The Equatin sin x+ βcs x= γ and a family f Hern Cyclic Quadrilaterals Knstantine Zelatr Department Of Mathematics Cllege Of Arts And Sciences Mail Stp 94 University Of Tled Tled,OH 43606-3390 U.S.A.
Intrductin On the average, intrductry texts in trignmetry may cntain five t ten printed pages f material n trignmetric equatins. Only in passing ne may cme acrss an equatin such as sin x + cs x = ; r, listed as an exercise in the exercise set. T study trignmetric equatins and trignmetry in general, in mre depth, ne must resrt t math bks n advanced trignmetry. There are nly a few f thse arund but they tent t be very gd and thrugh surces f infrmatin n the subject. Such are the tw bks listed in the references [] and []. In [3] and [4], amng ther material, trignmetric equatins (and systems f equatins) are studied in sme depth, and a variety f families f equatins are analyzed. In [3], many types f prblems f varying difficulty (bth slved and unslved) are presented. Let, β, γ be fixed real numbers and cnsider the equatin sin x + β cs x = γ () In Sectin 3, ur analysis leads t the determinatin f the slutin set S f Equatin (). In Sectin 3, via a simple cnstructin prcess, we exhibit the angle θ that generates the slutin set S f () in the case + β = γ (see Figure ) and with, β, γ > 0; the angle θ is part f an interesting quadrilateral Γ BΓ Γ. In Sectin f this paper, we generate a family f quadrilaterals Γ Γ Γ which B have three integer side lengths, the furth side length being ratinal, ne integral diagnal length, n ratinal diagnal length, and with the fur angles have ratinal tangent values and the fur vertices lying n a circle; hence the term cyclic in this paper s title. In Sectin 8, we will see that a certain subfamily f the abve family; cnsists slely f
Hern Quadrilaterals; quadrilaterals with integer side lengths and diagnals, as well as integral area. The Slutin Set f the Equatin sin x+ βcs x= γ;,, R βγ T find the slutin set S (a subset f R ), we use the well knwn half-angle frmulas, which are valid fr any angle r x (typically measured in radians r degrees) nt f the frm kπ + π; x kπ + π, k Z. Nte that as a simple calculatin shws, if the real numbers f frm kπ + π are members f the slutin set t (), then it is necessary that β + γ = 0. S, under the restrictin x kπ + π, by using the abve frmulas and after a few algebra steps we btain the equivalent equatin x x + + ( β γ) tan tan ( γ β) 0 { π π } { R, π ϕ, } =. () If β + γ = 0, then straight frm () ne sees that the reals f the frm kπ + π, k Z are slutins t (). The ther slutins are thse reals which are slutins t (). We have, Suppse β + γ=0; that is, γ = β. Then the slutin set S f () is given by: (i) S = R, if = β= γ=0. (ii) S = S, if =0 and β 0. (iii) S = S S, if 0, where U S = x x R, x= k +, k Z and S = x x x= k + k Z where ϕ is the unique angle with - π < ϕ < π and tan ϕ = β. ;
x Nw, assume β + γ 0. Then, equatin () is a quadratic equatin in tan ; and the trinmial f t t t () = ( β + γ) + ( γ β) is a quadratic plynmial functin with discriminant D given by ( ) D = 4( γ + β)( γ β) = 4( + β γ ). Clearly, when D 0, i.e., < + β < γ the trinmial () f t has tw cnjugate cmplex rts. When D = 0 + β = γ, the trinmial has the duble-real rt, r = r = r = β + γ ; and fr D > 0 (i.e., + β > γ ), the trinmial f () t has tw distinct real rts, namely We have (the reader is urged t verify), Suppse β + γ 0; then the slutin set S f () is given by: S (i) =, (empty set), if + β < γ. { } S x x R x k k Z (ii) =, = π + θ,, if + β = γ ; π π where θ is a unique angle such that - < θ < and tan θ =. β + γ U { π θ j} (iii) S = T T, if + β > γ ; where T = x x R, x= k + ; j π π j=,, and where θ j is the unique angle such that - < θ j < j+ + ( ). + β γ and tan θ j = rj; with r j =. β + γ When, βγ, are all psitive there is an bvius gemetric interpretatin;, βγ, can be the side lengths f a triangle, s the case + β < γ crrespnds t a triangle 3
with the angle Γ being btuse. The case + β = γ is the ne f a right triangle with Γ=90. In the case f + β > γ, all three angles are acute. 3 The Case + β= γ and a Gemetric Cnstructin Let, βγ, be psitive real numbers such that + β = γ. Let us lk at Figure. We start with the right triangle Γ BA with Γ B =, Γ A = β, AB = γ, BAΓ= ω, Γ BA = 90 ω, BΓ A = 90. The line segment BA lies n a straight line with divides the plane int tw half-planes. In the half-plane ppsite the pint Γ, we draw at B the ray perpendicular t BA and we chse the pint such that BΓ = Γ B =. We als extend the segment BA in the directin ppsite Γ frm the pint B and we pick the pint Γ such that AΓ = AΓ = β; thus, BΓ = β + γ. Frm the issceles triangle ΓBΓ we have BΓΓ = BΓ Γ= ϕ. In the issceles triangle, ΓAΓ it is clear AΓΓ = ω. Furthermre, ΓBΓ =Γ BA+ 90 = 90 ω+ 90 = 80 ω. Let BΓΓ = θ. Frm the right triangle B ΓΓ, it is clear that tanθ =. (), β + γ when + β = γ (the cnditin β + γ 0 is bviusly satisfied since, βγ>, 0). Nw, in the issceles triangle, ΓBΓ, the sum f its three angles must equal80 : ϕ+ (80 ω) = 80 ϕ = ω, with prves that the fur pints Γ, B, Γ Γ, lie n a circle with ΓΓ being a diameter, by virtue f Γ BΓ = 90. Therefre, since 4
these fur pints lie n a circle, we mush als have BΓΓ = BΓΓ ; that is, ϕ = θ. Altgether, ϕ = ω = θ Nte that angle ΓΓ Γ = ω + θ = ω And ΓBΓ =90 ω + 90 = 80 ω And s, ΓΓ Γ + ΓBΓ =80, cnfirming that ΓΓ Γ B is a cyclic quadrilateral. At this pint, we make an bservatin, namely ne that can als independently (frm angleϕ ) shw thatω = θ. Indeed, frm the right triangleγ BA, we have 0 ω 90 ;0 ω 45 and s 0<tan ω<. Als, tan ω= < < < <. Applying the duble- β tanω angle identity tan ω= leads t the equatin tan ω+ β tanω = 0. Using tan ω the quadratic frmula cmbined with + β = γ, leads t the tw pssible slutins γ β ( β + γ) γ β tan ω =, ;but nly is an actual slutin since 0< tanω <. Hence, γ β γ β tan ω =,but = γ + β because β γ β + γ + =.Therefre, tanω = = tan π θ (under 0< θ, ω< ) ω = θ. Next, bserve that since ΓΓ is a diameter, we must have ΓΓΓ = 90 and ΓΓΓ = ω + θ = ω. If we set 5
x =ΓΓ, y =ΓΓ, we have x=γγ.cs ω, and y= ΓΓ.sin ω and (frm the triangle ΓBA) β cs ω =,sin ω = γ γ We have, Γ φ B y φ γ. O(center) Γ β ω A θ ω ω x Γ FIGURE 6
In the quadrilateral ΓBΓ Γ in Figure, (i) The fur sidelengths are given by Γ B = BΓ =, β. + ( β + γ) ΓΓ = + ( β + γ), ΓΓ = x =. γ (ii) The tw diagnal lengths are given by BΓ = β + γ, β γ ΓΓ = y = γ. + ( + ). (iii) The tangents f the fur angles are given by, tan( ΓBΓ ) = tan(80 ω) = tan ω =, β tan( BΓΓ ) = tan(90 + ω) = ct ω = ( ) =, γ β β γ tan( ΓΓΓ ) = tan( ω+ θ) = tan ω =, β (3) β + γ tan( ΓΓ B) = tan(90 θ) = ct θ =. 4 A numerical example Let us take the perennial triple (, βγ, ) = (3,4,5). Frm (3) we btain 4 90 0 ΓΓ = 90 = 3 0, ΓΓ = x= =, Γ B = BΓ = 3, BΓ = 9, 5 5 3 90 9 0 3 3 ΓΓ = y = =,tan( ΓBΓ ) =,tan( BΓΓ ) = 3,tan ( ΓΓΓ ) =, 5 5 4 4 tan ΓΓ = 3. ( B) Frm tanθ = β γ =, with the aid f a scientific calculatr we btain + 3 ϕ = ω = θ 8.4349488 and by runding up, ϕ = ω = θ 8.435. 7
5 The case, βγ, Z +, +β= γ and a family f cyclic quadrilaterals with ratinal side lengths, ratinal diagnal lengths, and ratinal angle tangents. When, βγ, are psitive integers such that + β = γ, then the triangle Γ BA f Figure is a Pythagrean triangle and (, βγ, ) is a Pythagrean triple. Nte that in this case the real number ( β γ) + + will either be an irratinal number r it will be a + psitive integer. It is an exercise in elementary thery t shw if nc, Z then n c will be a ratinal number if and nly if, n + n c = k, fr sme k Z, s that c = k; we ffer an explanatin in Sectin 7 (see Fact ), thus; the abve square rt will be ratinal, if and nly if, it is a psitive integer; + ( β + γ ) = k ( ) + + β + γ = k, fr sme k Z. (4) On the ther hand, since (, βγ, ) is a Pythagrean triple (with γ the hyptenuse length), we must have the fllwing pssibilities:. δmn, β δ( m n ), γ δ( m n ); r alternatively = = = + (5a). δ( m n ), β δmn, γ δ( m n ) = = = + (5b) fr psitive integers δ, mn,, such that m>n,(m,n)=(i.e., m and n are relatively prime) and m+n (md ) (i.e., ne f mn, is dd, the ther even). The abve parametric frmulas describe the entire family f Pythagrean triples. Derivatin f these frmulas can be fund in mst intrductry number thery bks (r texts); fr example in [6]. 8
We will nly assume Pssibility, i.e., (5a); and cmbine it with (3) t btain a certain family f quadrilaterals ΓBΓ Γ ; see nte abut Pssibility (i.e., (5b)) at the end f this sectin. Cmbining (5a) and (4) yields, 4 δ m ( n + m ) = k (6) It is als an exercise in elementary number thery t prve that if + n abn,, Z and a is a divisr b n, then a must be a divisr f b. Again, refer t Sectin 7 f this paper, fr an explanatin (Fact ). Thus, since by (6), the integer + 4 δ m = ( δm) is a divisr f k. It fllws that k = δ ml, fr sme L Z and by (6), m + n = L k = δ ml (7) We can nw use (3), (5a), (6) and (7) t btain expressins in terms f mn,, δ and L; f the fur side lengths, the tw diagnal lengths and the fur tangent values f a special family f quadrilaterals: 9
Quadrilaterals ΓBΓ Γ Family F : (i) Sidelengths: Γ B = BΓ = δ mn, δ mm ( n) ΓΓ = δ ml, x =ΓΓ = L 4δ (ii) Diagnal lengths: BΓ = δ m, ΓΓ = y = L mn mn (iii) tan ( ΓBΓ ) = tan ω = = m n n m mn m tan ( BΓΓ ) = ctω = = n n mn tan ( ΓΓΓ ) = tan( ω+ θ) = tan ω = m n m tan ( ΓΓ B) = tan(90 θ) = ct θ =, n where δ, mnl,, are psitive integers such that m> n, ( mn, ) =, m+ n (md ) and m + n = L. mn (8) Finally, in view f ( mn, )= and the first equatin in (7), we see that ( mnl,, ) is itself a primitive Pythagrean triple which means, m= t t, n= tt, L= t + t r alternatively m= tt, n= t t, L= t + t (9a,9b) where t, t are psitive integers such that t > t,( t, t) =, t+ t (md) ; and always under the earlier assumptin m> n. Nte: If ne pursues Equatin (5b) (Pssibility ) in cmbinatin with (6), ne is led t the equatin δ.( m+ n)..( m + n ) = k Frm there, using a little bit f elementary number thery n arrives at 0
m + n = L k = δ ( m+ n) L This leads t a secnd family f quadrilaterals ΓBΓΓ, which we will nt cnsider here. We nly pint ut that in this case, the triple ( mnl,, ) will be a psitive integer slutin t the three variable Diphantine equatin X Y Z + =, whse general slutin has been well knwn in the literature. The interested reader shuld refer t [5]. 6 Anther numerical example If in (9b) we put t =, t =, then we btain L = 5 and m=4 > n=3 as required. Since the cnditin in (6) is satisfied if we take δ = 5, then we have, by (5a), = 0, β = 35, γ = 5 ; and frm (8) we btain the quadrilateral ΓBΓ Γ with the fllwing specificatins: B B x (i) Sidelengths: Γ = Γ = 0, ΓΓ = 00, = ΓΓ = 56. (ii) Diagnal lengths: BΓ = 60, y = Γ Γ = 9. ( B ) ( B ) 4 8 ( ΓΓΓ ) = ( ΓΓ B) = 4 8 (iii) tan Γ Γ =, tan ΓΓ =, 7 3 tan, tan. 7 3 Als, frm 4 3 tanθ = β γ = 3 =, and with the aid f a scientific calculatr, we find + 4 that ϕ = ω = θ = 36.86989765 ; ϕ = ω = θ = 36.87 Remark: Of the six lengths in (8), namely ΓB, BΓ, Γ Γ, x, BΓ,and y, fur are always integers as (8) clearly shws. These are the lengths ΓB, BΓ, Γ Γ, and BΓ.
On the ther hand, the lengths x and y are ratinal but nt integral unless δ is a multiple f L. This fllws frm the cnditins m+ n (md )and (m,n)=. These tw cnditins and m + n = L (see (7) r (8)), imply that the integer L must be relatively prime r cprime t the prduct mm ( n) as well as t the prduct 4nm ; the prf f this is a standard exercise in an elementary number thery curse. Therefre, accrding t (8), the ratinal number x will be an integer precisely with L is a divisr f δ. 7 Tw results frm number thery Let abn,, be psitive integers. n n Fact : If a is a divisr f b, then a is a divisr f b. th Fact : The integer a is the n pwer f a ratinal number if and nly if it is the pwer f an integer. These tw results can be typically fund in number thery bks. We cite tw surces. First, W. Sierpinski s vluminus bk Elementary Thery f Numbers (see reference [7] fr details); and Kenneth H. Rsen s number thery text Elementary Number Thery and Its Applicatins, (see [6] fr details). 8 A family f Hern Cyclic Quadrilaterals th n If we cmpute the areas f the triangles B ΓΓ, BΓ A and ΓAΓ by using frmulas (8) we find,
β δmnδ ( m n ) Area f (right) triangle Γ = = = δ. ( ) B A mn m n β.sin(80 ω) β sin ω β δ ( m n ) mn Area f (issceles) triangle ΓAΓ = = = γ= m + n β ( + γ) δmn δ( m + n ) + δ( Area f (right) triangle BΓΓ = = m n ) = 3 δ. nm. (0) BΓΓΓ The sum f the three areas in (0) is equal t the area A f the quadrilateral ( m n ) A = δ. mn. m n + + m m + n () In the winter 005 issue f Mathematics and Cmputer Educatin (see [8]), K.R.S. Sastry presented a family f Hern quadrilaterals. These are quadrilaterals with integer sides, integer diagnals, and integer area. Interestingly, a subfamily f the family f quadrilaterals described in (8); cnsists exclusively f hern quadrilaterals. First nte that, fr any chice f the psitive integer δ, the lengths ΓB, BΓ, Γ Γ, BΓ are always integral, while ΓΓ and ΓΓ, just ratinal. Clearly, if we take δ such that δ 0 (md L),then Γ Γ, ΓΓ will be integers as well; and as () easily shws, the area A will als be an integer, since by (7) L = m + n and s δ 0 (md L) => δ 0 (md L ) => δ 0 (md( m + n )). Cnclusin: When δ is a psitive integer multiple f L in (8); the quadrilaterals btained in (8) are Hern nes. The smallest such chice fr δ is δ =L. Frm (8) we then btain Γ B = BΓ = Lmn, Γ Γ = ml., x= Γ Γ = m( m n ), BΓ = Lm, Γ Γ = y = 4nm () 3
( m n ) And area A=L. mn m n + + m ; and since L = m + n we arive at, m + n ( ) A= mn m + n m n + m n + m m + n ( ). ( ).( ) A= mn m n + m mn + n + m + mn = 4nm 4 4 4 4 4 5 (3) Using (),(3), and (9a,9b) we arrive at the fllwing table(with tt being the smallest pssible; ne chice with even dd; the ther with dd, t even). t t t t t m n δ = L BΓ ΓΓ ΓΓ Γ B BΓ ΓΓ Area A 4 3 5 0 56 00 0 60 9 888 3 5 3 560 856 4056 560 3744 880 4. 5.5 δ mn n Als nte that frm tan θ = = = ; β + γ δ( m n ) + δ( m + n ) m 3 we btain (when n=3, m=4) tan θ= ; θ = ω = ϕ 36.86989765 ; 4 5 and (when n=5, m=) tan θ= ; θ = ω = ϕ.6986495. 9 References [] C.V. Durell and A. Rbsn, Advanced Trignmetry, 35 pp., Dver Publicatins, (003), ISBN: 04864397. [] Kenneth S. Miller, Advanced Trignmetry, Krieger Publishing C., (977), ISBN: 08875396. [3] Knstantine D. Zelatr, A Trignmetric Primer: Frm Elementary t Advanced Trignmetry, published by Brainstrm Fantasian Inc., January 005, ISBN: 0-97680--0. P.O. Bx 480, Pittsburg, PA 503, U.S.A. [4] Marins Zevas, Trignmetry (transliterated frm Greek) Gutenberg Press, Athens, Greece, (973), n ISBN. 4
[5] L.E. Dicksn, Histry f the Thery f Numbers, Vl. II, pages 435-437 (Als pages 46 and 47 fr the mre general equatin ***), AMS Chelsea Publishing, ISBN: 0-88-935-6; 99. [6] Kenneth H. Rsen, Elementary Number Thery and Its Applicatins, third editin, 993, Addisn-Wesley Publishing C., ISBN: 0-0-57889-; (there is nw a furth editin, as well). [7] W. Sierpinski, Elementary Thery f Numbers, Warsaw, 964. Fact can be fund n page 5, listed as Crllary t Therem 6a; Fact can be fund n page 6, listed as Therem 7. Fr a better understanding the reader may als want t study the preceding material, Therem t Therem 6 (pages 0-5). Als, there is a newer editin (988), by Elsevier Publishing, and distributed by Nrth-Hlland, Nrth- Hlland Mathematical Library, 3, Amsterdam (988). This bk is nw nly printed upn demand, but it is available in varius libraries. [8] K.R.S. Sastry, A descriptin f a family f Hern Quadrilaterals, Mathematics and Cmputer Educatin, Winter, 005, pp. 7-77. 5