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Theorem 0.1. For any r =06 , cos r is a transcendental number.

Proof. Since cos r = cos(−r), we can assume that r is positive and write r = s/t where s, t are relatively prime positive . We use a . Suppose that cos r is not a transcendental number. Then there exist a positive m and rational numbersa ¯j (0 ≤ j ≤ m) such that m j a¯j cos r =0. Xj=0 From the product-to-sum formulas for trigonometric functions, there exist rational numbers aj (0 ≤ j ≤ m) such that

m

aj cos jr =0. (0.1) Xj=0

By multiplying an integer, we may assume that aj (0 ≤ j ≤ m) are all integers. If a0 =6 0, let

t4mp+2p−2x2p−2(x2 − r2)2p(x2 − 4r2)2p ··· (x2 − m2r2)2p f(x)= , (0.2) (2p − 2)! where p is a positive prime to be specified later. If a0 = 0 and aj0 =6 0 for some 1 ≤ j0 ≤ m, we take f to be

4mp+2p−2 2p−2 t f (x)f (x) ··· f − (x)(x − j r) f (x) ··· f (x) f(x)= 0 1 j0 1 0 j0+1 m (2p − 2)!

2 2 2p where fk(x) = ((x − j0r) − (kr − j0r) ) for k =6 j0. In both cases, the following arguments are almost the same. Thus, we only give the proof for the case a0 =6 0. For 0

t4mp+2p−2(mr)4mp+2p−2 |f(x)| < . (0.3) (2p − 2)!

It can also be easily verified that for any k ≥ 0, f k(0) is an integer and f (2k+1)(0) = 0 (see [12, Chapter 2.2]).

2 By the elementary calculus, for any positive integer j,

jr ∞ ∞ ∞ f(x) sin(jr−x)dx = (−1)kf (2k)(jr)+ (−1)kf (2k)(0) cos jr+ (−1)kf (2k+1)(0) sin jr. Z 0 Xk=0 Xk=0 Xk=0

Note that f (2k+1)(0) = 0 for any k. Hence,

jr ∞ ∞ f(x) sin(jr − x)dx = (−1)kf (2k)(jr)+ (−1)kf (2k)(0) cos jr. Z 0 Xk=0 Xk=0

Then, by(0.1),

m jr m ∞ a f(x) sin(jr − x)dx = (−1)ka f (2k)(jr). (0.4) j Z j Xj=0 0 Xj=0 Xk=0

The right hand in above is an integer. Moreover, by(0.2), f (2k)(jr) is divisible by p for all k and j with one exception:

f (2p−2)(0) = s4mpt2p−2(m!)4p, if we choose p > max {m, s, t}. Next, by taking p > |a0|, the right hand of(0.4) (2p−2) consists of a sum of multiples of p with one exception, namely a0f (0). Hence, the right hand of(0.4) is a non-zero integer. However, by(0.3), the left hand of(0.4) satisfies

m m jr t4mp+2p−2(m2r2)2mp+2p−2 aj f(x) sin(jr − x)dx ≤ |aj|· jr · < 1, Z0 (2p − 2)! Xj=0 Xj=0

provided p is chosen sufficiently large. Thus we have a contradiction, and the theo- rem is proved.

Corollary 0.2. The trigonometric functions are transcendental at non-zero rational values of the arguments.

Proof. Because of cos 2r =1 − sin2 r and cos2r = (1 − tan2 r)/(1+tan2 r), sin r and tan r are transcendental numbers for any rational number r =6 0. Also, csc r, sec r and cot r are transcendental numbers for any rational number r =6 0.

3 References References [1] Butlewski, Z., 1948. A proof that em is irrational. Colloquium Math. 1, 197–198. URL: https://doi.org/10.4064/cm-1-3-197-198, doi:10.4064/cm-1-3-197-198. [2] Chrystal, G., 1959. : An elementary text-book for the higher classes of secondary schools and for colleges. 6th ed, Chelsea Publishing Co., New York. [3] Euler, L., 1744. De fractionibus continuis dissertatio. Commentarii academiae scientiarum Petropolitanae 9, 98–137. [4] Hardy, G.H., Wright, E.M., 2008. An introduction to the theory of numbers. Sixth ed., Oxford University Press, Oxford. Revised by D. R. Heath-Brown and J. H. Silverman, With a foreword by Andrew Wiles. [5] Hermite, C., 1873. Sur la fonction exponentielle. Comptes rendus de l’Acadmie des Sciences de Paris 77, 18–24. [6] Hurwitz, A., 1893. Beweis der Transcendenz der Zahl e. Math. Ann. 43, 220– 221. URL: https://doi.org/10.1007/BF01443646, doi:10.1007/BF01443646. [7] Koksma, J.F., 1949. On Niven’s proof that π is irrational. Nieuw Arch. Wiskunde (2) 23, 39. [8] Lambert, J.H., 1768. M´emoires sur quelques propri´et´es remarquables des quantit´es transcendantes, circulaires et logarithmiques. M´emoires de l’Acad´emie royale des sciences de Berlin 17, 265–322. URL: http://www.kuttaka.org/~JHL/L1768b.html. [9] Lindeman, F., 1882a. Uber¨ die ludolph’sche zahl. Sitzungsberichte der K¨oniglich Preussischen Akademie der Wissenschaften zu Berlin 2, 679–682. [10] Lindeman, F., 1882b. Uber¨ die zahl π. Mathematische Annalen 20, 213–225. [11] Niven, I., 1947. A simple proof that π is irrational. Bull. Amer. Math. Soc. 53, 509. URL: https://doi.org/10.1090/S0002-9904-1947-08821-2, doi:10.1090/S0002-9904-1947-08821-2. [12] Niven, I., 1956. Irrational numbers. The Carus Mathematical Monographs, No. 11, The Mathematical Association of America. Distributed by John Wiley and Sons, Inc., New York, N.Y.

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