The Equation αsin x+ βcos family of Heron Cyclic Quadrilaterals
|
|
- Damon Griffith
- 5 years ago
- Views:
Transcription
1 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 U.S.A.
2 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
3 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. ;
4 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
5 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
6 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
7 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
8 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 ΓΓ = 90 = 3 0, ΓΓ = x= =, Γ B = BΓ = 3, BΓ = 9, ΓΓ = y = =,tan( ΓBΓ ) =,tan( BΓΓ ) = 3,tan ( ΓΓΓ ) =, tan ΓΓ = 3. ( B) Frm tanθ = β γ =, with the aid f a scientific calculatr we btain + 3 ϕ = ω = θ and by runding up, ϕ = ω = θ
9 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
10 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
11 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
12 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 = ; ϕ = ω = θ = 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.
13 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,
14 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
15 ( 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 (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 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 ; 4 5 and (when n=5, m=) tan θ= ; θ = ω = ϕ References [] C.V. Durell and A. Rbsn, Advanced Trignmetry, 35 pp., Dver Publicatins, (003), ISBN: [] Kenneth S. Miller, Advanced Trignmetry, Krieger Publishing C., (977), ISBN: [3] Knstantine D. Zelatr, A Trignmetric Primer: Frm Elementary t Advanced Trignmetry, published by Brainstrm Fantasian Inc., January 005, ISBN: 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
16 [5] L.E. Dicksn, Histry f the Thery f Numbers, Vl. II, pages (Als pages 46 and 47 fr the mre general equatin ***), AMS Chelsea Publishing, ISBN: ; 99. [6] Kenneth H. Rsen, Elementary Number Thery and Its Applicatins, third editin, 993, Addisn-Wesley Publishing C., ISBN: ; (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
A little noticed right triangle
A little nticed right triangle Knstantine Hermes Zelatr Department f athematics Cllege f Arts and Sciences ail Stp 94 University f Tled Tled, OH 43606-3390 U.S.A. A little nticed right triangle. Intrductin
More informationLEARNING : At the end of the lesson, students should be able to: OUTCOMES a) state trigonometric ratios of sin,cos, tan, cosec, sec and cot
Mathematics DM 05 Tpic : Trignmetric Functins LECTURE OF 5 TOPIC :.0 TRIGONOMETRIC FUNCTIONS SUBTOPIC :. Trignmetric Ratis and Identities LEARNING : At the end f the lessn, students shuld be able t: OUTCOMES
More informationMODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b
. REVIEW OF SOME BASIC ALGEBRA MODULE () Slving Equatins Yu shuld be able t slve fr x: a + b = c a d + e x + c and get x = e(ba +) b(c a) d(ba +) c Cmmn mistakes and strategies:. a b + c a b + a c, but
More informationNUMBERS, MATHEMATICS AND EQUATIONS
AUSTRALIAN CURRICULUM PHYSICS GETTING STARTED WITH PHYSICS NUMBERS, MATHEMATICS AND EQUATIONS An integral part t the understanding f ur physical wrld is the use f mathematical mdels which can be used t
More informationTrigonometric Ratios Unit 5 Tentative TEST date
1 U n i t 5 11U Date: Name: Trignmetric Ratis Unit 5 Tentative TEST date Big idea/learning Gals In this unit yu will extend yur knwledge f SOH CAH TOA t wrk with btuse and reflex angles. This extensin
More informationLHS Mathematics Department Honors Pre-Calculus Final Exam 2002 Answers
LHS Mathematics Department Hnrs Pre-alculus Final Eam nswers Part Shrt Prblems The table at the right gives the ppulatin f Massachusetts ver the past several decades Using an epnential mdel, predict the
More informationMATHEMATICS SYLLABUS SECONDARY 5th YEAR
Eurpean Schls Office f the Secretary-General Pedaggical Develpment Unit Ref. : 011-01-D-8-en- Orig. : EN MATHEMATICS SYLLABUS SECONDARY 5th YEAR 6 perid/week curse APPROVED BY THE JOINT TEACHING COMMITTEE
More information1 PreCalculus AP Unit G Rotational Trig (MCR) Name:
1 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: Big idea In this unit yu will extend yur knwledge f SOH CAH TOA t wrk with btuse and reflex angles. This extensin will invlve the unit circle which will
More informationA Correlation of. to the. South Carolina Academic Standards for Mathematics Precalculus
A Crrelatin f Suth Carlina Academic Standards fr Mathematics Precalculus INTRODUCTION This dcument demnstrates hw Precalculus (Blitzer), 4 th Editin 010, meets the indicatrs f the. Crrelatin page references
More informationFunction notation & composite functions Factoring Dividing polynomials Remainder theorem & factor property
Functin ntatin & cmpsite functins Factring Dividing plynmials Remainder therem & factr prperty Can d s by gruping r by: Always lk fr a cmmn factr first 2 numbers that ADD t give yu middle term and MULTIPLY
More informationCalculus Placement Review. x x. =. Find each of the following. 9 = 4 ( )
Calculus Placement Review I. Finding dmain, intercepts, and asympttes f ratinal functins 9 Eample Cnsider the functin f ( ). Find each f the fllwing. (a) What is the dmain f f ( )? Write yur answer in
More informationPre-Calculus Individual Test 2017 February Regional
The abbreviatin NOTA means Nne f the Abve answers and shuld be chsen if chices A, B, C and D are nt crrect. N calculatr is allwed n this test. Arcfunctins (such as y = Arcsin( ) ) have traditinal restricted
More informationMath Foundations 10 Work Plan
Math Fundatins 10 Wrk Plan Units / Tpics 10.1 Demnstrate understanding f factrs f whle numbers by: Prime factrs Greatest Cmmn Factrs (GCF) Least Cmmn Multiple (LCM) Principal square rt Cube rt Time Frame
More informationCop yri ht 2006, Barr Mabillard.
Trignmetry II Cpyright Trignmetry II Standards 006, Test Barry ANSWERS Mabillard. 0 www.math0s.cm . If csα, where sinα > 0, and 5 cs α + β value f sin β, where tan β > 0, determine the exact 9 First determine
More informationSections 15.1 to 15.12, 16.1 and 16.2 of the textbook (Robbins-Miller) cover the materials required for this topic.
Tpic : AC Fundamentals, Sinusidal Wavefrm, and Phasrs Sectins 5. t 5., 6. and 6. f the textbk (Rbbins-Miller) cver the materials required fr this tpic.. Wavefrms in electrical systems are current r vltage
More informationDifferentiation Applications 1: Related Rates
Differentiatin Applicatins 1: Related Rates 151 Differentiatin Applicatins 1: Related Rates Mdel 1: Sliding Ladder 10 ladder y 10 ladder 10 ladder A 10 ft ladder is leaning against a wall when the bttm
More informationA solution of certain Diophantine problems
A slutin f certain Diphantine prblems Authr L. Euler* E7 Nvi Cmmentarii academiae scientiarum Petrplitanae 0, 1776, pp. 8-58 Opera Omnia: Series 1, Vlume 3, pp. 05-17 Reprinted in Cmmentat. arithm. 1,
More informationAlgebra2/Trig: Trig Unit 2 Packet
Algebra2/Trig: Trig Unit 2 Packet In this unit, students will be able t: Learn and apply c-functin relatinships between trig functins Learn and apply the sum and difference identities Learn and apply the
More informationBuilding to Transformations on Coordinate Axis Grade 5: Geometry Graph points on the coordinate plane to solve real-world and mathematical problems.
Building t Transfrmatins n Crdinate Axis Grade 5: Gemetry Graph pints n the crdinate plane t slve real-wrld and mathematical prblems. 5.G.1. Use a pair f perpendicular number lines, called axes, t define
More informationHomology groups of disks with holes
Hmlgy grups f disks with hles THEOREM. Let p 1,, p k } be a sequence f distinct pints in the interir unit disk D n where n 2, and suppse that fr all j the sets E j Int D n are clsed, pairwise disjint subdisks.
More informationTrigonometry, 8th ed; Lial, Hornsby, Schneider
Trignmetry, 8th ed; Lial, Hrnsby, Schneider Trignmetry Final Exam Review: Chapters 7, 8, 9 Nte: A prtin f Exam will cver Chapters 1 6, s be sure yu rewrk prblems frm the first and secnd exams and frm the
More informationChapter 3 Kinematics in Two Dimensions; Vectors
Chapter 3 Kinematics in Tw Dimensins; Vectrs Vectrs and Scalars Additin f Vectrs Graphical Methds (One and Tw- Dimensin) Multiplicatin f a Vectr b a Scalar Subtractin f Vectrs Graphical Methds Adding Vectrs
More informationRevision: August 19, E Main Suite D Pullman, WA (509) Voice and Fax
.7.4: Direct frequency dmain circuit analysis Revisin: August 9, 00 5 E Main Suite D Pullman, WA 9963 (509) 334 6306 ice and Fax Overview n chapter.7., we determined the steadystate respnse f electrical
More informationSOLUTIONS SET 1 MATHEMATICS CLASS X
Tp Careers & Yu SOLUTIONS SET MTHEMTICS CLSS X. 84 7 Prime factrs f 84 are, and 7.. Sum f zeres 5 + 4 Prduct f zeres 5 4 0 Required plynmial x ( )x + ( 0) x + x 0. Given equatin is x + y 0 Fr x, y L.H.S
More informationEASTERN ARIZONA COLLEGE Precalculus Trigonometry
EASTERN ARIZONA COLLEGE Precalculus Trignmetry Curse Design 2017-2018 Curse Infrmatin Divisin Mathematics Curse Number MAT 181 Title Precalculus Trignmetry Credits 3 Develped by Gary Rth Lecture/Lab Rati
More informationKansas City Area Teachers of Mathematics 2011 KCATM Math Competition GEOMETRY GRADES 7-8
Kansas City Area Teachers f Mathematics 2011 KCATM Math Cmpetitin GEOMETRY GRADES 7-8 INSTRUCTIONS D nt pen this bklet until instructed t d s. Time limit: 20 minutes Yu may use calculatrs. Mark yur answer
More informationCHAPTER 3 INEQUALITIES. Copyright -The Institute of Chartered Accountants of India
CHAPTER 3 INEQUALITIES Cpyright -The Institute f Chartered Accuntants f India INEQUALITIES LEARNING OBJECTIVES One f the widely used decisin making prblems, nwadays, is t decide n the ptimal mix f scarce
More informationCHAPTER 24: INFERENCE IN REGRESSION. Chapter 24: Make inferences about the population from which the sample data came.
MATH 1342 Ch. 24 April 25 and 27, 2013 Page 1 f 5 CHAPTER 24: INFERENCE IN REGRESSION Chapters 4 and 5: Relatinships between tw quantitative variables. Be able t Make a graph (scatterplt) Summarize the
More informationConcept Category 2. Trigonometry & The Unit Circle
Cncept Categry 2 Trignmetry & The Unit Circle Skill Checklist Use special right triangles t express values f fr the six trig functins Evaluate sine csine and tangent using the unit circle Slve tw-step
More informationAdmissibility Conditions and Asymptotic Behavior of Strongly Regular Graphs
Admissibility Cnditins and Asympttic Behavir f Strngly Regular Graphs VASCO MOÇO MANO Department f Mathematics University f Prt Oprt PORTUGAL vascmcman@gmailcm LUÍS ANTÓNIO DE ALMEIDA VIEIRA Department
More informationChapter Summary. Mathematical Induction Strong Induction Recursive Definitions Structural Induction Recursive Algorithms
Chapter 5 1 Chapter Summary Mathematical Inductin Strng Inductin Recursive Definitins Structural Inductin Recursive Algrithms Sectin 5.1 3 Sectin Summary Mathematical Inductin Examples f Prf by Mathematical
More informationANSWER KEY FOR MATH 10 SAMPLE EXAMINATION. Instructions: If asked to label the axes please use real world (contextual) labels
ANSWER KEY FOR MATH 10 SAMPLE EXAMINATION Instructins: If asked t label the axes please use real wrld (cntextual) labels Multiple Chice Answers: 0 questins x 1.5 = 30 Pints ttal Questin Answer Number 1
More informationMOCK CBSE BOARD EXAM MATHEMATICS. CLASS X (Paper 1) (AS PER THE GUIDELINES OF CBSE)
MOCK CSE ORD EXM MTHEMTICS CLSS X (Paper 1) (S PER THE GUIDELINES OF CSE) Time: 3 Hurs Max. Marks: 80 General Instructins 1. ll the questins are cmpulsry. 2. The questin paper cnsists f 30 questins divided
More informationThis section is primarily focused on tools to aid us in finding roots/zeros/ -intercepts of polynomials. Essentially, our focus turns to solving.
Sectin 3.2: Many f yu WILL need t watch the crrespnding vides fr this sectin n MyOpenMath! This sectin is primarily fcused n tls t aid us in finding rts/zers/ -intercepts f plynmials. Essentially, ur fcus
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 4: Mdel checing fr ODE mdels In Petre Department f IT, Åb Aademi http://www.users.ab.fi/ipetre/cmpmd/ Cntent Stichimetric matrix Calculating the mass cnservatin relatins
More informationInterference is when two (or more) sets of waves meet and combine to produce a new pattern.
Interference Interference is when tw (r mre) sets f waves meet and cmbine t prduce a new pattern. This pattern can vary depending n the riginal wave directin, wavelength, amplitude, etc. The tw mst extreme
More informationAn Introduction to Complex Numbers - A Complex Solution to a Simple Problem ( If i didn t exist, it would be necessary invent me.
An Intrductin t Cmple Numbers - A Cmple Slutin t a Simple Prblem ( If i didn t eist, it wuld be necessary invent me. ) Our Prblem. The rules fr multiplying real numbers tell us that the prduct f tw negative
More informationThermodynamics Partial Outline of Topics
Thermdynamics Partial Outline f Tpics I. The secnd law f thermdynamics addresses the issue f spntaneity and invlves a functin called entrpy (S): If a prcess is spntaneus, then Suniverse > 0 (2 nd Law!)
More informationDINGWALL ACADEMY NATIONAL QUALIFICATIONS. Mathematics Higher Prelim Examination 2010/2011 Paper 1 Assessing Units 1 & 2.
INGWLL EMY Mathematics Higher Prelim Eaminatin 00/0 Paper ssessing Units & NTIONL QULIFITIONS Time allwed - hur 0 minutes Read carefull alculatrs ma NOT be used in this paper. Sectin - Questins - 0 (0
More informationENGI 4430 Parametric Vector Functions Page 2-01
ENGI 4430 Parametric Vectr Functins Page -01. Parametric Vectr Functins (cntinued) Any nn-zer vectr r can be decmpsed int its magnitude r and its directin: r rrˆ, where r r 0 Tangent Vectr: dx dy dz dr
More informationHigher. Specimen NAB Assessment
hsn.uk.net Higher Mathematics UNIT Specimen NAB Assessment HSN50 This dcument was prduced speciall fr the HSN.uk.net website, and we require that an cpies r derivative wrks attribute the wrk t Higher Still
More informationCHM112 Lab Graphing with Excel Grading Rubric
Name CHM112 Lab Graphing with Excel Grading Rubric Criteria Pints pssible Pints earned Graphs crrectly pltted and adhere t all guidelines (including descriptive title, prperly frmatted axes, trendline
More informationExperiment #3. Graphing with Excel
Experiment #3. Graphing with Excel Study the "Graphing with Excel" instructins that have been prvided. Additinal help with learning t use Excel can be fund n several web sites, including http://www.ncsu.edu/labwrite/res/gt/gt-
More information4. Find a, b, and c. 6. Find x and y.
Grace Brethren Christian Schl Entering Trig/Analysis: Page f Summer Packet fr Students entering Trig/Analysis Review prblems frm Gemetry: Shw yur wrk!. Twice the cmplement f angle A is 35 less than the
More informationNWACC Dept of Mathematics Dept Final Exam Review for Trig - Part 3 Trigonometry, 9th Edition; Lial, Hornsby, Schneider Fall 2008
NWACC Dept f Mathematics Dept Final Exam Review fr Trig - Part Trignmetry, 9th Editin; Lial, Hrnsby, Schneider Fall 008 Departmental Objectives: Departmental Final Exam Review fr Trignmetry Part : Chapters
More information[COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t o m a k e s u r e y o u a r e r e a d y )
(Abut the final) [COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t m a k e s u r e y u a r e r e a d y ) The department writes the final exam s I dn't really knw what's n it and I can't very well
More informationPreparation work for A2 Mathematics [2018]
Preparatin wrk fr A Mathematics [018] The wrk studied in Y1 will frm the fundatins n which will build upn in Year 13. It will nly be reviewed during Year 13, it will nt be retaught. This is t allw time
More informationCurriculum Development Overview Unit Planning for 8 th Grade Mathematics MA10-GR.8-S.1-GLE.1 MA10-GR.8-S.4-GLE.2
Unit Title It s All Greek t Me Length f Unit 5 weeks Fcusing Lens(es) Cnnectins Standards and Grade Level Expectatins Addressed in this Unit MA10-GR.8-S.1-GLE.1 MA10-GR.8-S.4-GLE.2 Inquiry Questins (Engaging-
More informationIntroduction to Smith Charts
Intrductin t Smith Charts Dr. Russell P. Jedlicka Klipsch Schl f Electrical and Cmputer Engineering New Mexic State University as Cruces, NM 88003 September 2002 EE521 ecture 3 08/22/02 Smith Chart Summary
More information39th International Physics Olympiad - Hanoi - Vietnam Theoretical Problem No. 1 /Solution. Solution
39th Internatinal Physics Olympiad - Hani - Vietnam - 8 Theretical Prblem N. /Slutin Slutin. The structure f the mrtar.. Calculating the distance TG The vlume f water in the bucket is V = = 3 3 3 cm m.
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Prcessing Prf. Mark Fwler Intrductin Nte Set #1 ading Assignment: Ch. 1 f Prakis & Manlakis 1/13 Mdern systems generally DSP Scenari get a cntinuus-time signal frm a sensr a cnt.-time
More informationCambridge Assessment International Education Cambridge Ordinary Level. Published
Cambridge Assessment Internatinal Educatin Cambridge Ordinary Level ADDITIONAL MATHEMATICS 4037/1 Paper 1 Octber/Nvember 017 MARK SCHEME Maximum Mark: 80 Published This mark scheme is published as an aid
More informationTrigonometric Functions. Concept Category 3
Trignmetric Functins Cncept Categry 3 Gals 6 basic trig functins (gemetry) Special triangles Inverse trig functins (t find the angles) Unit Circle: Trig identities a b c 2 2 2 The Six Basic Trig functins
More informationPreparation work for A2 Mathematics [2017]
Preparatin wrk fr A2 Mathematics [2017] The wrk studied in Y12 after the return frm study leave is frm the Cre 3 mdule f the A2 Mathematics curse. This wrk will nly be reviewed during Year 13, it will
More informationMath Final Exam Instructor: Ken Schultz April 28, 2006
Name Math 1650.300 Final Exam Instructr: Ken Schultz April 8, 006 Exam Guidelines D nt pen this exam until are instructed t begin. Fllw all instructins explicitly. Befre yu begin wrking, make sure yur
More information7 TH GRADE MATH STANDARDS
ALGEBRA STANDARDS Gal 1: Students will use the language f algebra t explre, describe, represent, and analyze number expressins and relatins 7 TH GRADE MATH STANDARDS 7.M.1.1: (Cmprehensin) Select, use,
More informationUNIT 1 COPLANAR AND NON-COPLANAR FORCES
UNIT 1 COPLANA AND NON-COPLANA FOCES Cplanar and Nn-Cplanar Frces Structure 1.1 Intrductin Objectives 1. System f Frces 1.3 Cplanar Frce 1.3.1 Law f Parallelgram f Frces 1.3. Law f Plygn f Frces 1.3.3
More informationB. Definition of an exponential
Expnents and Lgarithms Chapter IV - Expnents and Lgarithms A. Intrductin Starting with additin and defining the ntatins fr subtractin, multiplicatin and divisin, we discvered negative numbers and fractins.
More informationRevisiting the Socrates Example
Sectin 1.6 Sectin Summary Valid Arguments Inference Rules fr Prpsitinal Lgic Using Rules f Inference t Build Arguments Rules f Inference fr Quantified Statements Building Arguments fr Quantified Statements
More information1996 Engineering Systems Design and Analysis Conference, Montpellier, France, July 1-4, 1996, Vol. 7, pp
THE POWER AND LIMIT OF NEURAL NETWORKS T. Y. Lin Department f Mathematics and Cmputer Science San Jse State University San Jse, Califrnia 959-003 tylin@cs.ssu.edu and Bereley Initiative in Sft Cmputing*
More informationDepartment of Economics, University of California, Davis Ecn 200C Micro Theory Professor Giacomo Bonanno. Insurance Markets
Department f Ecnmics, University f alifrnia, Davis Ecn 200 Micr Thery Prfessr Giacm Bnann Insurance Markets nsider an individual wh has an initial wealth f. ith sme prbability p he faces a lss f x (0
More informationCompetency Statements for Wm. E. Hay Mathematics for grades 7 through 12:
Cmpetency Statements fr Wm. E. Hay Mathematics fr grades 7 thrugh 12: Upn cmpletin f grade 12 a student will have develped a cmbinatin f sme/all f the fllwing cmpetencies depending upn the stream f math
More informationModule 4: General Formulation of Electric Circuit Theory
Mdule 4: General Frmulatin f Electric Circuit Thery 4. General Frmulatin f Electric Circuit Thery All electrmagnetic phenmena are described at a fundamental level by Maxwell's equatins and the assciated
More informationINSTRUCTIONAL PLAN Day 2
INSTRUCTIONAL PLAN Day 2 Subject: Trignmetry Tpic: Other Trignmetric Ratis, Relatinships between Trignmetric Ratis, and Inverses Target Learners: Cllege Students Objectives: At the end f the lessn, students
More informationChapter 2 GAUSS LAW Recommended Problems:
Chapter GAUSS LAW Recmmended Prblems: 1,4,5,6,7,9,11,13,15,18,19,1,7,9,31,35,37,39,41,43,45,47,49,51,55,57,61,6,69. LCTRIC FLUX lectric flux is a measure f the number f electric filed lines penetrating
More informationPhysics 2010 Motion with Constant Acceleration Experiment 1
. Physics 00 Mtin with Cnstant Acceleratin Experiment In this lab, we will study the mtin f a glider as it accelerates dwnhill n a tilted air track. The glider is supprted ver the air track by a cushin
More informationThe Electromagnetic Form of the Dirac Electron Theory
0 The Electrmagnetic Frm f the Dirac Electrn Thery Aleander G. Kyriaks Saint-Petersburg State Institute f Technlgy, St. Petersburg, Russia* In the present paper it is shwn that the Dirac electrn thery
More information(for students at grades 7 and 8, Gymnasium)
Kanguru Sans Frntières Kanguru Maths 009 Level: 7-8 (fr students at grades 7 and 8, Gymnasium) pints questins: ) Amng these numbers, which ne is even? 009 9 Β) 008 + 009 C) 000 9 D) 000 9 Ε) 000 + 9 )
More informationWYSE Academic Challenge Regional Mathematics 2007 Solution Set
WYSE Academic Challenge Reginal Mathematics 007 Slutin Set 1. Crrect answer: C. ( ) ( ) 1 + y y = ( + ) + ( y y + 1 ) = + 1 1 ( ) ( 1 + y ) = s *1/ = 1. Crrect answer: A. The determinant is ( 1 ( 1) )
More informationCS 477/677 Analysis of Algorithms Fall 2007 Dr. George Bebis Course Project Due Date: 11/29/2007
CS 477/677 Analysis f Algrithms Fall 2007 Dr. Gerge Bebis Curse Prject Due Date: 11/29/2007 Part1: Cmparisn f Srting Algrithms (70% f the prject grade) The bjective f the first part f the assignment is
More informationIntroduction to Spacetime Geometry
Intrductin t Spacetime Gemetry Let s start with a review f a basic feature f Euclidean gemetry, the Pythagrean therem. In a twdimensinal crdinate system we can relate the length f a line segment t the
More informationFlipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System
Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed
More informationA Few Basic Facts About Isothermal Mass Transfer in a Binary Mixture
Few asic Facts but Isthermal Mass Transfer in a inary Miture David Keffer Department f Chemical Engineering University f Tennessee first begun: pril 22, 2004 last updated: January 13, 2006 dkeffer@utk.edu
More informationPhysics 101 Math Review. Solutions
Physics 0 Math eview Slutins . The fllwing are rdinary physics prblems. Place the answer in scientific ntatin when apprpriate and simplify the units (Scientific ntatin is used when it takes less time t
More informationPhysics 2B Chapter 23 Notes - Faraday s Law & Inductors Spring 2018
Michael Faraday lived in the Lndn area frm 1791 t 1867. He was 29 years ld when Hand Oersted, in 1820, accidentally discvered that electric current creates magnetic field. Thrugh empirical bservatin and
More informationCHAPTER 2 Algebraic Expressions and Fundamental Operations
CHAPTER Algebraic Expressins and Fundamental Operatins OBJECTIVES: 1. Algebraic Expressins. Terms. Degree. Gruping 5. Additin 6. Subtractin 7. Multiplicatin 8. Divisin Algebraic Expressin An algebraic
More informationReview for the final exam (Math 127)
. Evaluate 3 tan tan 4 3 (b) (c) cs cs 4 7 3 sec cs 4 4 (d) cs tan 3 Review fr the final eam (Math 7). If sec, and 7 36, find cs, sin, tan, ct, csc tan (b) If, evaluate cs, sin 7 36 (c) Write the csc in
More information20 Faraday s Law and Maxwell s Extension to Ampere s Law
Chapter 20 Faraday s Law and Maxwell s Extensin t Ampere s Law 20 Faraday s Law and Maxwell s Extensin t Ampere s Law Cnsider the case f a charged particle that is ming in the icinity f a ming bar magnet
More informationx x
Mdeling the Dynamics f Life: Calculus and Prbability fr Life Scientists Frederick R. Adler cfrederick R. Adler, Department f Mathematics and Department f Bilgy, University f Utah, Salt Lake City, Utah
More informationSupport-Vector Machines
Supprt-Vectr Machines Intrductin Supprt vectr machine is a linear machine with sme very nice prperties. Haykin chapter 6. See Alpaydin chapter 13 fr similar cntent. Nte: Part f this lecture drew material
More informationMath 0310 Final Exam Review Problems
Math 0310 Final Exam Review Prblems Slve the fllwing equatins. 1. 4dd + 2 = 6 2. 2 3 h 5 = 7 3. 2 + (18 xx) + 2(xx 1) = 4(xx + 2) 8 4. 1 4 yy 3 4 = 1 2 yy + 1 5. 5.74aa + 9.28 = 2.24aa 5.42 Slve the fllwing
More informationChapter 9 Vector Differential Calculus, Grad, Div, Curl
Chapter 9 Vectr Differential Calculus, Grad, Div, Curl 9.1 Vectrs in 2-Space and 3-Space 9.2 Inner Prduct (Dt Prduct) 9.3 Vectr Prduct (Crss Prduct, Outer Prduct) 9.4 Vectr and Scalar Functins and Fields
More informationENG2410 Digital Design Sequential Circuits: Part B
ENG24 Digital Design Sequential Circuits: Part B Fall 27 S. Areibi Schl f Engineering University f Guelph Analysis f Sequential Circuits Earlier we learned hw t analyze cmbinatinal circuits We will extend
More informationFall 2013 Physics 172 Recitation 3 Momentum and Springs
Fall 03 Physics 7 Recitatin 3 Mmentum and Springs Purpse: The purpse f this recitatin is t give yu experience wrking with mmentum and the mmentum update frmula. Readings: Chapter.3-.5 Learning Objectives:.3.
More informationFundamental Concepts in Structural Plasticity
Lecture Fundamental Cncepts in Structural Plasticit Prblem -: Stress ield cnditin Cnsider the plane stress ield cnditin in the principal crdinate sstem, a) Calculate the maximum difference between the
More informationOn small defining sets for some SBIBD(4t - 1, 2t - 1, t - 1)
University f Wllngng Research Online Faculty f Infrmatics - Papers (Archive) Faculty f Engineering and Infrmatin Sciences 992 On small defining sets fr sme SBIBD(4t -, 2t -, t - ) Jennifer Seberry University
More informationPHYS 314 HOMEWORK #3
PHYS 34 HOMEWORK #3 Due : 8 Feb. 07. A unifrm chain f mass M, lenth L and density λ (measured in k/m) hans s that its bttm link is just tuchin a scale. The chain is drpped frm rest nt the scale. What des
More informationSolving Inequalities: Multiplying or Dividing by a Negative Number
11 Slving Inequalities: Multiplying r Dividing by a Negative Number We slve inequalities the same way we slve equatins, with ne exceptin. When we divide r multiply bth sides f an inequality by a negative
More informationREPRESENTATIONS OF sp(2n; C ) SVATOPLUK KR YSL. Abstract. In this paper we have shown how a tensor product of an innite dimensional
ON A DISTINGUISHED CLASS OF INFINITE DIMENSIONAL REPRESENTATIONS OF sp(2n; C ) SVATOPLUK KR YSL Abstract. In this paper we have shwn hw a tensr prduct f an innite dimensinal representatin within a certain
More informationPhysical Layer: Outline
18-: Intrductin t Telecmmunicatin Netwrks Lectures : Physical Layer Peter Steenkiste Spring 01 www.cs.cmu.edu/~prs/nets-ece Physical Layer: Outline Digital Representatin f Infrmatin Characterizatin f Cmmunicatin
More informationCode: MATH 151 Title: INTERMEDIATE ALGEBRA
Cde: MATH 151 Title: INTERMEDIATE ALGEBRA Divisin: MATHEMATICS Department: MATHEMATICS Curse Descriptin: This curse prepares students fr curses that require algebraic skills beynd thse taught in Elementary
More informationReview Problems 3. Four FIR Filter Types
Review Prblems 3 Fur FIR Filter Types Fur types f FIR linear phase digital filters have cefficients h(n fr 0 n M. They are defined as fllws: Type I: h(n = h(m-n and M even. Type II: h(n = h(m-n and M dd.
More information1 The limitations of Hartree Fock approximation
Chapter: Pst-Hartree Fck Methds - I The limitatins f Hartree Fck apprximatin The n electrn single determinant Hartree Fck wave functin is the variatinal best amng all pssible n electrn single determinants
More informationAP Physics. Summer Assignment 2012 Date. Name. F m = = + What is due the first day of school? a. T. b. = ( )( ) =
P Physics Name Summer ssignment 0 Date I. The P curriculum is extensive!! This means we have t wrk at a fast pace. This summer hmewrk will allw us t start n new Physics subject matter immediately when
More informationPAST EXAM PAPER & MEMO N3 ABOUT THE QUESTION PAPERS:
EKURHULENI TECH COLLEGE. N. 3 Mgale Square, Krugersdrp. Website: www. ekurhulenitech.c.za Email: inf@ekurhulenitech.c.za TEL: 0 040 7343 CELL: 073 770 308/060 75 459 PAST EXAM PAPER & MEMO N3 ABOUT THE
More informationCorrections for the textbook answers: Sec 6.1 #8h)covert angle to a positive by adding period #9b) # rad/sec
U n i t 6 AdvF Date: Name: Trignmetric Functins Unit 6 Tentative TEST date Big idea/learning Gals In this unit yu will study trignmetric functins frm grade, hwever everything will be dne in radian measure.
More informationGEOMETRY TEAM February 2018 Florida Invitational. and convex pentagon KLMNP. The following angle
February 0 Flrida Invitatinal Cnsider cnvex hexagn EFGHIJ measures, in degrees are interir angles t these plygns: me 00, mf 0, mg x 60, mh x 0, mi 0 mk 0, ml 00, mm 90, mn y 90, m y and cnvex pentagn KLMN
More informationLecture 6: Phase Space and Damped Oscillations
Lecture 6: Phase Space and Damped Oscillatins Oscillatins in Multiple Dimensins The preius discussin was fine fr scillatin in a single dimensin In general, thugh, we want t deal with the situatin where:
More informationECE 2100 Circuit Analysis
ECE 2100 Circuit Analysis Lessn 25 Chapter 9 & App B: Passive circuit elements in the phasr representatin Daniel M. Litynski, Ph.D. http://hmepages.wmich.edu/~dlitynsk/ ECE 2100 Circuit Analysis Lessn
More information