The Derivative of the Sine and Cosine. A New Derivation Approach

Transcription

1 The Deivative of the Sine and Cosine. A New Deivation Appoach By John T. Katsikadelis Scool of Civil Engineeing, National Technical Univesity of Athens, Athens 15773, Geece. jkats@cental.ntua.g Website: Keywods: Deivative; Sine; Cosine; Leibniz; Cotes; Eule. Abstact A new method is pesented fo finding the deivative of the sine and cosine using the discoveies of Leibniz in calculus between the yeas 1675 and 1677, namely the deivative of the poduct and the uotient of two functions as well as the chain ule, yet long befoe the discovey of the deivative of the sine by Roge Cotes in 1722 and Eule in Intoduction The seeds fo finding of the deivative of the sine and cosine have been sown by Roge Cotes ( ) in his wok Hamonia mensuaum [1] published posthumously in Cotes stated and poved the following lemma: Lemma I. Vaiatio minima cujusvis acus ciculais est ad Vaiationem minimam Sinus ejusdem acus ut adius ad Sinum complementi. which is endeed in English as [2] Lemma I. The small vaiation of any ac of a cicle is to the small vaiation of the sine of that ac, as the adius to the sine of the complement. The poof given by Cotes esults fom Fig. 1 (uoted fom [1]) by consideing the simila tiangles ACD and ECG. Thus we have EC AC = (1) EG AD

2 Figue 1. Using moden notation, the small vaiation of any ac of a cicle with adius becomes d( ) = CE, while the small vaiation of the sine of that ac is d(sin ) = CG. Hence, E. (1) is witten as d ( ) = (2) d(sin ) cos In 1748, Eule intoduced the tigonometic functions to teat the linea homogeneous diffeential euations [3] and simplified E. (2) using the unit cicle, yielding thus the deivative of the sine d(sin ) = cos (3) Since the poof given by Cotes might be objected as lacking in igo, the deivative was obtained late using the elation deived by Cauchy in 1822 [4] sin lim = 1 (4) 0 in which the measue of the angles is in adians. E. (4) esults fo small values of fom the ineuality sin 1 > > cos (5) which had been implied in the woks of Achimedes. Hence we obtain

3 d sin( +D) -sin (sin ) = lim D 0 D é sin( D / 2) ù = lim cos( +D/ 2) D 0 ê D /2 ú ë û = lim cos( +D/ 2) lim D 0 D 0 = cos sin( D / 2) D /2 This autho in his ecent pape [5] avoided the diffeentiation of the cosine because he wanted to use mathematics that was available befoe 1686, namely the yea when Newton published his Pincipia [6]. But this was still a challenge to develop a new method fo finding the deivative of the sine and cosine using mathematics available befoe The pesented pocedue uses the ules fo the diffeentiation of the poduct and the uotient of two functions of one vaiable as well as the chain ule. All these ules wee discoveed by Leibniz about 1676 [7]. (6) The new appoach fo finding the deivative of the sine and cosine We conside a moving paticle in the xy plane. Its position in pola coodinates detemined by the time dependent paametes t () and () t, while in Catesian coodinates by xt () and yt (), Fig. 2. j y O v et t () i vy () t = y D e v vx = x x Figue 2 We define the functions: x y c = cos =, s = sin = (7) Then we have

4 x = c, y = s, = x + y (8a,b,c)) Diffeentiation of Es. (8) with espect to t gives x = c + c (9a) y = s + s (9b) xx + yy = (9c) Multiplying Es. (9a) and (9b) by y and x, espectively, and subtacting them, we obtain yx - xy yc - xs = (10) Futhe, multiplying Es. (9a) and (9b) by x and y, espectively, adding them and using E. (8c), we obtain xx + yy = + xc + ys (11) which by vitue of E. (9c) becomes xc + ys = 0 (12) Es. (10) and (12) ae combined and solved fos and c. Thus we obtain x yx - xy 1 1 s =- =-cos ( x sin -y cos ) (13a) y yx - xy 1 1 c = = sin ( x sin -y cos ) (13b) Refeing to the Fig. 2, we have y cos - x sin = v = (14) whee v is the component of the velocity in the diection e, i.e., nomal to e. Hence, Es. (13a,b) by vitue of E. (14) and the use of the chain ule become d sin s = = cos (16a) o d cos c = =-sin (16b)

5 d sin d cos = cos (17a) =- sin (17b) Obviously, Es. (17a,b) give the deivative of the sine and cosine with espect to the angle measued in adians. Conclusions In this note a new method is pesented fo finding the deivative of the sine and cosine using mathematics available befoe 1686, the yea Newton published his Pincipia, namely the deivative of the poduct and the uotient of two functions as well as the chain ule discoveed by Leibniz between the yeas 1675 and Refeences [1] Cotes, R. (1722). Hamonia Mensuaum, Opea Miscellanea, Aestimatio Eoum, p. 3, Edito: R. Smith, Cambidge Univesity. [2] Gowing, R. (1983). Roge Cotes. Natual Philosophe, Cambidge Univesity Pess, p. 93. [3] Ince, E.L. (1956), Odinay Diffeential Euations, Dove Publications, New Yok. [4] Boye, C.B. (1947). Histoy of the Deivative and Integal of the sine, The Mathematical Teache, Vo. 40, pp [5] Katsikadelis, J.T. (2017). Deivation of Newton s law of motion fom Keple s laws of planetay motion, Achive of Applied Mechanics, DOI /s x. [6] Newton, Isaac (1686) Philosophiae Natualis Pincipia Mathematica, Royal Society Pess, London. [7] Boye, C.B. (1959). Histoy of Calculus and its Conceptual development, Dove Publications, New Yok.