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AUTHOR Brant, Vincent TITLE Trigonometry, A Tentative Guide Prepared for Use with the Text Plane Trigonometry with Tables. INSTITUTION Baltimore County Public Schools, Towson, Md. PUB DATE 65 NOTE 99p. AVAILABLE FROM Baltimore County Public Schools, Office of Curriculum Development, Towson, Maryland 21204 ($2.00)
EDRS PRICE EDRS Price MF-$0.65 HC-$3.29 DESCRIPTORS Curriculum, *Curriculum Guides, Instruction, Mathematics, *Secondary School Mathematics, *Teaching Guides, *Trigonometry ABSTRACT This teacher's guide for a semester course in trigonometry is prepared for use with the text "Plane Trigonometry with Tables" by E. R. Heineman. Included is a daily schedule of topics for discussion and homework assignments. The scope of each lesson and teaching suggestions are provided. The content for the course includes trigonometric functions, solution of right triangles, trigonometric equations and identities, oblique triangles, and inverse trigonometric functions. Also included are two supplementary units on special right triangles and set theory. (Author/CT) U.S. DEPARTMENT OF HEALTH. EDUCATION & WELFARE OFFICE OF EDUCATION THIS DOCUMENT HAS BEEN REPRO. DUCED EXACTLY AS RECEIVED FROM THE PERSON OR ORGANIZATION ORIG . INATING IT POINTS OF VIEW OR OPIN IONS STATED DO NOT NECESSARILY REPRESENT OFFICIAL OFFICE OF EDU CATION POSITION OR POLICY CURRICULUM GUIDE
TRIGONOMETRY A Tentative Guide Prepared for Use with the Text Plane Trigonometry with Tables by E. R. Heineman
Guide Produced by Vincent Brant Supervisor of Senior High School Mathematics
William S. Sartorius, Superintendent Tow son, Maryland 1965 2 BOARD OF EDUCATION OF BALTIMORE COUNTY Aigbu.rth Manor Towson, Maryland 21204
T. Bayard Williams, Jr. President
Mrs. John M. Crocker H. Ems lie Parks Vice-President
Mrs. Robert L. Berney Ramsay B. Thomas, M. D.
H. Russell Knust Richard W. Tracey, D. V. M.
William S. Sartorius Secretary-Treasurer and Superintendent of Schools Foreword
This tentative guide has been written to aid teachers in presenting an updated course in trigonometry.The function concept as a set of ordered pairs and the analytic approach are key features of this guide. The lessons as well as the homework assignments are given as sug- gestions.The teacher should feel free to modify the lesson structure as may be appropriate for the student in his class.This guide serves as a framework within which the teacher should base the course in trigonometry.If teachers desire to alter the course to a considerable degree, they should discuss this with their department chairman or supervisor.Since this guide is tentative and will be revised after being used in the classroom, teachers are encouraged to offer suggestions and criticisms which will strengthen the outline. Teachers should not overlook the excellent transparency pro- jectuals which are in each mathematics department.Visual and graphic means should be used whenever possible.
4 MATERIALS OF THE COURSE IN TRIGONOMETRY
Plane Trigonometry with Tables by E. R. Heineman will be used as the basic text. The references to the right of the scope in the outline indicate helpful or additional materials for the teacher and pupils.
Code Title of Reference RWM Rosenbach, Whitman, Moskowitz.Plane Trigonometry with Tables. New York: Ginn and Company. 1943 FSM Fr eilich, Shanholt, McCormack. Plane Trigonometry. Morristown, N. J.:Silver Burdett Company. 1957. B Brant, Vincent.Relations, Functions, and Graphs. Towson, Maryland: Board of Education of Baltimore County.1962. AK Aiken and Beseman. Modern Mathematics:Topics and Problems.(Teacher's Edition). New York: McGraw- Hill Book Company.
BW Butler and Wren. Trigonometry for Secondary Schools. Boston:D. C. Heath and Company. 1948.
M May, K. 0.Elements of Modern Mathematics. Reading, Massachusetts: Addison-Wesley Publishing Company. 1959. Br Brady Transparency Projectuals
5 CONTENTS
UNIT ONE THE TRIGONOMETRIC FUNCTIONS 1
UNIT TWO - SOLUTION OF RIGHT TRIANGLES .. . 32 UNIT THREE - LINE REPRESENTATION OF TRIGONOMETRIC FUNCTION VALUES 36
UNIT FOUR SIMPLE IDENTITIES AND EQUATIONS 0 52 UNIT FIVE FUNCTIONS OF TWO ANGLES 59 UNIT SIX TRIGONOMETRIC EQUATIONS 63 UNIT SEVEN LOGARITHMS 64 UNIT EIGHT LOGARITHMIC SOLUTION OF RIGHT TRIANGLES 67 UNIT NINE OBLIQUE TRIANGLES 68 UNIT TEN INVERSE TRIGONOMETRIC FUNCTIONS . 75 UNIT ELEVEN- COMPLEX NUMBERS 78 APPENDIX I- SPECIAL RIGHT TRIANGLES APPENDIX II- REVIEW OF ELEMENTARY SET CONCEPTS LESSON SCOPE AND TEACHING SUGGESTIONS The Trigonometric Functions UNIT ONE REFERENCECODE I PAGE 1 Introduction (1-1) wellaTrigonometry college as in other subject. isbranches a subject of which mathematics. is important Two in hundred physics years and ago, It was studied by sea captains for navigation, by surveyors engineering as it was H.... 1 practicaltoemphasis.Today,andEaster. chart the the However,measurementtheassistance colonial analytic with ofwilderness, treatment observatoriesaspect new navigational of of trigonometryand trigonometric in calculating methods, has functions declined the dates is in receiving importance. greater This course will stress this modern point of view as well as the by ministers to calculate the date of specialization of surveying, of Easter, the DirectedThe Rectangular Segments Coordinate (1-2) System (1-3)Coordinatesolution of axes triangles. : x-axis, y-axis H. ... 2 2, 3 The Distance Formula (1-4) QuadrantsRadiusOriginPlotting vector a point H.... 4, 5 H . SUGGESTED ASSIGNMENTpp. 5, 6 : Ex. 1, 5, 70 9, 10, 11,13 -1- LESSON 2 Trigonometric Angles (1-5) Students who have studied the SMSG Geometry with Coordinates have been taught SCOPE AND TEACHING SUGGESTIONS L. REFERENCECODE1 PAGE H.... 5 thatanStudentsa concept anangle angle iswho excludes theis thehave union union takenthe of zerotwoof the two distinctangle, course noncollinear and inrays regular which rays Geometry with a common have been endpoint. taught thatSuch angles equal to or greater than 180°. have a common endpoint. Such asItanglea concept shouldnoted of 180°. abovebe excludes pointed were the outnecessary zero to the angle student for and a postulational anglesthat the greater restrictions treatment of of degree angles. measure than 180° but includes an The measures-50°,likedifferentconceptstudent rotating and should will fromofothers anglesbeshafts besynthetic made quite taught ofwhich inmotorsuseful thisthatgeometry. were coursecertainin will nothis find work.permitted Forsinceextensions suchexample, the angles purposesin and synthetic engineers modificationsas 0°,in trigonometrygeometry180°, who 270°, dealof onthe with1750°, the areangle basisthings Thus trigonometry- will deal with terminalofcourse.ceptExperience the postulates.unless ray withhasteachers shown respect carefully that to studentsan explaininitial become ray the is distinction alsoquite useful confused in in the trigonometry, with purposes the angle of the con- The dynamic interpretation of an angle as the rotation of a LESSON(cont'd) 2 other,astextThesome usestext cases theoffersthe measure ideaas a the definition of name angle of the in of angle. two "angle" senses: For in example, terms of the"an symbolamount "0"of rotation. is used in" The SCOPE AND TEACHING SUGGEST IONS of the angle, and in other cases as the number of de- one,as the name of the angle, and the . REFERENCECODE ( PAGE isusedsymbolgrees well"angle."student as for orthe "0" radiansthe willreal were teacher havenumber asused tothe to judgeas which usemeasure the the in nameis whatfollowing the ofmeasure sense the angle,ideas the of symbol thein and clarifying angle. the "0" symbol is theused "m(0)"dual in this use were text. It of the angle. It would be more precise if the However, the of theThedifferent) magnitude trigonometricmeasure having of of aa the trigonometriccommonangle angle is theinendpoint. terms union angle of of issexigesimal two the rays,number (not or assigned radiannecessarily as TheasObserve those angle whichthat is stillthis are thedefinition greater union than of permits two 180 rays, orthe negative. notmeasure necessarily of angles different. 0, 180, as wellmeasure. It is still ber;MakeThea set the measureconcept. anangle agreement isof a an set angle withof points. isthe different class that than the the symbol angle. "0" as used in the text The measure is a num- difficultyItber.may has have been Drovided two shown meanings thethat above students -- oneideas asaccept are a set explained. this of pointsdual use and of the "0" other withoutThe as appropriate a much num- use may be judged from the context of the discussion. -3- LESSON (cont'd) 2 b.c.a. InitialVertexTerminal ray ray SCOPE AND TEACHING SUGGESTIONS REFERENCECODE I PAGE RWMH.... 10-12 7 d. Positive angle berReview x is thegreater following: than zero." Emphasize the translationThe notation of "x "x> 0" > 0"as for real numbers means "the real num- In similar fashion, stress the translation of "0 > 0" as "0 x is positive.is positive" e. StressNegative the angle translation of "0 < 0 " asActually,This is another it is the example number ofof the degrees dual useor radians of 0 mentioned which is before. positive. HSUGGESTED ASSIGNMENT "0 is negativep. 7 :" Ex. 1 - 17 LESSON 3 Special Right Triangles SCOPE AND TEACHING SUGGESTIONS . CODE1REFERENCE PAGECourseGeometry of The special right triangles such as the b.a.c. 30-60-905,12,133, 4, 5 triangle triangle triangle AppendixRightSpecialStudyTriangles I fully.mainingthatare used the students sidesextensively of maythe trianglein be able when to calculate they are mentally given Provided. sufficient practice in the 30-60-9045-45-90 and triangle this course. The teacher should review this topic care- the length of anythe one lengths side. of the two re- 45-45-90 triangles so fromfunctionsexperiences.isanglesThis essentially your skill in of department thewill special Geometry review be most angles. chairmanwork usefulCourse so inthat ofand finding Study. the loaned teacher the to values students can easily for provide this lesson. Use the unit in Appendix I on Special Right Tri- Copies of this unit may be obtained of the trigonometric refresher This Appendix I : SUGGESTED ASSIGNMENT pp. 125 - 127 : Ex. 1 - 10 LESSON 4 Radian Measure Up to this time students have been familiar with the measure of angles based SCOPE AND TEACHING SUGGESTIONS CODEREFERENCE I PAGE SMSG,RWMH....65-69 83-88 interchangeablypointcoursethistroducedupon topic theon, is sexagesimal tothesountil the importantteacher later.concept so that should system. thatthe of radianstudent it use should radian measure. becomes be measure presented You proficient willand at sexagesimal notethis with thattime. both the Frommeasures. measuretext thisdefers However, radian measure and its use throughout the It is now most important that students be in- FunctionsElementary243-245 .. lengthradianThe numberteacher equalis a special to shouldof the radians numberradius stress in of which a thatthe central circle.the measures radianangle tells isa centrala hownumber---not manyangle timeswhose an theangle. arc radius has A a is thelengthThe "wrappedsexagesimalcentral teacher equals angle around" shouldthe measure whoseradius, usethe arc aofarcis "radian 60;thelength of incentralthe contrast, wheel"equalscentral angle thedeviceangle. the subtended radius, sexagesimal with is string byless a than chord,to illustrate 60. whose that measure of M")-7'n,z '7.7 LESSON SCOPE AND TEACHING SUGGESTIONS \.. REFERENCECODE I PAGE (cont'd) 4 measureTheSinceandcircle, teacher "degrees" the in we anradius should may intuitive in ofusingsay: develop a circle the theapproach is containedrelationship suggested fashion. It is helpful to write out the words "radian" 2Tr times in the circumferencebetween of a radian and sexagesimalbelow. From this last basic formula the students can always 2TT radiansTr radians = = 360 degrees 180 degrees (multiplying by obtain the 1 ) relationships: 1 radian = 180 Tr degrees (multiplying by Tr1 ) 180 Tr radians = 1 degree Or (multiplying by 180 1 ) H.... pp. 68, 69 : Ex. 1 , 2, 3, 5, 6, 7, 9, 10, 11,17, 19, SUGGESTED57, 58, 59 ASSIGNMENT 21, 23, 25, 27, 29, LESSON 5 Length of a Circular Arc SCOPE AND TEACHING SUGGESTIONS REFERENCECODE' PAGEMUMH....69-73 88-90 Note: ifLinear time andpermits. Angular Velocity may be presented later H.... pp. 71-73 : Ex. 1, 2, 5, 6, 7, 13, 14, 18, 19 SUGGESTED ASSIGNMENT in the course, 6 Relations, Function, and Graphs ofThemodernas ordered specialtext mentions approach. pairs, ratios. but the Hence, for idea the of itmost awill trigonometric part be necessary treats trigonometric function to review as a concepts specialfunctions set of In this course, however, we shall emphasize a more AK.Functions,lations,B.and .1-139Graphs , Re- Inchairman.sheetsalgebrasets, this relations,lesson, of and elementary geometry. the functions, teacher concepts should and graphsshould review bewhich the obtained following students from ideas:have your studied department in This should be a rapid review. Student review b.a. SetSetcollection is notation an undefined of things term. or objects. We may describe a set as a 1. ExampleThe phrase : method the set of the (names of) the vertices of A. ABC. -3- LESSON SCOPE AND TEACHING SUGGESTIONS REFERENCECODE PAGE (cont'd) 6 2.3. ExampleThe ruleroster method : (listing)Using same method set as in (1) and (2) ; . Using same set as in (1) ; { A, B, C } c. The emptyThe setsymbol set which for contains the empty no elementsset is Ire is the empty set. { x I x is a vertex of AABC } d. Example:ofSetSubset A A is is ofalso a asubset givenan element ofset set Bof (denotedB. "-AC B" ) if and only if each element Let B = {1, 2, 3, 4, 5 } Then : AC B DAC = = {2,{1, { 4} -5,3, 4}3, 4, 1 } ; CC B .; DC B ; C CA ; e. Ais andTheCartesian the the setCartesian second of Productall orderedcoordinateProduct pairs of belongstwo in setswhich to A B. andthe Bfirst (not coordinate necessarily belongs different) to Also 4C A ; 4)C B ; 4C C ; 4 C D Then,Example:The ACartesian X LetB = A{(a, Product= 1),{ a, b, of c, A and B is denoted as A X B (read "A cross B" ) (b, 1), -(c, -1), } -9- ; Let B = { 1,2 } (a, 2), (h, 2), (c, 2) 1 LESSON SCOPE AND TEACHING SUGGESTIONS REFERENCECODE1 PAGE (cont'd) 6 f. ARelation relation fromis a set A toof B ordered is a subset pairs. of A X B. or Example: SomeLet A relations = {a, b, from c } ; Alet to B B = are: Relation{1, 2 } QR = { (a, 1)2) }, (b, 2) , (c, 1) } RelationFurthermore,Of course, T = A{ (a,X the 1)B emptyis also set,a relation. , (b, 1) , 4), (c, 1) } is a relation. g. TheofDomainThe the range domainordered and of range a of pairsrelation a ofrelation of a relationtheis the relation.is setthe ofset all of the all secondthe first coordinates coordinates Example:of the ordered Let Relationpairs of theT = relation. { (a, 2) , Then, RangeDomain of of T = {a, b, c } T = {2, 3, 5 } (b, 3) , (a, 5) , (c, 2) } LESSON 6 h. Function SCOPE AND TEACHING SUGGESTIONS . REFERENCECODE 1 PAGE (cont'd) firstAStated functionsame coordinate differently, first from coordinate. there A to no isB two oneis Hence,a orderedandspecial only the kindpairs one first of ofsecond coordinates relationa function coordinate. such may in that the have for ordered theeach However, the second co- ordinatepairsExamples: of themay function be the same.must be different. HD = ={ {(1, (1, a) a) , relation. , (2, b) , (1,(3, c) }} isis anot function. a function; it is a K =1 { = (1, a) , { (x,Y) Note:a constant Domain function. of K = { 1, 2, 3, (2, a) , (3, a) } is a function--- - } ;Rax3ge- of K = fal Note: Domain of I = the set of real numbers 1 y=x}Range of I - = the set of real numbers i. Tests'1. for a function AorderedThe relation ordered pairsis a pairfunction are testdifferent. prpvided the first coordinates of the 2. doesthelineThe graph not intersectsrelationvertical represent of the isline the relationa functiontesta graph function, in in exactly provided more but athanone relation. every point. one verticalpoint, then line the intersects graph If at least one vertical LESSON 6 J. Value of a function SCOPE AND TEACHING SUGGESTIONS used REFERENCECODE I PAGE (cont'd trueinTheTheand discussing whenteacher imprecisely,following discussing should functions. ideas leading be theshould very idea to careful befuzzy of emphasized "the and value vagueprecise when notions. with discussing the language a In the past, such language has been used loosely of the function. " This is especially function such as: 2.1. The function is f NOTis the "y fentire = = { 5x"(x, set y) of ordered pairs y= 5x } 3. Instead,secondanothersentence "ycoordinate. real assigns= 5x" number is to a sentenceeach 5 times real which asnumber large defines forused its theas corresponding a function, first coordinate This ingnate."y", notations: f = { (x,y) "5" or Thus,"f(x)" the are function different f namesmight also for thebe writtensecond in coordi- the follow- y= 5x } 4. "f(x)" f = { (x,(x, 5x)f(x) ) is translated as "f at x" or "1 of x" I x is a real number I f(x) = 5x } } LESSON(cont'd)6 5. carefulIn describing to use thea function, followingSCOPE such language: as AND f above, TEACHING the teacher SUGGESTIONS"the should function be f defined by y = x", REFERENCECODE PAGE whenideasThe theteacher throughout unit shouldon graphing the capitalize course. the trigonometric on every opportunity functions to is"the use taught. functionthese y = x" Do not say: This will result in extra dividends (The teacherWritten mimeographedshould Assignment ditto copies copies of fromthis assignment the department or obtain chairman) SUGGESTED ASSIGNMENT Z.1. RAConstruct = { 0,r, t,2, k 4,the } 6and cartesian } andB = S{ =productproduce6, { 16, 1, 3,1526 ofof 5 RA} } XX SB where where 3. followingRDEachConstruct =is {the a of} set and theparts: theof Efollowing realcartesian= { numbers.1 } relations product ofis Ea relationX D where from R to R where In each example, complete the d.b.c.a. StateDraw thewhetherthe domainrangegraph the ofof of relation thethe the relation.relation. relation. is a function. -13- (cont'd)LESSON 6 SCOPE AND TEACHING SUGGESTIONS SUGGESTED ASSIGNMENT (continued) I REFERENCECODE I PAGE 4.5. R = { i.,:., y) I y = x } I 7,6. TS U= {= (x,{ (x, S(x)y) y) ) I y = x2 } y > 5x } 5(x) = 12- x } 9.8. WV == {{ (x,(x, V(x)y) ) I y = I V(x) = 2x - 5 }x3 } 12.110. 1 . GYX = { (x, G(x)y) 1 ) I x2 + y2 = 25xy } = 12 } I G(x) = 5 } 13. "ReviewH =of { Elementary(x,Study y) Assignment: Set Concepts" Study Mimeographed Sheet, I x = 3 } -14- LESSON 7 Definitions of the Trigonometric Functions of a General Angle (1-7) SCOPE AND TEACHING SUGGESTIONS CODEREFERENCE PAGEH. ..7 -10 whenprevioustheinThespecial a trigonometric mannerspeakingteacher lesson. set consistentshouldof of Observe orderedthe functions presentfunction with pairs,that theas the andthe follows: but definitionsconcept itstext then value. does revertsof ofmentionfunction the to trigonometric loose asthe reviewed function expressions functions inas the a The teacher should define sine (or sin) = {(O, sin 0) Observe that (1) I sin 0 = -Y- (3)(2) Sinof0The is the 0asymbol isnumber---aangle a number in"sin" degrees numberindicates---- the or numberradians.that a set measures of which ordered theis assigned pairsmagnitude (4) Sinas 0the is NOTsecond the coordinate function. by the defining sentence sin 0 = r (5) Each ordered pair of the sine functionforSin the has 0 isfirst real thethe.name and numbersVALUE second of ofthe coordinates.the second function coordinate. at 0. LESSON SCOPE AND TEACHING SUGGESTIONS 7 cosine ( or cos ) = 0 cos 0) I cos 0 = r } (cont'd) tangent ( or tan) { (0, tan 0) I tan 0 = } cotangent (or cot) = { (0, cot 0) I cot 0 = -c- } I sec 6 cosecantsecant (or (or sec) scs) = { (0, scs 0), = { (0, sec 0) J csc 0 = r } LESSON(cont'd) 7 SCOPE AND TEACHING SUGGESTIONScot,Thus, sec, the and trigonometric csc. functions are called sin, cos, tan, REFERENCECODE) PAGE functionsEmphasizecotcoordinatesThe 0, values sec consists0, theof ofand the factthe csctrigonometric oforderedthat all0. the real domainpairs) numbers functions are of eachsin0 for 0,(or of which:cos secondthe 0,above the tan 0, divisionupondependsStresscorresponding the theby solely lengths zerofact thatisuponratios impossibleof the thethe have value sidesmagnitude a nonzero(notof used the defined). intrigonometricof denominator,thethe ratios.angle and functions sincenot Consequences of the Definitions (1-8) 2.1. ReciprocalTheorem onrelations coterminal angles: The value of the trigonometric function H....10,11 angleof an anglecoterminal is equal the to given the value angle. of the trigonometric function H.... p.14; Ex. 1, SUGGESTED ASSIGNMENT 2, 6, 7, 19, 21, 22, 23, 25, 26, 27, 29, 30 of any -17- REFERENCE (41 (contid)LESSON 8 theytheTheSigns followingteacher presentationdiscover of the should values diagramsthe ofgeneralization provideof this the and topic trigonometric ancharts. inopportunity theand text formulate functions is forpoor discovery the and rule. should in this not topicbe used. by presenting SCOPE AND TEACHING SUGGESTIONS Have the students complete the chart so that Instead, CODE I PAGERWMH....11 17,18 Function value Quadrant + = + I + II = + III IV cosin s e 0 T+ 4- _L + cotetan o T-+-T 4- cscsec 0 TT-+ ++ -18- LESSON SCOPE AND TEACHING SUGGESTIONS S. REFERENCE (cont'd) SUMMARY CHART FOR SIGNS OF FUNCTION VALUES sin 0 > 0 areall function positive values CODE ( PAGE tan.0Qcsc IIIII >0 >0 0 cosQO IVI 0 > 0 Use of inequality symbols 1. sin 0 > 0 means that the value of the sine is positive. cot 0 > 0 sec 0 > 0 TheObservesamesymbol. interpretation object the incorrect as", of or "sin language"is 0another = -1-" used meansname by the for."that text a Direct number in Examples your is the students 31-38 to rewrite2. This is incorrect,sin because 0 < 0 means equality that in the a sentence value of the sine is negative, means "names thesame as the positive on page 15. exampletheseThe inuse proper31 of above. set form; intersection example is helpful31 should in solvingbe properly problems written such The presentationsin 0 < 0 andof the tan solution > 0. might be as follows: as as /44 REFERENCE cont'd)LESSON 8 Now,Let x be the quadrant or quadrants in which the given angle terminates. NM={xI ={xjtane>0} sine<0}={III, ={I, III IV} SCOPE AND TEACHING SUGGESTIONS CODE PAGE Hence, MnN={xIsin0<0}n{xItanO>0} = {{Ill, } } n { 1, III} Representation of the quadrants by set notationentireThenLet R Rplane.be X the R isset the of setreal of numbers. ordered pairs of real numbers, and its graph is the Hence, A = { (x, y) to theThe right graph of ofthe A y is axis. the set of all I x > 0 }; YA. s of the plane or the half-plane -20- A4ie...4. (contLESSON8 B = { (x,y) toThe the graphleft of ofthe B y is axis. the set of all points of the plane or the half-planeSCOPEI x AND < 0 } TEACHING SUGGESTIONS CODE!REFERENCE PAGE DC =={ { (x,y)(x,y) aboveThe graph the x of axis. Cis the set of all points of the plane or the half-plane 1I yy >< 0 } Hence, AnB(1 C C= ={ {(x,y) (x, y) belowThe graphthe x axis.of D is the set of all points of the plane or the half-plane I x <> 0 and y > 0 }; its graph is Quadrant II.I. An BD n = D { =(x, { (x,y) y) I x < 0 and x > 0 and y < 0 }; its graph is Quadrant IV. y < 0 }; its graph is Quadrant III. -21- eisil`41\ whichThe teacherare not shouldin the text. either ditto or write on the board the problems below SCOPE AND TEACHING SUGGESTIONS HOMEWORK ASSIGNMENT REFERENCECODE1 PAGE 1. underIn(a) what the quadrants following must conditions: the angle whose measuresin 0 > 0; is 0 terminate (b) cos 0 < 0; (c) tan 0 < 0; (d) sec 0 > 0 2. belowExpress(a)(Refer 390° as to thefunction page function 11 values in values the of text angles of for sin, model between cos, tanillustrations) zero of the and angles 360. given (b) 890° (c) -34° (d) -590° In(2)(I) problems RewriteIdentify 31 thethe - 38problemquadrant on inin correctwhich thenotation0, terminatesgiven usingangle, in inequalities. whose order tomeasure satisfy is the conditions. page 15, LESSON SCOPE AND TEACHING SUGGESTIONS 1. REFERENCECODE PAGE 9 Function values of speciNra7n-gre-W ricTheofIt is anglesfunction wisetext does atby this valuesusing this time in theof differentto suchratios provide angles of sections. the further assides 330°, practice of -the 135°, 30-60-90 in 600°finding andand the the45-45-90 function quadrantal triangles. values In this guide, however, the trigonomet- RW/v1H....21-23 21-26 50-52 LaThechangeangles this following section ably.are treated teachers ideas at areone should important time. use radianin this andlesson: sexagesimal measure inter- 2.1. The relatedthesegments horizontal (or referencereference) representing (or x) angle) triangleaxis theand as asabscissathe the the positive terminal triangle and acute theside formed ordinate ofangle the by betweengiven ofdrawing a pointangle. on the 4.3. TheThe 45-45-90 30-60-90terminal triangle triangle side. as as a aspecial special related related triangle. triangle. concerningtheThe students teacher the shouldcomplete procedure present the inchart thefinding diagramsafter5. the a suitable function and charts explanation values. on the nexthas been pageThe made and unit letintersect circle and the the circle. coordinates of the points where the x and y axes I. REFERENCE LESSON 9 A. SCOPE AND TEACHING SUGGESTIONS -1-16: CODE} PAGE (cont'd) A The function ...-----"-*/ values of angles related /\ to an angle of A radians or 30°. I / 150° 210° , 330° -..": ..- il_ / 30° la % 4-3 _1 - NTT' -1 ;.Av30°i 2 ----"2 0° /T --... -1 COMPLETEFunction Angle THE CHART .6----trI or 30° -r bn5 or 150° --tr 67or 210° 1 I 6 tr or 330° cossin 0 0 Nu- 1 _ cottan 0 4-3. I32 - \Fr43 2 cscsec 0 0 24 r1 LESSON(cont'd) 9 B. The function values of angles related to an angle SCOPE AND TEACHING SUGGESTIONS of radians or 60°. . REFERENCECODE ! PAGE 0 6O 1 NTT" 41 COMPLETE THEFunction CHART sin 0 Angle ir or31 60° g 3 ir2 or3 120 ° nor43 240 ° nor35 300 ° tanco 0 s 0 4 3 -2- 12 -2- 1 seccot 0 0 r - 4-3 3 csc 6 -25- LESSON(cont'd) 9 C. The function values of angles related to angle of 4 radians or 45°. SCOPE AND TEACHING SUGGESTIONS .REFERENCECODE I PAGE 45\+1 COMPLETE THE CHART 1 0 3 7 Function------.,... sin 0 Angle NIT- 1 -4 n or 45- 2 :--i ir or 135° -4.5 ir or 225° 7-1 ir or 315° tanco s 0 e A. 1 - NT-2 2 seccot 0e 4.--2-- 1 csc e -26- REFERENCE LESSON (coned) 9 D. QuadrantalUsevector a unit and angles;circle the circle.and that point is, angles P representing whose measure the is 0, SCOPE AND TEACHING SUGGESTIONS intersection of the radius 7r 7r, T.37r radians CODE' PAGE 10 027r -1 Tr 37r 02 OrdinateAbscissa (y)(x) 10 1 10 -1 1 Radius vector (r) 1 3 F unction Angle 0 or 0° 1 '.. rr or2 90° 7r or 180° 1 = - 1 7r2 or 270° cos°sin 6 xry 0 =-1 1 .n =1 ' -I tancot 0 0 --=--yxy_r 1 = 001 0 notfinecl de- 01 -defined not cscsec 0 0 -27- REFERENCE r1LESSON (cont'd 9 haveannounceThestudy. teacher the charts a shortmay completed wishquiz toon letthis in the class topic students and for assignthe complete next additional day, the or charts the problems teacher for homework formay home wish and toIn that case, the suggested assignment is given below. SCOPE AND TEACHING SUGGESTIONS CODE I PAGE PimH....H. p.23: .p. p. 61, 69: Ex. 62: Ex. 1,Ex. 37, 2, 1,39, 5, 2, 41,6 6, 9,43, 10 45, 49, 51 SUGGESTED97 ASSIGNMENT
fI LESSON SCOPE AND TEACHING SUGGESTIONS REFERENCECODE I PAGE 10 theGiven Other One Trigonometric Trigonometric Functions. Function Value of aninterestedpracticedStudents Angle, should To inin theDrawfinding previousbe the shown the Angle second lessons. that and this coordinate Find topic actually of an orderedreverses pair the procedurewhen the first In prior sections, students were primarily coordinate was given; that is,To find sin 30° means to fill in the slot in the ordered pair (30°, I 1 ) ofIn anthis ordered new section, pair when students the second areTo concerned find coordinate 6 when with sinis findinggiven; 6 = for the example: first coordinate 1 means to fill in the slot in ( , 1 NotetheThe inverse that teacher ordered of shouldsin wouldpairs take in require suchcare notan interchangeto call this the would inverse give function rise to such process ordered since (2)(1) the interchangeresulting set of of the ordered first and pairs second also becoordinates a function M. n ..284-294 pairs as (-2 30°), 21 ' 150°), 21 390°), (-2' 510°), 1etc. interchangingSinceThesuitable followingthe firstrestrictions, thecoordinates procedure coordinates a function isin helpful thewould ordered may notin result.the result pair solution ofin aa functionfunction. of a problem must suchbe different, as (- (- Of course, with Given: sin = 1 ; locate the terminal side of the angle. -29- LESSON(cont'd) 10 Procedure for solution of: Given1. sin Determine the quadrant or quadrants from the sign of the function value. SCOPE AND TEACHING SUGGESTIONS = ; draw the angle in standard21 position. REFERENCECODE1 PAGE 2. theKeepWriteQuadrant Thissine emphasizing thewillvalue II.defining enable is positive the thesentence importance student and henceof to the assignof thesin this anglefunction, lengthsfirst must step. toand liea relatedthe in Quadrantgiven ratio as In the above problem (or reference) I or below. vectorOftriangle. course, is defined the fraction to be a cannot positive be quantity. represented as sin 8= yr = +2+1 ; Hence y +1 ; r = +2 However, in the tangent or -2-1 because the radius 3. atorpositivecotangentMake must a or drawing alwaysratio, negative either be us" values.a gpositive the the numerator parts quantity. found or in denominator part 2. can In the secant and cosecant ratio, the numer- take on either \I +1 +1 1 2 V 4.5. HencetostudentsThe the third30-60-90 the shouldrelated side maytriangle. recognizeangle be foundis 30°, the by and ratio the 8 ofPythagorean= 30°the andsides 150°. 1 relation. : 'TT2 as belonging In this case, H....p. 17 ; Ex. 1, 3, 5, 7, 9, 11 SUGGESTED ASSIGNMENT -30.- LESSON 11 Reinforcement of Lesson 10 SCOPE AND TEACHING SUGGESTIONS REFERENCECODE I PAGE toRWM....The copy exercises these pp. on in20, the RWM 21:board Ex. are or 1,excellent 3, SUGGESTED ASSIGNMENTditto them. 17, 19, 21, 24(a) and more challenging. You will want 1213 ReviewTest for test
-31- 1 LESSON SCOPE AND TEACHING SUGGESTIONS REFERENCECODE1 PAGE 14 Trigonometric Functions of an Acute Angle (2-10) Solution of Right Angles UNIT TWO RWMH....18,19 28-30 Cofunctions (2-11) Theorem 2:1: IftheAny two cofunction trigonometric acute angles, value function A of and its complementaryB, value have of any an functionacute angle. angle value is equalof one to RRWMH....20 WM .7.030 trigonometricHence,The topic it will of functions.bevariation deferred of tothe a function later section values dealing at this with point line is too limited in scope. Aequal and B to are the complementary. confunction value (Converse of the other, of Theorem then the 1) angles representations of RWM H..H....p.19: Exercise 5: Examples 1,2,3,4 p.p.31: 23: Exercise Examples 6: Examples 1, 3, 5, 12, 9, 10,17 11, 13, SUGGESTED ASSIGNMENT 14 LESSON 15 GivenTables an of Angle, Trigonometric To Find the Function Value of Values One of Its SCOPE AND TEACHING SUGGESTIONS (2-14) Functions (2-15) 1. REFERENCELCODE I PAGEH...H.... 23,24 24 Given the Function Value of an p.25 : Exercise 7; Examples 1, 2, 3, 5, 6, 7, 9, 10, 11, SUGGESTEDAngle, To Find ASSIGNMENT the Angle (2-16) 16 Interpolation (2-17) SUGGESTED ASSIGNMENT 14,15,17,18,19, 21, 22,23 H. .. 25-27 17 H... p.27, 28: Exercise 8: Examples 1, 2, 3, 9, 10, 11, 13 17, 18,19,21,22 ApproximationsInterpolation (Second and Significant day for further Figures reinforcement (2-18)An excellent of treatment of this topic may be found in Butler and Wren skills) II...13WH.. .. . 41-4428-3025-27 H....H. p. 30: Exercise 9: Examples 1, 2,.. p. 27, 28: Exercise 8: Examples 5, 6, 7, 14, 15 SUGGESTED ASSIGNMENT 25, 26, 27, 29,3, 305, 6, i, 9, 10, 11 LESSON 18 The Solution of Right Triangles (2-19) mightInsist uponbe patterned a neat format after the for following: the solution of the right triangles. SCOPE AND TEACHING SUGGESTIONS Such a format REFERENCECODE)H.... PAGE 30-33 Given: LA = 38°50' b = 311 38°501 Ca To find: LB = ca= = (1) To find LB (2) To find a b = 311 (3) To find c LBLB = =90°- 51°101 38°50' a =TanA 250311b Tan ( Tan =b0.8050) A 38°50' 2 cCos - A = b 0,7790311Cos A b c H.... pp. 33-34.: Exercise 10 : Examples 1,3,5,7, 9, 15 SUGGESTED ASSIGNMENT c = 399 LESSON19 Angles of Elevation and Depression (2-20) (2-20) SCOPE AND TEACHING SUGGESTIONS I. H. REFERENCE.34-36CODE PAGE Bearing of a Line H....pp. 36, 37: Exercise 11 : Examples 1, 3, 5, 7, SUGGESTED ASSIGNMENT 11, 13 H....34-36 20 Angles of Elevation and Depression; Bearing of a Line (Second Day) H.... pp. 36-39: Exercise 11: Examples 14, 15, 21, 22, SUGGESTED ASSIGNMENT 23, 28 2221 TestReview
iI LESSON SCOPE AND TEACHING SUGGESTIONS REFERENCECODE I PAGE Introduction Line Representation of Trigonometric Function Values UNIT THREE inthepossibleofasPreviously, athe lengthsunit radiusratio andcircle, of oftenvectorthe atwo andsingle trigonometric convenientsegments thenof linea pointrepresenting segment. represented toon functions represent the Thisterminal the byoftrigonometriccan the thethe betrigonometricside generalabscissa, done of the by angleratios radiusrepresentingordinate function havewith vector. or beendenominator valuesthe the expressed length angle as It is riclinemaythisIt 1.is identities materialrepresentations unfortunatechart the throughsince variation that it thus providesthegraphic oftextdeveloped the which methods.an trigonometric interesting willis excellent The be used referencesgraphic function into manydevelop method values.by respects RWM the by Furthermore,basic which andchooses trigonomet-FSM the to studentpro- omitthe andThenForvidesin betterthe an values,the excellentsec tan classes might andmight assignpresentation bethe be discussed teacherdiscussed the line ofmay in representation inthis class classwish topic. with withto present the the of csc cot the the assignedassigned cos line values representation forfor for homework.homework; homework. of the -36- LESSON For slower classes, the teacher may wish to develop SCOPE AND TEACHING SUGGESTIONS all of these in class. REFERENCECODE I PAGE theSometeachersimultaneously. best teachers way should to prefer presentemphasize This to is itteach toleft the the totan following students and sec inideas: simultaneously his class. (1) domain and range of the function teacher preference and to his judgment of In any case, the and the cot and csc Askwhich your you department should use chairman in presenting for the this excellent topic. transparency projectuals (2)(3) amplitude graphical and representation periodicity -37- LESSON 23, Line Representation of the Sin Function Value SCOPE AND TEACHING SUGGESTIONS . R CODE;EFFT-!.17.Tr7EH.. PAt .. 76-82,115- A. Geometric derivation Given: thepointUnit unit P circle theradiuscircle. intersection 0 withvector angle and of 6, RWFS.M.. M. .61,63232-234 123239-14466-68 2.1. sinDraw 0 PM= PM _L OX = PMPO B. Definition: The line representation of the sine is the ordinate of point P on the theunit x circle axis; withnegative, radius if vectormeasured OP. below the x axis. It is positive= PM if measured above 1 C. Alepresentation of the sine value in each quadrant I\ sin (sin 0 (-11 -38- LESSON (cont'd)23 D. Complete the chart of variation for the sine values SCOPE AND TEACHING SUGGESTIONS i REFFar,NCECODE1 PAGE FunctionAngle 0 1 41 31 21 23 1T 3 ZIT5 34 23 35 7 E. Graphic representation of the sine function1. value of sin Sine curve (sinusoid) defined by y = sin 0; domain; range .5 .71 3.2. Graph of (a) the function defined byFeriod; y = amplitude; continuity (c)(b) thethe functionfunction defineddefined byby y y = = 3 2 sin sin 0 0 21 sin 0 4. ComparisonGraph of of (a) the the graphs function of (a), defined by y = sin 1 0 (b)(c) the the function function defined defined by by y y= =sin sin 2 30 0 (b), (c) with respect to apaislitude and period. 2 NOTE:5. To save time, the teacherComparison mayGeneralizations wish to of let the one graphsconcerning row of students(a), the period and amplitudeanothercoordinate row 3(b),axis toetc. make , and comparisons let the students and drawgeneralizations. the graphs on (b), (c), with respect to amplitude graphoneand 3(a),set period., of LESSON 23 SCOPE AND TEACHING SUGGESTIONS SUGGESTED ASSIGNMENT REFERENCECODE I PAGE (coned) BeRepeat sure theto include classroom the followingdiscussionA. ideas: forGeometric the line derivationrepresentation (Hint: for Observe cos the abscissa of the point. values. H....FSM..61-63 76-78, 115-123 66-6882, B.D.C. DefinitionGraphicLine representation representation of the of thecos cosinevalue incurve. each quadrant RWM 232234 239-145 2.1. CosinePeriod; curve amplitude; defined continuity by y = cos 0; domain; range 1 3. period.GraphCompare of (a) the the graphs function of (a),defined by y = (c)(b) the function defined by y = 32 cos 0 (b), (c) with respect to amplitude and cos 0 4. CompareGraph of (a)the the graphs function of (a), defined (b), (c) by with y = cos respect to (b)(c) the the function function defined defined by by y y= =cos cos 20 30 21 0 period and 5. amplitude.cosMake curves. a generalization concerning amplitude and -40- period for the . REFERENCE LESSON 24 IV. Line representation of the tangent functionGeometric derivation SCOPE AND TEACHING SUGGESTIONS /\ Given: intersectionangleUnit circle 0, point 0 of with Pthe the CODERWMFSM.H.... 182,83 61-63,65235-237PAGE 69-71 2.1. Draw tangent TV at point of contactDraw T. PM OX l/ This is the unitand radius thestandard unit vector circle. position of 246 4.3. ExtendtheTan tangent 0 radius = PM in linevector representation OP intersecting of functions TV at R OM 7.6.5. APM PM0 7\./AR TO TanOM 0 = OTTR OTTR B. 8.Definition: Tan 0 = TR pointcircletangentThe1 line ofand (whose tangencyrepresentation the positive point to the of side ofcontactpoint the of oftangent the is intersection alwaysx axis) value= themeasuredTR is ofintersection the the length radius from vector of theof thethe belowtiveextended (+) the if to xmeasured meetaxis. the abovetangent. the x axis, negative -41- The sign of the function is posi- (-) if measured LESSON SCOPE AND TEACHING SUGGESTIONS REFF:RENCECODE I PAGE (cont'd) 24 C. Represent tion of the tangent function in each /I\ tan e (+) .,.\ quadrant l tan el (-0 D. Complete the chart of variation for the tangentnegativevaluespropertyfinedI(D) of resultfunction.since thisrotation, is unit.indicatedbecauseit involves as shown 90° as division tancan below. 90°be approachedby= + zero.) co. In from either Note that at 7112 or 90° the value of tangent is unde- The dual11/ positive and negativemany texts this undefined (Refer to achart positive or -42- LESSON 24 E. Craphic representation of the tangent SCOPE AND TEACHING SUGGESTIONS function 1. REFERENCECODE I PAGE (cont'd) 2.1. TangentPeriod; curve:discontinuity y = tan at 0 2ir and 32 ir Repeat the classroom discussion SUGGESTED ASSIGNMENT for the line representation of cot values, Be sure to present the following A.C.B. Geometric derivation RepresentationDefinition of the cot value in each quadrant ideas: (NOTE: The above assignment is given on pages D.E. line for the teacher's use. )asymptotesGraphicChart of representationvariation in cotangent of the cotangent values 42 and 43 of this out-function period, -43- LESSON 24 VI. Line representation of the cotangent function SCOPE AND TEACHING SUGGESTIONS CODEREr.E.PT:NrEH.... r1 GE 83 (coned) A. Geometric derivation W K Given: Unitthepoint circle unit P the0radiuscircle. with intersection anglevector 0, and of RWM.238-240 247 2.1. Draw QK tangent to circle 0 at pointDraw of PMcontact Q. 1/ _L OX This is the 4.3. functions.cotExtendstandard e = radius position vector of theOP cotangent to intersect in tangent line representation QK at W of OM 6.5. APMO "-N./ LIQW 0 MPOM OQQW QWMP 8.7. cot 0e = QWOQ 1 9. cot 0 = QW -44- LESSON SCOPE AND TEACHING SUGGESTIONS . REFERENCECODE I PAGE (contid)24 B. Definition: themeasuredofThe(extended,section the intersection line tangent representationof to the theif (whose necessary). unit ofright the circle pointorhorizontal of left theand Itof from iscotangent contactthe positive tangent thepositive pointis alwaysvalueif and measured sideof the contactis theof theradius the inter- length to yto theaxis)vector C. Representation of the cotangent function in each quadrant cot (+) right; negative, if measured to the left.cot (-D tft cot e (+) cot o t\ D. Complete the chart of variation for thechart cotangent I (D) offunction. this unit,) (Refer to E. 2.Graphic1. representation of the cotangentPeriod;Cotangent function discontinuity curve defined at 0 by and y =Tr cot radians 0 LESSON 25 V. Line representation of the secant function SCOPE AND TEACHING SUGGESTIONS . CODEREFERENCE1-1.... PAGE 84 A. Geometric derivation R Given: Unitvectorthe circle intersection and 0 thewith unit angle of circle. the 8, unit point radius P RWM 235-37 247 2.1. Draw TV tangent to circle 0 at pointDraw of PMcontact _L OX V CJ 5.4.3. D OMP r-/ dR TO Extendsec 0 = radius vector OP intersecting TV at R OMOP 7.6. OMOPsec @ - OTOR OTOR 9.E. sec @@= = OR OR 1 REFERENCE LESSON(coned) 25 B. Definition: ifdirectionradiusThe sectiontheSCOPE line radius vector representationwithas AND thevector extended,the TEACHINGdirection tangent. must of measuredbe of the SUGGESTIONS extendedthe secant given from valuethrough radius the is origin vector; the lengthorigin to negative, its inofinter- thethe It is positive if measured in the same CODE I PAGE C. Representation of the secant function in each quadrant1\sec 0 oppositesect the directiontangent in of the the standard given radius position. vector in order to inter- /;sec 0 (-) D. I(D) Completeof this unit.) the chart of variation for the tangent function. (Refer to chart E. Graphic representation of the secant2. function1. Period;Secant discontinuity curve defined at sr and by y = sec C -- ; asymptotes.3ir 1 LESSON SCOPE AND TEACHING SUGGESTIONS I. REFERENCECODE' PAGE (cont'd) 25 RepeatBecosecant sure the to value. classroom include the discussion following forideas: the line representation of the SUGGESTED ASSIGNMENT D.C.B.A. RepresentationGeometric derivation of cosecant value in eachDefinitionChart quadrant of variation in the cosecant values (NOTE: The above assignment is given onE. pages 46 and 47 of this period,Graphic asymptotes; representationoutline discontinuity. of for the the cosecant teacher's function; use. ) . -48- VII. Line representation of the cosecant function SCOPE AND TEACHING SUGGESTIONS . CODEREFERENCE 1 PAGE A. Geometric derivation Given: Unit circlethepoint unit 0 P with theradiuscircle. intersectionangle vector 0, and of RWMFH... SM 238-3984 . 24761-63 1. Draw PM I OX 3.2. OK tangent to circle 0 at point of Extendcsccontact 0 = radiusQ vector OP to intersect tangent QK MPOP at W 6.5. AMPS, r--../AQW0 PMOP OQOW OW 3.7. csc 0 = OWOQ 1 9. csc 0 = OW -49- LESSON SCOPE AND TEACHING SUGGESTIONS CODEREFERENCE i PAGE (cont'd) 25 B. Definition: vectorfromradiusandThesame thethe linevectormust directionoriginhorizontal representation be (extended extendedto theas tangent. the intersection to throughgivenof meet the radius cosecantthe the of horizontal originthe vector; radiusvalue in negative thetangent) isvector theopposite length extended measured direc- of the It is positive if it is measured in the ,if the radius C. Representation of the cosecant function in each quadrant taltion tangent of the ingiven the standardradius vector position. in order to intersect the horizon- A ,- Acsc 0 cos 0 + 1 I A csc C (-) 4 csc 0 (- ) -,----( / o 0 9 \ . -- ...... --... ..- Ilig o -N\ D. K Completechart theI(d) chart of this of unit.)variation for the cosecant function.\I v (Refer.,. to v E. GraphicZ.1. representation of the cosecant curvePeriod,Cosecant discontinuity, curve defined asymptotes by y = csc 0 l LESSON 26 ofThis functions, lesson should and sketching be devoted of graphs. to further practice with line representations SCOPE AND TEACHING SUGGESTIONS 1 REFERENCECODE i PAGE H.... pp. 84-86: Exercise 21, Examples 1, 2, 3, 5, 6, 7, SUGGESTED A SSIGNMENT 27 Review for test 39, 41, 42, 43 28 Test
-r 1 1 _ LESSON SCOPE AND TEACHING SUGGESTIONS UNIT FOUR 1. REFERENCE CODE1 PAGE 2.9 I. The Fundamental Relation (3-2) Simple Identities and Equations H.. . 40-44 thesectionStudentsaalgebraicThe morework eight intuitiveshoulddone find orbasic geometric inthese emphasize theidentities and lastderivations graphic unitmeans. given work on approach, thesimple Sinceonwith line fractions theand this geometric guide and will radicals. approach capitalize page 41 of the text may be proved by representations of functionsatisfying. values. The drills in this presents upon Pythagorean Id entitie s A. sin Geometric20 derivation + cos20 = 1 . by thesin2 Pythagorean Theorem 0 -52- + cos 0 2 = 1 LESSON (cont'd 29 therotation,The derivation line positive function of theor representations negative.identities geometrically are valid forfirst, the and general then showingangle of the alge- SCOPE AND TEACHING SUGGESTIONS It can be seen that the reason for approaching any kind of .REFERENCECODE I PAGE powerfulbraicanglesis wise interpretation in tomethod, differentlet the capable students secondquadrants of proveis dealing due and theseto also withthe forrelations fact angles quadrantal that in forthe any themselvesgeometric angles. amount of approach by rotation. considering is 2' The following Ita variousfigures showquadrants the verity and quadrantal of the identity, positions. sin 0 + cos /N A 0 + cos20 = 1, in the cos 06 cos 6. 0 1 1 sin 6 ri (-) ".:)1--4/ co 0 ---, i I I -- (70°i ..,....----T----- . A \ .,,k,.-cos--.. 0 sin 0(0)1 7 ji\P / 7,-1, / I / '.,(0) 4---, '----y---1° cos 0 ; /-
Certain right triangles appear frequently in problems of the physical world such as engineering and in problems of related mathematics courses such as trigonometry.These special right triangles may be classified into sets of triangles, each set containing only triangles that are similar to one another. You should be able to recognize the special right triangles discussed in this unit, to understand the relationships that exist for each set, and to apply these relation- ships in the solution of problems.
I.The 3, 4, 5 right triangles.
Since32-I-42=52,the converse of the Pythagorean Theorem tells us that a triangle whose sides have measures 3, 4, 5 is a right triangle.In a similar manner, we can conclude that if any positive number k is used as a constant of proportionality, then a triangle whose sides measure 3k, 4k, 5k is a right tri- angle.For example, any triangle whose sides measure 6, 8, 10 or 150, 200, 250 is a member of this set of right triangles. Example 1.Find the length of the hypotenuse of a right triangle if the length of thei two legs of the triangle are 18 and 24. You should note that the triangle is a 3, 4, 5 right triangle 1 with 6 as the constant of proportionality since 18 = 6 4.There- fore, the length of the hypotenuse of the triangle is 6 5 or 30. Example Triangle ABC is a right triangle with L C as the right angle. AC = 75 and BC = 45.Find the length of AB. Since the length of AC is 15 5 and the length of BC is 15 3, the triangle is a 3, 4, 5 triangle with 15 as the constant of proportionality.Therefore, AB must equal 15 4 or 60. II.The 5, 12, 13 right triangles. Explain why a triangle whose sides have measures of 5, 12, 13 is a right angle.Explain why a triangle whose sides have measures of 5k, 12k, and 13k is a right trianglek is some positive number. Explain why all these triangles are similar.Give other examples of three numbers which are the respective meas- ures of the sides of triangles in this set of similar right triangles. Example 1.Find the length of the hypotenuse of a right triangle if the lengths of the two legs of the triangle are 15 and 36. You should note that the triangle is a 5, 12, 13 right triangle with 3 as the constant of proportionality since 15 = 3 5 and = 3'12.. Therefore, the length of the hypotenuse of the right triangle is 3 13 or 39. Example 2. Triangle ABC is a right triangle with Z C as the right angle. If the hypotenuse of the triangle has a measure of 78 and one of the legs of the triangle has a measure of 72, find the measure of the other leg of the triangle. Since 78 = 6 13 and 72 = 6 12, the triangle is a 5, 12, 13 triangle with 6 as the constant of proportionality.Therefore, the measure of the other leg is 6 5 or 30.
III.The 30, 60, 90 triangles. Unlike the sets of triangles in (I) and (II), the triangles in this set are usually designated by measures of angles rather than lengths of sides.Suppose that in AABC, m L C = 90, mL B = 60, mL A = 30. We know by Theorem 5 7 on page 202 of your text, that the length of BC is one-half the length of AB. If we let m (BC) = a, then m (AB) = c = 2a, and by the Pythagorean Theorem 2 2 a2+ b =(2a) 2 2 a2+ b = 4a b2= 3a2 b = a4-3
Thus we see that if AABC isa 30, 60, 90 triangle (a, b, c) p (1,NTT, 2). We can also prove the converse of this statement.If AA'B'C' is a tri- angle with (B'C', C'A', A'B') p (1, NrST 2), then AA'B'CI is a 30, 60, 90 tri- angle. We know that LIA'B'C' is similar to A ABC by the SSS Similarity Theorem. Therefore, m L Al = 30, m L B' = 60, and m L C' = 90 because corresponding angles of similar triangles are equal in measure. The above discussion can be summarized in the following.theorem and corollaries: Theorem A. Triangle ABC is a right triangle with m Z A = 30, mL B = 60, mL C = 90 if and only if (a, b, c) P (1,NiT, 2). Corollary 1.In a right triangle whose acute angles have measures of 30 and h0, the shorter leg is one-half the hypotenuse.(830 = ) 2 Corollary 2.In a right, triangle whose acute, angles have measures of 30 and 60, the longer leg is equal to one-half the hypotenuse multiplied by 3,(s = h^17) Example 1. Find the length of the altitude of an equilateral triangle if the length of one of the sides is 8. The altitude of an equilat eral triangle bisects the vertex angle making a 30, 60, 90 triangle. (4, x, 8)p(1, 457 2). The constant of proportionality is 4. Therefore, x = 4 'TT The problem can also be solved by using Corollary 2.
h 60 2
8 '43 s60= 2 s60 = 4 Tr' Example 2. Find the measures of the sides of a 30, 60, 90 triangle if the length of the longer leg of the triangle is 10. (a, 10, c) p (1, 47, 2). The constant of proportionality is10 Nrr 10 2 10, or Therefore, a = and c = 3 10 Nrcand 20 NTS- c 3 3
IV.The 45, 45, 90 triangles. Since the triangles in this seehave two angles that are equal in measure, they are sometimes called isosceles right triangles. We can prove that the lengths of the sides of any isosceles right triangle are proportitinal to (1,1, Nrir. In AABC. m L C= 90, mL A=mLB= 45.If we let the length of. AC = a, then the length of BC = a. Why? By the Pythagorean Theorem a 2 2 c2= a+ a
2 . 2 c =2a = a we see that if 6,ABC is a 45, 45, 90 triangle We shall now prove the converse of this statement.If AAIBICI is a tri- angle with (a', b', c') p (1,1, VII, then 'AAIBICI is a 45, 45, 90 triangle. We know that is similar to &ABC by SSS Similarity Theorem. Therefore, m L C = 90, m L A = 45, m L B = 45 because corresponding angles of similar triangles have equal measure. The above discussion can be summarized in the following theorem and corollary. Theorem B.Triangle ABC is a right triangle with m L A = m L B = 45 and m C 90 if and only if (a, b, c) (1,1, 42-). Corollary.In an isosceles right triangle, either leg of the tri is equal to one-half the hypotenuse multiplied by 4-2, (s = h 45
Example 1. Find the length of the hypotenuse of a right isosceles triangle if each leg of the triangle has ameasure of 6. (6,6,c) 13. (1,1, .,4-2-1 The constant of proportionality is 6. Therefore, c = 6 4-7 Example 2.Find the length of each leg of an isosceles right triangle if the length of the hypotenuse is 5. (a,a,5) p (1,1, NrEj The constant of proportionality is5 a
5 5 Therefore, a = ,or 2 The problem can also be solved by using the corollary. h s 45 2 5 NTT s45- 2- Problem Set In each of the following exercises, the lengths of a leg and the hypotenuse of a right triangle are given. Which of the measures belong to a triangle similar to the 3, 4, 5 triangle? to the 5,12, 13 triangle? to the 1, Nri; 2 triangle? to the 1,1, 47-triangle? 2.In each part of Problem 1, find the length of the side which is not given.
3.Complete the table for the right isosceles triangle in the diagram. B .111.11001111.a (a) 10 (b) 5 (c) 9 a (d) 6 (e) 3 Nri C A (f)
4.Complete the table for the 30-60-90 triangle in the diagram.
a b B
18 a 94-3 20 12 NT-3-- C b In each of the following exercises the length of one segment in the adjacent plane figure is given.Find the lengths of the remaining segments.
m(AB) m(BC)m(CD)m(AD) m(3B) m(AC) (a) 8 (b) 2 (c) 4 (d) 9 (e) 10 N3 B(1) 8'15
Use the three corollaries in this unit to find the length of AB and BC in each of the following plane figures.You should do all work mentally and write only the answer. C 7.In the diagram, BC1 AC and mt.A = 30
(a)If m(AB) =6,find m(AC). (b)If m(AB) =4Nrrrind m(AC). (c)If m(AC) =9,find m(AB). (d)If m(AC) =6Nfr"3",-find m(AB),
8.Repeat problem 7 if mLA = 60.
.In the diagram, &ABC is an equilateral triangle which is inclined at an angle to plane K. The measure of the dihedral angle C-AB-X is 60; CX is perpendicular to plane K; CD is per- pendicular to AB. Find the measure of each of the following segments if the measure of AB = 6 (a)AC (d) CX (b) BC (e) AX (c) CD (f) BX
Repeat problem 9 if the measure of the dihedral angle is 30;(b) 45.
.pt 80
BALTIMORE COUNTY PUBLIC SCHOOLS
APPENDIX II
REVIEW OF ELEMENTARY SET CONCEPTS by Vincent Brant Supervisor of Senior High School Mathematics
July 1965 92 REVIEW OF ELEMENTARY SET CONCEPTS
1.Member Each object in a set is called a member (or element) of the set.
2. The symbol " " means "belongs to" or "is a member of" or "is an element of."
3.Set Notation (1) The phrase method Example:the set of the (names of the) vertices of ,AABC (2) The roster (listing) method Example:Using the same set as in (1) above: { A, B, C } (3) The rule method Example: Using the same set as in (1) and (2) above: { xx is a vertex of AABC This is read as "the set of all x such that x is a vertex of triangleABC."
4.Infinite set A set which is unending is called an infinite set.
5.Finite set A set in which the members can be listed and the listing terminates is called a finite set.
6.The empty set The set which contains no members is called the empty set or the null10 set. The symbol for the empty set is " 4)"
7.Equal sets Two sets are equal if and only if they have the same member s.
8.Subset Set A is a subset of set B (denoted "ACB ") if and only if each member of A is also a member of B. Every set is a subset of itself; that is, if A is a set, then AC A. The empty set is a subset of every set; that is, if A is a set, then .4)C A. Set A = set B if and only if AC B and BC A.
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1k Set A is a proi-er subset of set B if and only if there is at least one member of B which is not a member of A.
Examples: Let A ={1,3,4 } B ={1,2,3,4,5 } C ={4} D ={2,5,4,3,1} E ={3,5,6 } Then, AC B ; CC B AC D and DC A; hence A = D A and C are proper subsets of B AC A ; BC B ; CC C ; DC D ; EC E 4) is a subset of every set A, B, C, D, E. But E Q B (E is not a subset of B)
9.Disjoint sets Two sets which have no members in common are called disjoint sets.
10.Inter section The intersection (denoted by" n 11) of two sets A and B is the set consisting of all the members common to A and B. Symbolically, A nB ={x I x C A and x c= B} If A and B are disjoint sets, then An B = If two sets are said to intersect (verb form), then their intersection has at least one member; that is, their in- tersection is non-empty. Examples: Let A = { 1, 2, 3, 4, 5 } B = { 3, 4, 6, 7, } C = { 1,3, 4, 8 } D = { 10, 11, 12 } Then, Bn C = { 3, 4 An c = { 1,3, 4 } Bn D = cl)
11.Union The union (denoted by "U " ) of two sets A and B is the set consisting of all the members which are in at least one of the two given sets A and B. Symbolically, AU B={x I x E A or x B} Observe that the connective "or" in mathematics is used in the inclusive sense; that is, x is a member of A or x is a member of B or x is a member of both A and B.
-3- 94 . Examples: Let A ={1,2,3,4,5 } B ={3,4,6,7} C ={1,3,4,8} Then, AU B = { 1, 2, 3, 4,5,6,7 } BU C= { 1,3, 4,6,7,8 } 12.Cartesian The Cartesian Product of two sets A and B (not necessarily Product different) is the set of all ordered pairs in which the first coordinate belongs to A and the second coordinate belongs to B. The Cartesian Product of A and B is denoted as; A X B (read "A cross B") Example: Let A = { a, b,c,} and B = { 1, 2 } Then, A X B = { (a, 1), (b, 1), (c, 1 ), (a, 2), (b, 2),(c, 2) } 13.Relation A relation is a set of ordered pairs. or A relation from A to B is a subset of A X B. Example: Let A = { a, b,c }; let B = { 1, 2} Some relations from A to B are: Relation R = { (a, 1),(b, 2), Relation Q = { (a, 2) } Relation T = { (a, 1),(b, 1), Of course, A X B is also a relation. Also, the empty set, 4, is a relation. 14, Domain of a The domain of a relation is the set of all the first coor- relation dinates of the ordered pairs of the relation. 15.Range of a The range of a relation is the set of all the second coor- relation dinates of the ordered pairs of the relation.
Example:Let Relation T = { (a, 2),(b, 3),(a, 5),(c, 2) } Then, Domain of T = { a, b, c } Range of T = 5} r-
-4- 95 16.Function : A function from A to B is a special kind of relation such that for each first coordinate there is one and only one second coordinate. Stated differently, no two ordered pairs of a function may have the same first coordinate.Hence, the first coordinates in the ordered pairs of a function must be different.However, the second coordinate may be the same.
Examples:D = { (1, a),(2, b),(3, c),} is a function. E = { (1, a),(2, b),(1, c) is not a function; it is a relation. K = { (1, a),(2, a),(3, a) } is a function-- in a particular, a constant function. Note: Domain of K = { 1, 2, 3 } Range of K= { a }
I = { (x, y) I y =x } is a function. Note: Domain of I is the set of real numbers. Range of I is the set of real numbers.
17.Tests for a function : (a) The ordered pair test A relation is a function provided every first coordinate is different. (b) The vertical line test A relation is a function provided every vertical line intersects the graph of the relation in exactly one point. If at least one vertical line intersects the graph in more than one point, then the graph does not represent a function, but a relation.
18.Comparison of f and f(x) : In the function, f = { (x, y)I y = 5x } (a) the function f is the entire set of ordered pairs.
(b)f(x) is not the function.
(c)Ilylt,"f(x) ",H5x" are different names for the second coordinate.Thus the function f might be written in different forms, as:
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f = { (x, f(x) ) I f(x) = 5x } or f = { (x, 5x)I x is a real number } (d) The function is not "y = 5x". Instead, "y = 5x" is a sentence which defines the function.This sentence assigns to each real number used as first coordinate another real number 5 times as large for its corresponding second coordinate. "f(x)" is read "f at x" or "f of x" or "the value of the function f at x". The function, f = { (x, y)I y = 5x } is read: the function f defined by y = 5x HOMEWORK ASSIGNMENT
Study assignment: Study pages 1- 5 in this Review of Elementary Set Concepts Written assignment:
1.Construct the cartesian product of A X B where A = { r, t, k } and B = { 6, 16, 1526 }
2.Construct the cartesian product of R X S where R = { 0, 2, 4, 6 } and S = { 1,3,5 }
3.Construct the cartesian product of E X D where D = { } and E = { 1 } Directions for Examples 4 - 13: Each of the following relations is a relation from R to R where R is the set of real numbers.In each example, complete the following parts:
a.Draw the graph of the relation. b.State whether the relation is a function. c.State the domain of the relation. d.State the range of the relation.
4.R = { (a, y)I y = x }
1 5.S = { (x, S(x) ) I S(x) = 2 x }
6.T = { (x, y)y > 5x
7.U = { (x, y)y =x2 }
8.V = { (x, V(x) )V(x) = 2x - 5} 9. W = { (x,y)j y =x3 }
10. X = { (x,y) x2+y2= 25 }
11.Y = { (x, y)I xy = 12 }
12. G = { (x, G(x) ) I G(x) = 5 13. H = { (x, y)I x = 3 }
-7- Trigonometric Functions Let P with coordinates (x, y) be a point on the terminal side of an angle with measure 0 in standard position.Let r = qx 4 + y-2 represent distance of P from the origin. P (x, y)
x 1.The Sine Function
Sine (or sin) = { (0, sing) I sin 0 = } Observe that: (1)0 is a number----a number that measures the magnitude of the angle in degrees or radians.
(2)sin 0 is a number----the number which is assigned as the second coordinate by the defining sentence sin 0 = Y-r (3)sin 0 is NOT the function. sin 0 is the name of the second coordinate in the sine function. sin 0 is the VALUE of the sine function at O. (4) Each ordered pair of the sine function has real numbers for the first and second coordinates. These ideas apply as well to the remaining trigonometric functions.
2.The Cosine Function
cosine (or cos) = { (0, cos 0) I cos 0 = }
3.The Tangent Function tangent (or tan) = { (0, tan 9)I tan 0 =I }
4.The Cotangent Function cotangent (or cot) = { (0, cot 0) I cot 6 = Y 5.The Secant Function
} secant (or sec) = { (0, sec 0) I sec 0=13.--c-
6.The Cosecant Function
cosecant (or csc) = { (0, csc 0) I csc v = -8- 99