SOME TRIGONOMETRIC IDENTITIES RELATED TO POWERS OF COSINE AND SINE FUNCTIONS

Transcription

1 Folia Mathematica Vol. 5, No., pp. 4 6 Acta Uiversitatis Lodziesis c 008 for Uiversity of Lódź Press SOME TRIGONOMETRIC IDENTITIES RELATED TO POWERS OF COSINE AND SINE FUNCTIONS ROMAN WITU LA, DAMIAN S LOTA Abstract. Some decompositio of certai basic symmetric fuctios of -th power of cosie sie fuctios is preseted here. Next the applicatios of these decompositios to geeratig may ew biomial trigoometric idetities are discussed.. Itroductio I this paper we wish to ivestigate differet trigoometric idetities coected with the followig trigoometric idetities previously preseted i [8]: si x + cos x π 6 + cos x + π 6 + cos x+ + cos x π 3 + cos x + π 3 cost,,..., 5. It is atural to relate these purpose to decompositios of the followig four symmetric fuctios of the -th powers of cosie sie fuctios: C + x, ϕ : cos x + ϕ + cos x ϕ, C x, ϕ : cos x ϕ cos x + ϕ, S + x, ϕ : si x + ϕ + si x ϕ, S x, ϕ : si x + ϕ si x ϕ. for N. The term decompositio meas here the trigoometric polyomial form of these fuctios. We ote that some other decompositio of the fuctios 5 usig the Chebyshev polyomial of the secod id i the separate paper [7] is discussed. The basic decompositios of fuctios 5 are preseted i Sectio i two ext sectios are applied to geeratig of trigoometric idetities, startig from the simplest forms see Sectio 3 Lemma 8 to Istitute of Mathematics, Silesia Uiversity of Techology, Kaszubsa 3, Gliwice Pol. roma.witula@polsl.pl. Istitute of Mathematics, Silesia Uiversity of Techology, Kaszubsa 3, Gliwice Pol. d.slota@polsl.pl. AMS subject classificatios: 6C05, B37, B83.

2 4 Roma Witu la, Damia S lota mai idetities show i Lemma 0 Corollaries 8, 0,, 4, respectively. Moreover at the ed of the Sectio a iterestig applicatio of these decompositios to obtaied some otrivial biomial idetities is also give.. The basic decompositio Lemma see [3]. The followig two classical idetities hold 6 C + x, ϕ 7 C x, ϕ / 0 / 0 cos ϕ cos x si ϕ si x. The proof of the above idetities by iductio follows will be omitted here. The sequece of the followig idetities ca easily be deduced from idetities 6 7 the formulas 4 are ow see [], the formula 5 is probably ew: 8 S + x, ϕ 0 0 cos ϕ cos x cos ϕ cos x + ; 3 S + x, ϕ 9 cos ϕ si x ; 0 3 S x, ϕ 0 si ϕ cos x ; 0

3 Some Trigoometric Idetities Related to S x, ϕ 0 si ϕ si x ; si x si x ; 0 3 si x S + x, 0 0 cos x ; 4 cos x C + x, 0 / cos x 0 ; / C + x + π 8, π 4 cos x π 8 + cos x π cos π cos x + π 4 0 cos 4x + π + cos 8x + π 4 cos x π + cos 6x + π si4x cos8x six + cos6x si0x cos4x si8x + cos3x I the Tables below the first seve decompositios of the type 6 of every fuctio C +, C, S + S are preseted.

4 44 Roma Witu la, Damia S lota Table.. The first seve decompositios of every fuctio C + C C + x, ϕ C x, ϕ cosϕ cosx siϕ six + cosϕ cosx siϕ six 3 `3 cosϕ cosx + cos3ϕ cos3x `3 siϕ six + si3ϕ si3x 4 ` cosϕ cosx + cos4ϕ cos4x `4 4 siϕ six + si4ϕ si4x 5 `0 8 cosϕ cosx + 5 cos3ϕ cos3x+ `0 8 siϕ six + 5 si3ϕ si3x+ + cos5ϕ cos5x + si5ϕ si5x 6 ` cosϕ cosx+ `5 6 siϕ six+ +6 cos4ϕ cos4x + cos6ϕ cos6x +6 si4ϕ si4x + si6ϕ si6x 7 `35 3 cosϕ cosx + cos3ϕ cos3x+ `35 3 siϕ six + si3ϕ si3x+ +7 cos5ϕ cos5x + cos7ϕ cos7x +7 si5ϕ si5x + si7ϕ si7x Table.. The first seve decompositios of every fuctio S + S S + x, ϕ S x, ϕ cosϕ six siϕ cosx cosϕ cosx siϕ six 3 `3 cosϕ six cos3ϕ si3x `3 siϕ cosx si3ϕ cos3x 4 `3 4 4 cosϕ cosx + cos4ϕ cos4x `4 4 siϕ six si4ϕ si4x 5 `0 8 cosϕ six 5 cos3ϕ si3x+ `0 8 siϕ cosx 5 si3ϕ cos3x+ + cos5ϕ si5x + si5ϕ cos5x 6 `0 6 5 cosϕ cosx+ `5 6 siϕ six +6 cos4ϕ cos4x cos6ϕ cos6x 6 si4ϕ si4x + si6ϕ si6x 7 `35 3 cosϕ six cos3ϕ si3x+ `35 3 siϕ cosx si3ϕ cos3x+ +7 cos5ϕ si5x cos7ϕ si7x +7 si5ϕ cos5x si7ϕ cos7x At the ed of this sectio it will be give the applicatio some of above decompositios to geeratig a iterestig sets of biomial idetities see also [, 4]. Corollary. Sice for x, ϕ 0 we have C x ϕ x, ϕ! x + ϕ ! x + ϕ x ϕ x ϕ,

5 Some Trigoometric Idetities Related to so by 6 the followig formula hold: lim x,ϕ 0 C x, ϕ lim x ϕ x,ϕ 0 i.e., / 0 / 0 si ϕ ϕ. si x Corollary. From it ca be easily derived the formula six + x + + +r r x r, r! r 0 hece we get the sequece of the followig biomial idetities: + r for r,,...,, !, !, !, etc., sice we have: six 6 5 x 6 x + x Similarly, from 3 we deduce the formula six + x x r +r x r, r! r0 0 x,

6 46 Roma Witu la, Damia S lota which by 6 implies the secod sequece of the followig biomial idetities:, 0 r 0, for r,,...,, etc. 0!, 0 + +!, !, Corollary 3. From 4 we obtai 3 x + x / + r r x r, / r! which implies: r0 0 etc. / 0 / 0, 4 3, Corollary 4. We have the formula see [4, 6]: / P x x, 0

7 Some Trigoometric Idetities Related to where P x deotes the -th Legedre polyomial. Hece, by 4 for eve N, we get π P cos ϕ dϕ π ! π 4 4!!.! 0 O the other h, we have [6]: P cos ϕ 4 + a, cos 0 for some a, Q, which by 7 implies the idetity: 4! 4 0!!! Simple trigoometric idetities We metio oly the simple form of trigoometric idetities to discuss here. Simultaeously this approach leads i Sectio 4 to direct our cosideratios to oly symmetric trigoometric idetities with respect to the phase traslatios. Lemma. Fix a, ϕ, ψ R. The si f a,ϕ,ψ x : a } x cos + si x + ϕ + si x + ψ cost x f a,ϕ,ψ x is idepedet of x uder this values a, ϕ, ψ iff either: or Proof. First, we ote that: ϕ ψ + π a 0 ϕ + ψ π a cosϕ cosϕ }. f a,ϕ,ψ x cost f a,ϕ,ψ x 0 f a,ϕ,ψ 0 0..

8 48 Roma Witu la, Damia S lota Hece, we get: siϕ + siψ 0 siϕ + ψ cosϕ ψ 0 ϕ + ψ π or ϕ ψ + π for some Z. If ϕ ψ + π the: f a,ϕ,ψ x : a si x cos x } + which implies a 0. If ϕ + ψ π the, we have: a si x + S + } f a,ϕ,ψ x x, ϕ cosϕ + a si x + si } a cos x + S + ϕ x, ϕ a cosϕ cos x + cos ϕ the proof is completed. Corollary 5. We have: si a } x cos + cos x + ϕ + cos x + ψ cost x iff either: or ϕ ψ + π a 0 ϕ + ψ π a Proof. Set ϕ : ϕ + π ψ : ψ + π Lemma 3. We have: a cos 4 x + S 4 + x, ϕ cost cosϕ cosϕ i the Lemma. }. iff either a ϕ ± π 6 + π, Z or a ϕ ± π + π, Z; a si 4 x + S 4 + x, ϕ cost iff either a ϕ ± π 3 + π, Z or a ϕ π, Z. Proof. The assertio from the followig decompositio follows: 4 a cos 4 x + S 4 + x, ϕ cos ϕ 3 4 cos ϕ + a cos 4 x 4 cos ϕ 4 cos ϕ 3 cos x + cos 4 ϕ.

9 Some Trigoometric Idetities Related to a si 4 x + S 4 + x, ϕ cos ϕ 3 4 cos ϕ + a si 4 x si ϕ 4 cos ϕ si x + si 4 ϕ. Let us set: Lemma 4. We have: g a,ϕ,ψ x : a cos 3 x + cos 3 x + ϕ + cos 3 x + ψ. g a,ϕ,ψ x cost is idepedet o x g a,ϕ, ϕ x cost Proof. We have: either ϕ + ψ π for some Z or a 0 ϕ ψ + π for some Z. either a l+ ϕ l π, l Z, or a 0 ϕ l + π, l Z. g π a,ϕ,ψ ga,ϕ,ψ π si 3 ϕ + si 3 ψ 0 ϕ + ψ ϕ ψ siϕ + siψ 0 si cos 0 ϕ + ψ π ϕ ψ + π, Z g a,ϕ,ψ ϕ ga,ϕ,ψ ψ a cos 3 ϕ cos 3 ψ 0 a 0 cosϕ cosψ 0 ϕ + ψ ϕ ψ a 0 si si 0 a 0 ϕ + ψ l π ϕ ψ l π, l Z. Hece, we coclude that either ϕ + ψ π, Z, or ϕ ψ + π, Z a 0. I the secod case g a,ϕ,ψ x 0. By 6 we get: g a,ϕ, ϕ x C 3 + x, ϕ + a C+ 3 x, 0 4 cos3 ϕ + a cos3 x cosϕ + a cosx,

10 50 Roma Witu la, Damia S lota which is idepedet o x iff cos3 ϕ + a 0 cosϕ + a 0 a cosϕ cos3 ϕ cosϕ 0 a cosϕ siϕ si ϕ 0 a cosϕ si ϕ 0 a cosϕ ϕ π for some Z either a l+ ϕ l π, l Z, or a 0 ϕ l + π, l Z. Now, let us set: h a,ϕ x : a cos 3 x + cos 3 x ϕ + cos 3 x + ϕ a C 3 + x, 0 + C+ 3 x, ϕ. Lemma 5. We have: h a,ϕx cosx cost h a,ϕx cosx 6 si ϕ cosϕ a + T 3 cosϕ 0 a cos3ϕ; ta : 3 a + a + a 3 a T 3 ta a; 3 ta cosϕ for some a, ϕ R a a + ϕ π, Z, where T 3 x deotes the third Chebyshev polyomial of the first id see [5]. Proof. It follows from 6 for 3: h a,ϕ x cosx cos3 x cos3 ϕ + a cosx + 3 cosϕ + a cost which implies. We have a cos3 ϕ h a,ϕx cosx 3 cosϕ cos3 ϕ, T 3 ta a+ + a 3 3 a+ + a a+ + a [ a+ + a 3 ] a + + a [ a+ + a ] a a 3 a. where a + : 3 a + a a : 3 a a.

11 Some Trigoometric Idetities Related to We ote that ta is a odd fuctio. Moreover we have: t a 6 3 a + a 3 a a. a Hece, we deduce that ta is a decreasig fuctio o, ] cosequetly a icreasig oe o [,. Should also be oticed, that t. Lemma 6. We have: C 5 + x, ϕ cost cosx ϕ + π, Z. Proof. Sice C + 5 x, ϕ cos5ϕ cos5 x + 5 siϕ si4ϕ cos 3 x + so we obtai: C 5 + x, ϕ cost cosx Now, let us set: Lemma 7. We have: F a,ϕ,ψ x cost + 0 si 4 ϕ cosϕ cosx, cos5ϕ 0 si4ϕ 0 ϕ + π, Z. F a,ϕ,ψ x : a cos 6 x + cos 6 x + ϕ + cos 6 x + ψ. a ϕ π ψ l π for some, l Z. Proof. If F a,ϕ,ψ x cost the F a,ϕ,ψ x 0, i.e.: a cos 5 x six + cos 5 x + ϕ six + ϕ + cos 5 x + ψ six + ψ 0. Hece, for x 0 x π, respectively, we get: 8 cos 5 ϕ siϕ + cos 5 ψ siψ 0, 9 si 5 ϕ cosϕ + si 5 ψ cosψ 0. Subtractig 9 from 8, we obtai: si4ϕ + si4ψ 0, i.e. si ϕ + ψ cos ϕ ψ 0 which implies that either 0 ϕ + ψ π

12 5 Roma Witu la, Damia S lota or ϕ ψ π 4 + π for some Z. O the other h, if we realize the operatio: si 4 ϕ 8 cos 4 ϕ 9, i.e. si 4 ϕ cos 5 ψ siψ cos 4 ϕ si 5 ψ cosψ 0, siψ siϕ+ψ siϕ ψ si ϕ cos ψ+cos ϕ si ψ 0, operatio: si 4 ψ 8 cos 4 ψ 9, i.e. si 4 ψ cos 5 ϕ siϕ cos 4 ψ si 5 ϕ cosϕ 0, 3 siϕ siϕ+ψ siϕ ψ si ϕ cos ψ+cos ϕ si ψ 0 we assume that holds, the from 3 we get siϕ+ψ 0, i.e.: ϕ + ψ l π, for some, l Z, so: ϕ ψ π 4 + π ϕ π 8 + l π + π 4 ψ l π π 4 π 8. Hece, we obtai: F a,ϕ,ψ x a cos 6 x + C 6 + x + l π, π 4 + π a + cosx + cos4x + cos6x cos π 4 + π cos x + l π + + cos 3 4 π + 3 π cos 6 x + 3 l π 5a a + l 4 cos π 4 + π cosx a cos4x + a 5 + l cos 3 4 π + 3 π cos6x which, from the liear idepedece of trigoometric system, implies a 0, i cosequece, F a,ϕ,ψ x is ot cost, cotrary to our assumptios. If we suppose ow that 0 holds, i.e. ϕ + ψ π + l π, l Z, the: F a,ϕ,ψ x : a cos 6 x + cos 6 x + ϕ + si 6 x ϕ F a,ϕ,ψ ϕ F a,ϕ,ψ ϕ

13 Some Trigoometric Idetities Related to i.e.: cos 6 ϕ + si 6 ϕ siϕ 0 ϕ π, Z. Hece: F a,ϕ,ψ x a + cos 6 x + si 6 x cost. Next let us assume that ϕ + ψ l π, l Z, the: F a,ϕ,ψ x a cos 6 x + C 6 + x, ϕ 5 a + cos6ϕ cos6x a + cos4ϕ cos4x a + cosϕ cosx +... hece 4 F a,ϕ,ψ x cost a cosϕ, cos6ϕ cos4ϕ cosϕ. We ote that cost cost cos3t cost cost + 0, cost cost cost + 0, which implies cost. Thus, from 4 it follows that: a cosϕ a ϕ π, Z. Now, let us set: f x : si x + cos x π 6 + cos x + π 6, g x : cos x + cos x π 3 + cos x + π 3. We have the followig idetities: f x g x `0 3 cos6x 3 `0 3 + cos6x 3 ` cos6x 3 ` cos6x 7 `4 5 5 cos6x 7 ` cos6x 3 ` cos6x + cosx 3 ` cos6x + cosx

14 54 Roma Witu la, Damia S lota f x + g x f x + g x ` cosx 3 ` cosx 45 ` cosx 5 ` cosx 969 ` cosx 97 ` cosx The form of f x + g x for is more complicated there are at least three terms i the respective decompositio, for example we have: f x + g x cos x + cos4 x. Remar. We have also C + x, π 6 + C + x, π 3 + cos x + cos x 0, for,, for, 3, + 3 cos x, for 4, + 5 cos x, for 5. The ext result idicates the directio i which attempts at geeralizig certai results from Sectio 3 should follow. Lemma 8. Let ϕ, ψ R Θ x C + x, 0 + C + x, ϕ + C + x, ψ, N, x R. If Θ x cost for some two differet values of N at least for oe odd value of N, the Θ x C + x, 0 + C + x, 5 π + C+ x, 4 5 π, N, x R. If Θ x cost, for two differet eve values of N, the Θ x C + x, 0 + C+ x, 5 π + C+ x, 4 5 π, N, x R. Proof. Directly from decompositio 6 it follows, that if Θ x cost Θ l x cost for, l N, < l, so, i view of the liear idepedece of the trigoometric system oe of the followig three coditios holds: cosϕ + cosψ 5, cosϕ + cosψ,

15 Some Trigoometric Idetities Related to wheever + l 0; or cosϕ + cosψ 6, cos3ϕ + cos3ψ, wheever + l ; or cosϕ + cosψ 7, cos4ϕ + cos4ψ, wheever + l. Ad 5 The give system implies: cos ϕ + cosϕ cosψ + cos ψ 4, cos ϕ + cos ψ 3 4, i.e. cosϕ cosψ 4. So, system 5 is equivalet to the followig system: cosϕ + cosψ 8, cosϕ cosψ 4. The solutios cos ϕ cos ψ of 8 form the set of the roots of polyomial x + x 4, i.e. the set } 9 cos ϕ, cos ψ 4 ± 5 } cos 5 π, cos 4 5 π}. Ad 6 We have: cosϕ + cosψ, T 3 cosϕ + T3 cosψ, i.e. cosϕ + cosψ, 4 cos 3 ϕ + cos 3 ψ. By trasformig the secod equatio we obtai, respectively: 4 cosϕ + cosψ cosϕ + cosψ 3 cosϕ cosψ, cosϕ cosψ, cosϕ cosψ 4, so the system 8 holds the equalities 9 are satisfied.

16 56 Roma Witu la, Damia S lota Ad 7 Coditio 7 from coditio 5 for ϕ : ϕ ψ : ψ follows. Hece, we get: } 30 cosϕ, cosψ cos 5 π, cos 4 5 π}. Now, the assertios of Lemma 8 from 9 30 follows. 4. Some geeralizatios Let us ow step dow to preset the aouced geeralized trigoometric idetities of the same ature as idetity. Each oe should be preceded by essetial techical lemmas describig the values of some trigoometric sums. Lemma 9. Let, r N, r. The the followig equality holds: 3 σ r : 0 exp i r π i ta, wheever r N, + i cot, wheever r N, where N N deotes the set of eve odd positive itegers. Proof. We perform the followig trasformatios: σ r exp i exp i i exp i exp exp i i r exp exp i i r exp i exp cos, for r N, i i exp si, for r N, which implies the desired idetity.

17 Corollary 6. Let, r N. The: Some Trigoometric Idetities Related to r 0, wheever r, 3 + cos π, wheever r, 33 r si π 0, wheever r, rπ ta rπ cot, wheever r r N,, wheever r r N. I the ext Lemma, idetity shall be geeralized. derived o the grouds of 6, 7 Corollary 6. The Lemma is Lemma 0. Let us set for, r N: 34 Φ + r,x : C r + x, 0 + C + r π x, 35 Φ r,x : C r π x,. The we have for r N : 0, r <, r r + r cos x, r < 3, Φ + r,x r r + r cos x + r + r 6+3 r cos 3 x, 3 r < 5,

18 58 Roma Witu la, Damia S lota for r N: 8 r ` r/ r, r <, ` r r/ r +! >< Φ + r,x >: + r r ` r + + r r 4+ r r 8+4 r/ r +! r cos ` x, r < 4, r cos ` x +! r cos `4 x! We have for r N r : r / 36 Φ r,x r 0 for r N r : r/ 37 Φ r,x r 0 r cot r ta Lemma. The followig idetity hold: exp i e i, 4 r < 6. r π si r x r π si r x. + e i e i +r e i e i e i + e i +r + e i 38 ei e i rπ rπ i si e i + cos sec i ta, for + r N, ei e i + cos e i + cos + sec + i ta, for + r N.

19 Some Trigoometric Idetities Related to Corollary 7. We have: 39 +r + cos 40 si 0 sec +r ta. Corollary 8. Usig idetities 6, 7 Corollary 7 we fid: Ξ + r,x : C r + π 4 x, 4 r/ r 0 Ξ r,x : Cr r +r + sec r/ +r r Lemma. We have: 43 exp 0 + e i + e i i cos cos + sec e i i si 0 x, r π cos r x π r ta e i + e i e i r π si r x. +r e i + e i i ta, for + r N, e i + e i cos sec i ta, for + r N. Corollary 9. We have: 44 +r sec + cos

20 60 Roma Witu la, Damia S lota 45 si ta. 0 Corollary 0. Usig idetities 6, 7 Corollary 9 we obtai: Ψ + r,x : C r + x, 0 + C r + π x, r/ r lr π r + l cos cos r x + 0 l + r r r r r/ r r π +r r sec cos r x r r r r Φ r,x : Cr r/ r Lemma 3. We have: exp i 48 e i 0 π x, r ta e i r π si r x. e i e i 0, wheever r is eve r r r N, e i i csc, wheever r is odd. Corollary. We have: r r N, 49 cos r r N, 0 otherwise.

21 50 Some Trigoometric Idetities Related to wheever r is eve, si csc wheever r is odd. Corollary. By 6, 7 Corollary we get: 5 + r,x : C r + π x, 0, wheever r is odd, 5 r,x : 0 C r x, π 0 wheever r is eve, r/ r r csc r π si r x wheever r is odd. Lemma 4. Let N, r Z r. The we have: exp i e i 53 +r e i cos + e i + e i e i cos e i, wheever + r is eve, + e i cos i si cos e i cos sec i si, wheever + r is odd. Corollary 3. We have: 54 cos cos, wheever + r is eve, cos sec, wheever + r is odd, 55 si +r si.

22 6 Roma Witu la, Damia S lota Corollary 4. We have: Θ + r,x : C r + 56 r r/ 0 r/ r r 0 sec x, π r cos r π cos r x, for r N, [ csc r π r π Θ r,x : Cr r/ +r ] cos r x, for r N. 0 x, r si π si r x. Remar. All the trigoometric idetities of 3 56, o the bases of idetities 8, also may be give for S + S fuctios. Refereces [] I. S. Gradshtey, I. M. Ryzhi, Tables of Itegrals, Series, Products, Academic Press, New Yor, 980. [] A. Harmaci, Two elemetary commutativity theorems for rigs, Acta Math. Acad. Sci. Hugar , pp [3] P. S. Modeov, Problems o a Special Course of Elemetary Mathematics, Sovetsaya Naua, Moscow, 957 i Russia. [4] J. Riorda, Combiatorial Idetities, Wiley, New Yor, 968. [5] T. Rivli, Chebyshev Polyomials from Approximatio Theory to Algebra Number Theory, d ed., Wiley, New Yor, 990. [6] P. K. Sueti, Classical orthogoal polyomials, Izdat. Naua, Moscow, 976 i Russia. [7] R. Witu la, D. S lota, Decompositio of Certai Symmetric Fuctios of Powers of Cosie Sie Fuctios, It. J. Pure Appl. Math , pp. -. [8] R. Witu la, D. S lota, O Modified Chebyshev Polyomials J. Math. Aal. Appl , pp