chapter 1 - analytic trigonometry

8/21/2019 Chapter 1 - Analytic Trigonometry

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BETU1013 Technical Mathematics Chapter 1

BETU1013 Technical Mathematics

ONLINE NOTES

Chapter1

Analytic Trignmetry

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Faculty of Engineering Technology

(FTK)


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CHAPTER 1: ANALYTC TR!"N"#ETRY

Reference&'(ng) C* +,010-)Algebra and Trigonometry +,the.*-* /iley

Steart) "* +,01,-) Calculus +the.*-* Br2s$Cle) Cengage Learning

Aler) S* +,01,-)Algebra & Trigonometry) /iley

"$%ecti&e'&

1* Eplain the cncepts 4 trignmetry*,* Use apprpriate meth.s t sl5e trignmetry pr6lems*

Content':

1*1 Angle an. Their Meas(re

1*, 7ight Triangle Trignmetry

1*3 Cmp(ting the 8al(es 4 Trignmetric 9(nctins 4 Ac(te Angle

1*: Trignmetric 9(nctins 4 ;eneral Angles

1*< Unit Circle Apprach= %rperties 4 the Trignmetric 9(nctins1*> ;raphs 4 the Sine an. Csine 9(nctins

1* ;raphs 4 the Tangent) Ctangent) Csecant) an. Secant 9(nctins1*? Trignmetric I.entities

1*@ S(m an. i44erence 9rm(las

1*10 (6leAngle an. al4Angle 9rm(las1*11 %r.(ct tS(m an. S(m t %r.(ct 9rm(las

1*1, The In5erse Trignmetric 9(nctins

1*13 Trignmetric ED(atins

1Analytic Trigonometry

11 AN!LE AN* THER #EA+RE

A ray, r half-line, is that prtin 4 a line that starts at a pint n the line an. eten.s

in.e4initely in ne .irectin* The starting pint 4 a ray is calle. its &erte. See 9ig(re

1*

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I4 t rays are .ran ith a cmmn 5erte) they 4rm an angle /e call ne 4 the rays

4 an angle the initial 'i/e an. the ther the ter0inal 'i/e The angle 4rme. is i.enti4ie.

6y shing the .irectin an. am(nt 4 rtatin 4rm the initial si.e t the terminal si.e*

I4 the rtatin is in the c(nterclc2ise .irectin) the angle is o'iti&e2 i4 the rtatin is

clc2ise) the angle is negati&e See 9ig(re ,*

An angle is sai. t 6e in 'tan/ar/ o'ition i4 its 5erte is at the rigin 4 a rectang(lar

cr.inate system an. its initial si.e cinci.es ith the psiti5exais* See 9ig(re 3*

/hen an angle is in stan.ar. psitin) the terminal si.e ill lie either in a D(a.rant) in

hich case e say that lie' in that 3ua/rant, r ill lie n thexais r theyais) in

hich case e say that is a 3ua/rantal angle 9r eample) the angle in 9ig(re :+a- lies

in D(a.rant II) the angle in 9ig(re :+6- lies in D(a.rant I8) an. the angle ( in 9ig(re :+c-

is a D(a.rantal angle*

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/e meas(re angles 6y .etermining the am(nt 4 rtatin nee.e. 4r the initial si.e t

6ecme cinci.ent ith the terminal si.e* The t cmmnly (se. meas(res 4r anglesare degrees an. radians*

egrees &

One cmplete re5l(tin

One D(arter 4 a cmplete re5l(tin ne right angle

One .egree eD(als >0 min(tes) i*e* *

One min(te eD(als >0 secn.s) i*e* *

7a.ians &

One cmplete re5l(tin ra.ians One ra.ian is the angle s(6ten.e. at the center 4 a circle 6y an arc 4

the circle eD(al in length t the ra.i(s 4 the circle*

1 ra.ian

Note: ra.ians

ra.ian 1 ra.ian

egrees 0 30 :< >0 @0 1?0 3>0

7a.ians 0

Note& Let an. 6e psiti5e angles*I4 )they are cmplementary angles*

I4 )they are s(pplementary angles*

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14 R!HT TRAN!LE TR!"N"#ETRY

9r any ac(te angle 4 a right angle. triangle OAB+4ig(re shn-

9(rther) ) )

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Co0le0entary Angle'2 Cofunction'

T ac(te angles are calle. co0le0entaryi4 their s(m is a right angle*

Theorem&C4(nctins 4 cmplementary angles are eD(al*

Note& The 4(nctins sine an. csine) tangent an. ctangent) an. secant an. csecant are

c4(nctins 4 each ther*

Trigono0etric Ratio' of Allie/ Angle':

9r +ac(te-

Sme (se4(l res(lts t nte&

Tratis

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15 C"#P+TN! THE 6AL+E "F TR!"N"#ETRC F+NCT"N "F

AC+TE AN!LE

Co00only +'e/ Ratio' 57

, 89

, an/ 7

Angle'The trignmetric ratis assciate. ith the angles 57

, 89

, an/ 7

are (se. 4reD(ently

in pr6lems in5l5ing trignmetry* Their eact 5al(es can 6e easily 6taine. (sing

either an eD(ilateral triangle +4 si.e t (nits- +4ig(re 1- r an issceles rightangle.

triangle +4ig(re ,-*

+9ig(re 1- +9ig(re ,-

Sme i.ely (se. ratis&

sin>03

,

= cs>01

,

= tan>0 3 =

sin301

,

= cs30 3

,

= tan301

3

=

sin:< 1,

,,

= = cs:< 1,

= tan:< 1 =

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18 TR!"N"#ETRC F+NCT"N "F !ENERAL AN!LE

The ign' of the Trigono0etric Function'

The Cartesian aes .i5i.e a plane int : D(a.rants&

@00 1stD(a.rant 1?0@0 ,n.D(a.rant

,01?0 3r.D(a.rant 3>0,0 :thD(a.rant

y,n.D(a.rant 1stD(a.rant

x+ve- x+Fve-

y+Fve- y+Fve-

x3r.D(a.rant :thD(a.rant

x+ve- x+Fve-y+ve- y+ve-

Emply a rectang(lar cr.inate systems r Cartesian systems&

r

a=sin y

r

b=cs

ba=tan

a

b

b

r

a

r=== ct=sec=csc

The trignmetric rati 4 any angle is then 6taine. 6y .etermining the D(a.rantcnnecte. ith the angle) the sign 4xr yithin that D(a.rant an. the assciate. ac(te

angle ma.e ith the +ve+r -ve-xais* Irrespecti5e 4 in hich D(a.rant the angle lies)

r is alays ta2en as +ve*

A (se4(l ai. is the .iagram shing hich trignmetric ratis are +vein each D(a.rant&

y

Sine All

+sin, cos, tan-

x

Tangent Cosine

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Th(s) a- sin,:0 3r.D(a.rant tangentnly psiti5e sin sin *,:0 >0 0?>>0 = =

6- cs+ - 30 :thD(a.rant cosinenly psiti5e

cs+ - cs * = =30 30 0?>>0

;ua/rantal Angle': 3>0),0)1?0)@0)0

Coter0inal Angle'

T angles in stan.ar. psitin are sai. t 6e cterminal i4 they ha5e the same terminalsi.e*

Eample&

9r eample) the angles >0G an. :,0G are cterminal) as are the angles :0an. 3,0G*

Note:

1* is cterminalith k , ) kis any integer*,* The trignmetric 4(nctins 4 cterminal angles are eD(al*

Eample& -,sin+sin k =

Reference Angle'

Let .ente a nnac(te angle that lies in a D(a.rant* The ac(te angle 4rme. 6y theterminal si.e 4 an. either the psiti5e ais r the negati5e ais is calle. the

reference angle4r *

9ig(re :3 ill(strates the re4erence angle 4r sme general angles* Nte that a re4erence

angle is alays an ac(te angle* That is) a re4erence angle has a meas(re 6eteen 0G an.

@0G*

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Theore0: (Reference Angle')

I4 is an angle that lies in a D(a.rant an. i4 is its re4erence angle) then

ctctsecseccsccsc

tantancscssinsin

===

===

here the F r sign .epen.s n the D(a.rant in hich lies*

19 +NT CRCLE APPR"ACH2 PR"PERTE "F THE TR!"N"#ETRC

F+NCT"N

E&en < "// Proertie'

7ecall & A 4(nctin is sai. t 6e eveni4 f f+ - + - = 4r all in the .main 4 f) an.

a 4(nctin is sai. t 6e oddi4 f f+ - + - = 4r all in the .main 4 f*

Theore0:sin+ - sin+ - = cs+ - cs+ - = tan+ - tan+ - =

csc+ - csc+ - = sec+ - sec+ - = ct+ - ct+ - =

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1 !RAPH "F THE NE AN* C"NE F+NCT"N

The !rah of y ='inx

= ,0)sin xxy

Characteristics 4 the Sine 9(nctin &

main & all real n(m6ers

7ange & 1 1y

%eri. & ,

Symmetry thr(gh rigin & sin+ - sin =

O.. 4(nctin

x intercepts & ****) ) ) ) ) ) )****** , 0 , 3

y intercept &

ma 5al(e & 1 ) cc(rs at x = ***** ) ) ) ) ***3

, ,

chapter 1 - analytic trigonometry

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