree ϕ(n)/ such that ψ n (x + x ) = x ϕ(n)/ Φ n (x). From [4, Theorem ] and its proof, we have Proposition.. Let α [, ]. If π arccos α = k/n Q with k, n Z, n > and gcd(k, n) =, then α = cos(kπ/n) is an algebraic integer of degree ϕ(n)/ whose minimal polynomial is ψ n (x). Note in addition that the cases where n = and are trivial for cos(kπ) =, cos(kπ) = (±) k. A special case of Theorem. is the following result, which is a slight extension of the main result in []. Corollary.. The value of cos(a b c ), where a, b and c are nonnegative integers, is an algebraic integer of degree 5. Moreover, cos(a b c ) is an algebraic integer of exact degree 5 if and only if gcd(c, 0) =. Proof. The result follows at once from Proposition. by noting that a b c = (60 a + 60b + c)π Proposition. also enables us to answer the problem( posed at the end of the last section, namely, to determine when ) π arccos a+b d c is rational for a, b being any integers, and much more. Theorem.. Let α [, ] Assume that π arccos α = k/n, k Z, n N, gcd(k, n) =. Then (i) the number α = cos(πk/n) is an algebraic integer of degree if and only if n =,,, 4, 6; in such cases, all the values taken by α are 0 π, 0 π 4 6π 4, π 6 0π 6 π 4π ;

5 ALGEBRAIC TRIGONOMETRIC VALUES (ii) the number α = cos(πk/n) is an algebraic integer of degree if and only if n = 5,, 0, ; in such cases, all values of α are 5 π 4 5 π 6π 4π 5 0 0, 5 + 4π 4 5 6π π π 5 0 0, π 4π 6π 0π, π π 0π 4π ; (iii) the number α = cos(πk/n) is an algebraic integer of degree if and only if n = 7, 9, 4, ; in such cases all values of α are cos π 7 π 7 0π 4 cos 6π π 4 4 4π 7 π 4, 0π 7, cos π 6π 4 4 6π 7 π 7, cos π 9 cos 4π 9 6π 9 4π 9 4π 0π π, 6π, cos π 4π π 0π 9 9 ; (iv) the number α = cos(πk/n) is an algebraic integer of degree 4 if and only if n = 5, 6, 0,, 0; in such cases all values of α are ( (5 ) 5) ( 5 + 6(5 + ) 5) ( + 5 6(5 ) 5) ( 5 6(5 + ) 5) π π 6π 4π , 4π 6π π π , π π 4π 46π , 4π 5 6π 5 π 0π 4π π , 6π 6π 0π π , π π π π , 6π 4π 4π 6π , π 46π π 6π, 0π π 4π π 5π 0 0, 4π.

6 4 PINTHIRA TANGSUPPHATHAWAT Proof. The following table gives the values of n for the first four values of ϕ(n)/. The determination of these values can be found in the paper [5] which gives an elementary approach to solve the equation ϕ(x) = k. ϕ(n)/ n, 4, 6 5,, 0, 7, 9, 4, 4 5, 6, 0,, 0 Using the values of n in the preceding table, the corresponding minimal polynomials are shown in the next table. n Φ n (x) ψ n (t) (t = x + x ) x + x + t + 4 x + t 5 x 4 + x + x + x + t + t 6 x x + t 7 x 6 + x 5 + x 4 + x + x + x + t + t t x 4 + t 9 x 6 + x + t t + 0 x 4 x + x x + t t x 4 x + t 4 x 6 x 5 + x 4 x + x x + t t t + 5 x x 7 + x 5 x 4 + x x + t 4 t 4t + 4t + 6 x + t 4 4t + x 6 x + t t 0 x x 6 + x 4 x + t 4 5t + 7 x x 4 + t 4 4t + 0 x + x 7 x 5 x 4 x + x + t 4 + t 4t 4t + The stated values of α are computed by finding all the roots of ψ n (t) using Mathematica. Remark.4. The results in Proposition., Corollary., and Theorem. can be transformed to those of hyperbolic cosine function by noting that cosh(z) (zi). 4. Algebraic sine and tangent values The following results are taken from [6, Theorems.9 and.]. They were first considered by Lehmer [4]; however, the original Lehmer s result for the sine function was incomplete. Proposition 4.. Let α [, ]. I. If π arcsin α = k/n Q with k, n Z, n >, n 4 and gcd(k, n) =, then α = sin(kπ/n) is an algebraic integer of degree ϕ(n), ϕ(n)/4 or ϕ(n)/ according as gcd(n, ) < 4, gcd(n, ) = 4 or gcd(n, ) > 4.

7 ALGEBRAIC TRIGONOMETRIC VALUES 5 II. If π arctan α = k/n Q with k Z, n >, n 4 and gcd(k, n) =, then α = tan(kπ/n) is an algebraic integer of degree ϕ(n), ϕ(n)/ or ϕ(n)/4 according as gcd(n, ) < 4, gcd(n, ) = 4 or gcd(n, ) > 4. The cases when n = and n = are trivial for 0 (kπ) (kπ) = tan(kπ) = tan(kπ), and the case when n = 4 is also trivial for sin(kπ/) {, } and tan(kπ/) = 0 or undefined. Using Proposition 4. and Theorem., all algebric values of degrees < 5 of the sine and tangent functions are given in the following theorem, whose proof is omitted. Theorem 4.. Let α [, ]. A. Assume that π arcsin α = k/n, k Z, n N, gcd(k, n) =. Then (A) the number α = sin(πk/n) is an algebraic integer of degree if and only if n =,, 4, 6, ; in such cases, all the values taken by α are 0 0 π, π 4 6π 4, 4π π, 0π 6π ; (A) the number α = sin(πk/n) is an algebraic integer of degree if and only if n =, 6,, 0; in such cases, all the values taken by α are π 6π 5 π π 0 4π 6 4π π 4π 0 46π 0 0π π 0 6π 0 0π 6, 4π, π 0, 4π 0 ; (A) the number α = sin(πk/n) is an algebraic integer of degree if and only if n =, ; in such cases all the values taken by α are cos π 7 cos 6π 7 π π 6π 46π 0π π 54π, 66π, cos 6π 6π 5π 4π 50π 4, cos π 9 cos π 9 cos 0π 6π 50π π π 5π 70π 4π π 46π 74π, 6π, 6π ;

8 6 PINTHIRA TANGSUPPHATHAWAT (A4) the number α = sin(πk/n) is an algebraic integer of degree 4 if and only if n = 5, 0, 6,, 60; in such cases all values of α are ( + 5) + 4 (5 5) π π 4π π 9π 60 60, ( 5) + 4 (5 + 5) 4π 46π 4π π 06π 60 60, ( + 5) 4 (5 5) π 6π π π π 60 60, ( 5) 4 (5 + 5) 4π 6π 94π π 60 46π 60, π 0π π π 6π , 6π 4π 4π π 0π , π 6π 4π 4π π 5, 6π 0 4π 5 π 5 6π 5 π 0, π 4π 5π 4π π, 0π π 50π 6π 46π. B. Assume that π arctan α = k/n, k Z, n N, gcd(k, n) =. Then (B) the number α = tan(πk/n) is an algebraic integer of degree if and only if n =,, ; in such cases, all the values taken by α are 0 = tan 0 = tan π, = tan π 4 = tan 5π 4 = tan π 4 = tan 7π 4 ;

9 ALGEBRAIC TRIGONOMETRIC VALUES 7 (B) the number α = tan(πk/n) is an algebraic integer of degree if and only if n =, 6,, 6, ; in such cases, all the values of α are 4π = tan = tan 0π 6, = tan π 6 = tan π = tan π 4π 0π π, = tan 4π 6 = tan 0π 6 π π = tan = tan 6 6, + = tan 6π π 0π 6π , = tan π 6π π 46π, + = tan 0π = tan 4π = tan 4π = tan π ; (B) the number α = tan(πk/n) is an algebraic integer of degree 4 if and only if n = 5, 0, 0,, 40, 4; in such cases all the values taken by α are = tan π = tan 6π = tan 4π 0 = tan π 0 = tan 4π 5 = tan 6π 0 = tan π 5, = tan π 0, = tan π π π π , = tan 6π 6π 4π 4π , 4 + = tan π 4π 0π 6π, 4 + = tan 6π π 6π 5π, 4 + = tan 0π 4π π 54π, = tan 4π 46π π 50π. 5. Final remarks. As is easily checked from Proposition 4., II, there are no third degree algebraic values for the tangent function.. The result about all possible rational values of the sine, cosine and tangent functions derived in [] is properly contained in Theorem., (i), and Theorem 4., (A), (B). Acknowledgements The author wishes to thank the Thailand Research Fund for financial support, and also the anonymous referee for all valuable suggestions.

10 PINTHIRA TANGSUPPHATHAWAT References [] M. Aigner and G. M. Ziegler, Proofs from THE BOOK, Springer-Verlag, Berlin, 99. [] E. W. Cheney, Introduction to Approximation Theory, Chelsea, New York, 9. [] R. W. Hamming, The transcendental character of cos x, Amer. Math. Monthly 5 (945), 7. [4] D. H. Lehmer, A note on trigonometric algebraic numbers, Amer. Math. Monthly 40 (9), [5] N. S. Mendelsohn, The equation ϕ(x) = k, Math. Mag. 49 (976), 7 9. [6] I. Niven, Irrational Numbers, The Carus Mathematical Monographs, Wiley, New York, 956. [7] I. Niven, H. S. Zuckerman, and H. L. Montgomery, An Introduction to the Theory of Numbers, Wiley, New York, 99. [] J. M. H. Olmsted, Rational values of trigonometric functions, Amer. Math. Monthly 5 (945), [9] H. Pollard and H. G. Diamond, The Theory of Algebraic Numbers, The Carus Mathematical Monographs 9, The Mathematical Association of America, Washington, 975. [0] J. L. Varona, Rational values of the arccosine function, Cent. Eur. J. Math. 4 (006), 9. Department of Mathematics, Faculty of Science and Technology, Phranakhon Rajabhat University, Bangkok 00, Thailand address: pinthira@hotmail.com; pinthira@pnru.ac.th