Introduction to Trigonometry (Zambak)

144 Pages·2008·17.85 MB·English

by  İlker Tantürk

Introduction to
TRIGONOMETRY

iter Tare

Zambdk

, ee OS on

Tint hook wax matraducnay to miganametry and tricanonerne ayplications for bok
school students. Uefore berianing tis Hook, sticents sould be foliar sith she basic
Droperties of of angles, mianpler and chctes, and be able to manipulete and fastorise
‘lgebraveexpretione The boak ve dided ont fre sections, structnted ae flo,

‘he frst section covers the concepis ef angle, are and circle ant sets basie properties
I also mroduoes the degree and radian waite of augle mesure
he secvor section 8 we meroduction to right triangle tipgnometey, and introduces
Gey raver e cizauleea OheSy sed thd ebarat Ted wie ee
‘he third secion deals with eigonomenry oa the wait circle, and ead students to a
understanding of the sie maun tgonancere funenons and thes properes
Tie fourth sectym bua on tc matertal ofthe premons sections aad presents the mein
ISSEECISTT oar an TIS ins gen OHA
‘mang examples of aw they are used
The fifth seton concludes the book with a look at some applied trigonometry problems,
Drinaioy togette the waterial that sadn base sbuaied during tbe course
{me ok has heen designed 29 beam effete tearing aid, and mice allo he features
‘af the Zembake Moduler Sytem bigh whoo! made teaching veriee
SRS aera snc cheer pwnd al ere te dear tet aaa
ov eooepts and math covered previously 1m the text
SelEesr Check Yourself sertions at Hey potnts tw te text allow students to chee? there
understanding, aod busld student confidence for further study
sercises at the ret af each rection aive sludeats a chance to ase the skills wna
lechnges they have learned (sole both pure and applied prodtems
A chapler semmara at the end of the book provides « conecive cextew of the min
antent of the chapter. Included tn the sumovary ere « set af ccucept check questions
which as students (0 expla hey concepts and wdeas in their own words
eciew tests at hs end of te book cover she material in the hole bok, ant lp tw
-repare students fr axems

Mam Jie a collagen wer of vont help dari be eit af tix boot va
to thank everybody who helged me ot Zambak publieations, especally Mustafa Kurt
‘tod Cem Giray Special tents also go tamil Gesimagla for hi patent Iypeseticg cod
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Definition boxes phe Jormat deseriptions of new
‘concepts Theanen lover vichude propositions that

an be proved. The injormasion tm these buses is

} ery important for Jurther understanding and for)
L sofviag examples

Lxamoles inciade problems related tothe pic ant

tess solani, veh explanations The examples aie

numbered, 40204 ca fd hem casi te bok

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rndertanding af whet you hove just stacied Linergiange bass seang
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avon, g6 throught qour working agate end eee,
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reminds you af mere! sear és ated 19 te rape yox
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fo tee all rhe problems wich ao vor Reve a ras
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‘question mean the gueeion J8 Jor studenrs 'b0
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tnvthe materat you studied. Go bach aver the mate tf
tou are not sure about an avsner ta Concept Check
‘queen After the Concrh Cherk there aie Chapter
Remew Tests im smcreaseag order of augpentty with
tnnitipte chovee questions. The aaswer key for theve teste
t5-at the back ofthe boob
TRIGONOMETRY

FUNDAMENTALS OF
TRIGONOMETRY ....cese ses s10
AC THEUNIT CRE “e
| ANGlES an cimecrion ie

I. The Coneapt ot Rie 13

3 Directed arcs fr

S2Radien 2 17
fate Measure aot’
', PRIMARY OIRECTED ANGLES a
4. Coremnal Angles 2

21 Printer Directed
‘angles ane ares as

EXERCISES 1

AIGHT TRIANGLE TRIGONOMETEY .

1A. TRIGCHOMETRG RATION
2 Specil Tested wid Ratios. 32
|a Thlecnomern IDENT 2
2. Simpihang ngonemehie
s.veriving tridenomatic
Hdenitttes
Ente pe eee Gramie

EXERCISES 2

TRIGONOMETRIC FUNCTIONS
OF REAL NUMBERS

A TRIGONOMETRIC FUNCTIONS
ihe Sine Function es
The Cosme Func 51
Tie Fe nh mE
{ca-canne tmcancxaetmc vals 60

1 tiga ometric Values of
Quasi Angles 0
2Usiing a Ketorence Arle a

4. trigerometric¥alues Uf
Or anion vows

EXERCISES 3
TRIGONOMETRIC THEOREMS

AND FORMULAS. sevussscres e078

‘A. TRiconowertic THEOAENE
1. Ihe Law of Cosien
2. Theta ol Sinee
‘fa teanale
4. Farther Teeaerens Ctiorel
18 TwiconoueT#Ic FORMA
Suen
nemo)
fenton)
Hthed fee incr
Formulae

BEG Fanta
Sieatni se)

Peeeedusum terrain
enh Bore

EXERCISES 4
APPLIED THGONDMELKY
PXPRCISES 5

concarr cHucK

(CHAPTER REVIEW TEEY 1 |
INTRODUCTION

‘The tmanch of mathemanes whch we roday call yonomery develazed fram she study of he
danenishaps between the sides and angles of a nght mands and also fom the studs of salar
luuuiges. dae past uiexe was no couvv9t uf xououmctne funcuen alliough people hue how te
apple mgonomeme aropents to sve silerentyaoleas,

‘Tigonouuy was fist used for cartogrpin, as.rououy surveying ned naigaton
Jor sample, lie Lpypllas used properties of siullar Wiaugls tee wit aad
Surveyane mid We consteuciion of pyramids. The Babvioniaus also used
tfteles an ef rhelewgmns of me canvas emse subverted the aces

jate in aati eevee olloislionaire

fice othr fimerons and hed discovered and proved several base theorems of
UWgonoumetr Jor Sous trues it a Ha pluie aud Uraes ou we Surtuce of a
Avedte Coan en iita star eee aim ore

Imgomometre and spherteal mgonmner. Several mamnemaricans mugeesred using — 1 tmaread ot
r= G0: ths ested paeduces the wodura values uf the tngouenetew fuuetons, Tae Muslis alse

Jnuoduved Ue gular tsangle in splirical ulgonomeun. Al ef these disvorenes
Ue Ouveu 3f Mocea fr Uae five Ua pravers required bv Musa to. Blush
Scwnnsts alse profieed verv accurate mgonomeme tales. For exanvgle.rhett
fables ofsme and tangent values constucted for inewements af YON of dese
‘vere accurate for better than one past m 700 athe,

“Sui Vets Buzcant was the fra sehelar who pur forward rhe Basile af sphencal
sand plane trapometn. He analzed the ome Sanction seensaealy ane develoyed
le eluvang steuuues

sma)

fan atom hetons ae

sm 2e 2am a-cos., and

vuropean senclars devwea the same formula ustrg met awn methors after mans centuries and ata
1 have work, The ere astionamier Nasir at Tim ar Tus tots the Bcoh of the Transrersal Figure
whieh was the fist teatueat of slave and spheneal tigmauste as an independent aethcuna Deal
The Wenern Wild became asare of Mum migmametr through the
tuuilanone of Arabiecetenemy ln dboaks, begunnin'm the 12th cent
‘The fuse max Wester noxk en saigonomctry was vantea Wy the Corman
asunemer and udemammn Joho Mle, wy was kucwn as
eyiowontuaus. la he ext cealury We Guns aatzauouee Geostes
Jraehim, knionm as hens, nredupad the meen concept af mesnomeme
Famcrions as arin mnstend of as rae lengths af cetam Imes. hs believed
tae the term tmigonomety wie Saet used ky the Gensaan mathemanean

7, ssaouene Daxhdomaeus Taneus who paissd apse of xk
lous tar was later tensed ant called nigonemetnia sive de amen

Iriangulae ix 1600, Va word is dete ous two Grsek woels, Iigunon au. neti whieh wie

trhuige an uieasureaueut avepeclivds: The Ateach suathuuLean Laagois Vidke iauodueed Uae

lar mangle mo saheneal mesiomer and stated the mulnple ae forlas

for an ax andeos ex m terms of powers of an x and cxe x

te Scottish maheuatiian John Nagiee as amie nortan coatsbuious
the developinens nf mzonoment m she Ith cents: Napier decreed memory ands
calles Napier aalogies foe slag uliqae spre rignales A uot exe si
cenrun alter Navies rork mas pushed. Isaac Newton mented atTerencal and
Jlepesl caleulus, Que of he fouusdaives of this work was Newta's
‘epueseut un of diferent Kunetions ws lial seis ol pumers ox, hus Newton
developed « scies Bor sin x and similar sures tor cos x aud tue a. With the
amvention of cletlue. the tgonometne fnictone wexe tcken over mito aualene where they sal lay

Important roles m both pur: and eyphet mathemancs. Tinally, an the 16th
fenmury the Swiss warnemaeian Leonhard Huet defhed the mgsnomerte
fuxcuons <n terms of complex aumbers. Ths made the shoe subject of
Uuigonoamotr just one of Use many appliatous of complex muubers ant sowed.
Unt tie basic lays uf igenomtey were sims comsecucnees of uae anianotie of
‘many other arcas of pure and applied meth=matis.

‘Today there are im mam branches of ruonowers, namely lene mgonomen
and sphenicd tmgonometn: Fle mgonomeny cocems mages m a oo
aleare the properes at taatoron. te propaga ortgnt and orherpevioa¥e phenamana. ¥prerial
monomer deals yath mangles an the surfaces of phere, Ths fm of migonomert: 6 wal used
tw astrououn and for lus-ranwe sigue.

1a dus Louk ve wil bogus bur sth uf txgunounet by louie a pine yunoaset alas basse
pricigtes

A. THE UNIT CIRCLE

_ FUNDAMENTALS OF TRIGONOMETRY

‘A eaele a the set of pants whneh are all rhe same distance
‘rom a fed pant m a plane, The fixed poms the center of
Ie ste, ana rhe airines of me pons Is clled the race
ofthe cate

‘The standasd equanon of ¢ quele wath radus r and center
case a GN

sii
Tn rhe ccondmate plane, 2 etcle which fas 1s center C atthe orig an wadsus | amt
called 3 unm que

Subsuuuuue dhe eoudumts of se ongun O10, 0) an Whe standaré equation ofa arele wes Us
D1 Oy = Presa bgt. Thasts rhe equnnon ofa mr once

‘Tus we can wnate the set of posta whch
au wnat eele ae

C= (ey aye Rand t= 0

I other words, « unit cle ws made up ofall
the poms ¢r, 4) whose and y components
sonst the equanon v7 1 a" =. The
unter of dis ile i he rian ad its caus
We car Kind Une vintereepts and griercepts
of the ere in the courdiaate plus by using

tory 0.x + 0° Lsea’ ~ Lat
So he x-imteroepts of We unt eile
sre ACL ©) ad 471,

fore 00+ gf Lsog'~ Landy Li
So rey meerceprs of the umt ence ate
CO, Iyand B'0. bs