Conceptual Trigonometry Part I A Companion to S L Loney by Chandra Sekhar Kumar IIT Solution

367 Pages·2019·1.421 MB·English

by  Chandra Sekhar Kumar

Conceptual Trigonometry
Part I
A Companion
to
S. L. Loney’s
Plane Trigonometry Part I
By
Chandra Shekhar Kumar
IntegratedM.Sc. inPhysics,IITKanpur,India
CEO&Co-Founder,AncientKriyaYogaMission
CTO&Co-Founder,AncientSciencePublishers
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Preface
Back in 1990, solving the problems and exercises given in the
text-bookof PlaneTrigonometryPartIbyS.L.Loneyhadater-
rorizing effect on me, irrespective of the outcome of the countless
hours, full of perspiration and inspiration, laced with joy and sur-
rendering to the sheer beauty and elegance of each problem, sub-
problem, ... woven with multi-concepts.. Whenever stuck, I used
torevisetheconceptsembeddedinthetext-bookandrelatedrefer-
ences,monographs,takeabreakandstartallover... anirresistible
journey... backnforthbetweentheclassicsofLoneyandothers.
Astimegrew,Iendedupstockingahugepileofsheetscomprising
ofmynotesasanendeavortosolveanddevourtheentirebookand
beyond (needless to mention that I laid my hands on everything I
couldinmypursuit).
Somewherein2005,Istartedcollatingandorganizingmynotes
toinstillcoherenceandcapturetheeleganceintheflow.
Thepresentworkisanoutcomeofthispursuit,whichwillserveas
acompleteguidetoprivatestudentsreadingthesubjectwithfewor
noopportunitiesofinstruction. Thiswillsavethetimeandlighten
the work of Teachers as well. This book helps in acquiring a bet-
terunderstandingofthebasicprinciplesofPlaneTrigonometryand
in revising a large amount of the subject matter quickly. Care has
beentaken,asintheforthcomingones,topresentthesolutionswith
multi-concepts and beyond in a simple natural manner, in order to
meetthedifficultieswhicharemostlikelytoarise,andtorenderthe
workintelligibleandinstructive.
Thisworkcontainsseveralvariationsofproblems,solutions,meth-
ods,approachestoenrich,strengthenandenliventheinherentmulti-
concepts.
Ancient Science Publishers ChandraShekharKumar
July,2018.

List of Chapters
Preface i
1 MeasurementofAngles 1
1.1 SexagesimalandCentesimalMeasure. . . . . . . . . . 1
1.2 CircularMeasure . . . . . . . . . . . . . . . . . . . . . . 7
1.3 TheRadian. . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 MeasurementofAnyAngleinRadians . . . . . . . . . 14
2 Trigonometrical Ratios for Angles Less Than A Right
Angle 21
2.1 TrigonometricalRatios. . . . . . . . . . . . . . . . . . . 21
2.2 RelationsandTrigonometricalRatios . . . . . . . . . . 29
2.3 ValuesofTrigonometricalRatios . . . . . . . . . . . . . 36
3 SimpleProblemsinHeightsandDistances 41
3.1 SimpleProblems . . . . . . . . . . . . . . . . . . . . . . 41
4 ApplicationsofAlgebraicSignstoTrigonometry 51
4.1 Tracingthechangesintheratios. . . . . . . . . . . . . 51
5 Trigonometrical Functions of Angles of Any Size and
Sign 55
5.1 AnglesofAnySizeandSign. . . . . . . . . . . . . . . . 55
6 GeneralExpressionsforAllAnglesHavingAGivenTrigono-
metricalRatio 65
6.1 GenericValues . . . . . . . . . . . . . . . . . . . . . . . 65
6.2 TrigonometricalEquations . . . . . . . . . . . . . . . . 73
7 Trigonometrical Ratios of The Sum and Difference of
TwoAngles 85
7.1 AdditionandSubtractionTheorems . . . . . . . . . . . 85
7.2 ProductFormulae. . . . . . . . . . . . . . . . . . . . . . 88
7.3 ConverseFormulae . . . . . . . . . . . . . . . . . . . . . 96
7.4 TangentofTheSumofTwoAngles. . . . . . . . . . . . 100
8 TheTrigonometricalRatiosofMultipleandSubmulti-
pleAngles 103
8.1 MultipleAngles . . . . . . . . . . . . . . . . . . . . . . . 103
8.2 SubmultipleAngles . . . . . . . . . . . . . . . . . . . . . 114
LISTOFCHAPTERS
8.3 Anglesof9◦,18◦,36◦,81◦ . . . . . . . . . . . . . . . . . 128
9 IdentitiesAndTrigonometricalEquations 133
9.1 TangentofTheSumofAngles . . . . . . . . . . . . . . 133
9.2 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.3 TrigonometricalEquations . . . . . . . . . . . . . . . . 145
10Logarithms 159
10.1 Characteristics,MantissaandProperties . . . . . . . . 159
11TablesofLogarithmsAndTrigonometricalRatios,Prin-
cipleofProportionalParts 165
11.1 ProportionalParts . . . . . . . . . . . . . . . . . . . . . 165
11.2 LogarithmicSines,TangentsAndSecants . . . . . . . 172
12RelationsBetweenTheSidesandTheTrigonometrical
RatiosofTheAnglesofAnyTriangle 177
12.1 Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
12.2 SidesAndAnglesofATriangle . . . . . . . . . . . . . . 180
13SolutionofTriangles 195
13.1 SolutionofRight-AngledTriangles . . . . . . . . . . . . 195
13.2 SolutionofNon-Right-AngledTriangles . . . . . . . . . 197
13.3 TwoSidesandIncludedAngle . . . . . . . . . . . . . . 202
13.4 AmbiguousCase . . . . . . . . . . . . . . . . . . . . . . 213
13.5 OneSideTwoAnglesvs. ThreeAngles . . . . . . . . . 220
14HeightsandDistances 225
14.1 InaccessibleObjectandDistantPoints . . . . . . . . . 225
14.2 AngleSubtendedatTwoPoints . . . . . . . . . . . . . . 236
15PropertiesofATriangle 259
15.1 AreaofaGivenTriangle . . . . . . . . . . . . . . . . . . 259
15.2 TheCirclesConnectedWithATriangle . . . . . . . . . 264
15.3 OrthocenterandPedalTriangle . . . . . . . . . . . . . 272
16OnQuadrilateralsAndRegularPolygons 291
16.1 AreaofaQuadrilateral. . . . . . . . . . . . . . . . . . . 291
16.2 RegularPolygons . . . . . . . . . . . . . . . . . . . . . . 296
17TrigonometricalRatiosofSmallAngles,AreaofACir-
cle,DIPofTheHorizon. 307
17.1 RatiosofSmallAngles . . . . . . . . . . . . . . . . . . . 307
17.2 AreaofACircle . . . . . . . . . . . . . . . . . . . . . . . 311
17.3 DipofTheHorizon . . . . . . . . . . . . . . . . . . . . . 314
18InverseCircularFunctions 319
18.1 IdentitiesandEquations . . . . . . . . . . . . . . . . . . 319
19OnSomeSimpleTrigonometricalSeries 335
19.1 SimpleSeries . . . . . . . . . . . . . . . . . . . . . . . . 335
20Elimination 347
LISTOFCHAPTERS
20.1 EliminationofUnknownQuantity . . . . . . . . . . . . 347