8/12/2019 TRIGONOMETRIC Ratio & Identities Theory_H

1/14

HI_JU

"hik`sjfuhirpjys`es.`k" 9

b8ofs.fbarj6 vuq`fr ,di ndWnbaof,(_r`ldkdhatr`e rit`d & `oakt`t`as)

'fEk 'b8fofs.fbabr'kfs xzjo v{fgfsi [b8fofs.f* rBff [babr* ns bayog euf lUI rBff cnof vBfW lfsrf lU & [[,o b8fJfqt ojJfqtfvfsi vfUg ofs.ffsi of af`u**X

ofs.f Iklna 7ofs.f kj xcW bog.f os fgbAJfo beQkq os nf s{f 4fqafd oj af` lUI dfNrbdo bog.f `zfgbAJfo Jfqtf vfUg bog.f oj 4fw.fWuos efk vibra bNBfbr ofs.f oj vbQra Jfqtf olyfrj lUI 4fq.fWu beQkq 'fj"fW olyfrf lUI 6bk 4fqafd oj bk'ff dfafdrW(ikt`endefw`sa)lUX rfs ofs.f /fufSao lfsrf lU rBff 6bk 4fqafd oj bk'ff kb{f.ffdrW (endefw`sa) lUX rfs ofs.f T.ffSaolfsrf lUI

ofs.f os af`u oj `)br6f Uystahs Bdr Haisurahakt db Iklnas 7,o ofs.f buAu `)br6fsi asi af`f tf norf lUI

9. "ffb"Ro&`)br bezbR'f&`)br 7cn `)br asi ,o nA w.fW dmfj6 4fqafd of52=

9Jffx ,o bHxzj (),,o bHxzj of

2=

9Jffx

,o bauR ( ) vfUg ,o bauR of2=

9Jffx ,o nUo.H ()olyfrf lUI

,o naofs.f 8 0=, 9 8 2=, 98 2=

3. 'fbro&`)br sip& `)br 7cn `)br asi ,o nA w.fW dmfj6 4fqafd of?==

9Jffx ,o xzsH (l),,o xzsH of

9==

9Jffx ,o

bauR vfUg ,o bauR of9==

9Jffx ,o nUo.H olyfrf lUI

,o naofs.f 8 9==l6 9l 8 9== 6 9 8 9==

ufsR 7 "ffb"Ro&`)br (Uaxilas`hin systah)asi bauR ,di nUo.HX 'fbro&`)br (Eaktas`hin Uystah)ns bJfQu lUIkfsufsi `)br6fsi asi bpQl Jfj bJfQu lUI

5. dmfj6 `)br E`reunir Uystah Sio`ik Haisurahaktbonj dmf pf` }fgfX btnoj yAefcW b8f_6f os egfeg lUX osQkz `g euf6f x6f ofs.f ,o gsbH6u olyfrf lUI cn `)br assiaf` oj cofcW gsbH6u (e)lUI

pwbo cofcW b8f_6f os dmf oj `bgb/f 3lfsrj lUX cnby, `zfgbAJfo Jfqtf of ,o nA w.fW pOog 3gsbH6u ofs.f eufrf lUIbdNrfg wdWo rb8f_6f os ,o dmf asi ryAefcW os pf` }fgf euf6s x6s ofs.f oj af` 9gsbH6u lfsrj lUI 6l Ffr lU bo ,o lj

dmf os nafu pf` dmf os osQkz `g nafu ofs.f vQrbgr ogrs lUI vr% yAefcW of ,o pf` dmf os osQkz `gr

gsbH6u

of ofs.f euf,xfI cn ofg rb8f_6f os ,o d mf asi yAefcW of pf osQkz `g gsbH6u ofs.f eufrf lUX

rfs 8r

6f 8 r

8/12/2019 TRIGONOMETRIC Ratio & Identities Theory_H

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HI_JU

"hik`sjfuhirpjys`es.`k" 3

9

I

C

9D

9

8 9 rio`ik

(`)

9

I

C

9D

9

8 9 rio`ik

(``)

3

I

C

9D

9

8 3 rio`ik

(```)

3

I

C

9D

9

8 3 rio`ik

(`v)

ufsR 76bk ofscW bpQl u bk6f lfs tebo ofs.f oj af` lfsX rfs 6l afuf tf norf lU bo 6l af` gsbH6u asi lUhkflg.ffBfW 8 91 91 gsbH6u

#dmfj6 Jffx of {fs8fYy %

{fs8fYy 83

9r3 dxW cofcW

gsbH6uX bHxzj ,di xzsH os ejp nAeQ/f Sanit`dk catwaak rio`ik, oalraa iko lrioa 7

3 rio`ik 8 0= 8 9==l

Q6wuofs.ffsi os by, b8fofs.fbarj6 vuq`fr _r`ldkdhatr`e Sit`ds bdr Ieuta Iklnas 7afuf ,o 4fwb.fWr bog.f D], DIns zfgAJf lfsrj lUX vfUg D] oj bNBfbr ro 4fwarj lUX cn `zofg ofs.f ID]eurf lUI

4fwb.fWr bog.f asi ,o beQkq ]ysi vfUg ]ns fgbAJfo bog.f DI`g yAEf ]HVfjipsiI

naofs.f b8fJfqt HD]asi D] bdo.fW lUX ]HyAe rBff DH vf/ffg lUX

ve ofs.f ID]os by, b8fofs.fbarj6 vuq`fr 6f Yyu cn zofg `bgJffb"fr bo6s tfrs lUiI

D]

H], `.a.

o.f

yAe,ofs.f ID]of s`kalUI

D]

DH, `.a.

o.fW

vf/ff,ofs.f ID]of eds`kalUI

DH

H], `.a.

vf/ff

yAe, ofs.f ID]of tiklaktlUI

H]

DH, `.a.

yAe

vf/ff, ofs.f ID]of edtiklaktlUI

DH

D], `.a.

vf/f

o.fW, ofs.f ID]of saeiktlUI

H]

D], `.a.

yAe

o.f, ofs.f ID]of edsaeiktlUI

btn af8ff }fgf eds`kacofcW ns oa lfsrf lUXvBffWr~ 9 eds ID], ID]of Rarsao U`kaolyfrf lU vfUg btn af8ff }fgf s`kacofcW ns oa lfsrf tfrf lUX vBffWr~9 s`k ID], ID]of Edvarsao s`kaolyfrf lUI

nJfj b8fofs.fbabr6 vuq`fr dfNrbdo niV6f, lUI nib{f@r asi cu vfC vuq`frfsi ofs a'f% cn zofg byVff tf norf lU&s`k ID], eds ID], tik ID], edt ID], edsae ID], sae ID], vars ID],rBff edvars ID]

8/12/2019 TRIGONOMETRIC Ratio & Identities Theory_H

3/14

HI_JU

"hik`sjfuhirpjys`es.`k" 5

ofs.f Sos by, b8fofs.fbarj6 vuq`fr7ve la b8fofs.fbarj6 vuq`fr oj `bgJff"ff ofs ,o ofs.f gsbH6u os nQkJfW asi vfxs e>+frs lUi vfUg hQlsi b8fofs.fbarj6 Yyu os7` asi v/66u ogrs lUiI (dmfj6 Yyu)afuf ,o cofcW dmf (b8f_6f 9cofcW)lUX btnof osQkz buksW'ffio v{ffsiof awybeQkq lUI afuf ](i, c)dmf `g ofscW beQkq cn zofg lU boofs.f ID ]8xgsbH6u vBffWr~ cn zofg pf` I]oj yAefcW 8 x

ns la e d s x 8 i vfUg s ` k x 8 cofs `bgJffb"fr ogrs lUiI pwbo DH],o naofs.f b8fJfqt lU vr% DH3 + H]3 8 D]36fi3 + c3 8 9 cn `zofg dmf `g h`bNBfr zS6so beQkq os by,i3 + c3 8 9 6f eds3x + s`k3 x 8 9

pwbo ,o nA w.fW pOog dmf os osQkz `g 3ofs.f eufrf lUI IDC 83

, IDE 8 rBff

IDO 83

5,

3

os w.ffo xq.ft ofs.f prqBff'f ofs.f olyfrs lUiI beQkqvfsi I,C,E rBff O

os buksW'ffio a'f% (9,=), (=, 9), (9, =)rBff (=, 9)lUI vr% prqBff'f ofs.ffsi os by, la olnors lUi &eds = 8 9 s`k = 8 =,

eds3

8 = s`k

3

8 9

eds 8 9 s`k 8 =

eds3

58 = s`k

3

58 9

eds 38 9 s`k 38 =

ve 6bk la bNBfbr D]ns ,o nA w.fW pOog ysiX rfs la `qu% zfgbAJfo bNBfbr D] `g df`n vf tfrs lUi cn zofg la cnbu"o"fW `g `lqprs lUi bo 6bk xbonj 3os w.ffo xq.ft ns e>+rf lU (6f 4fRrf lU)rfs s`karBff eds`kaYyufsi of afu

`bgdbrWr ulji lfsrf lUX cn `zofgs`k (3k+ x) 8 s`k x , k W, eds (3k+ x) 8 eds x, k W

vfUg s`k x 8 =,6bk x 8 =, , 3, 5.....vBffWr~ te x,of `w.ffo xq.ft lU rBff eds x 8 =,6bk

x 8 3

,

3

5,

3

1, .....vBffWr~eds xof afu 'fwQ6 lfs tfrf lU te x,

3

of bd"fa xq.ft lUI

s`k x 8 = x 8 k ,tlf kofscW w.ffo lUI

eds x 8 = x 8 (3k + 9)3

,tlf kofscW w.ffo lUI

ve la vQ6 b8fofs.fbarj6 vuq`frfsi ofs s`ka vfUg eds`kaYyu os 7` asi D6Or ogrs lUiI

edsae x 8 xs`k

9

, xk, tlf kofscW w.ffo lUI

sae x 8xeds

9, x(3k + 9)

3

,tlf kofscW w.ffo lUI

tik x 8xeds

xs`k, x(3k + 9)

3

,tlf kofscW w.ffo lUI

edt x 8xs`k

xeds, x k,tlf kofscW w.ffo lUI

la zkb'fWr og pqos lUi bo nJfj dfNrbdo xos by, s`k3x + e d s3x 8 9

btnos vuqnfg 9 + tik3x 8 sae3x {x(3k + 9) 3

6 kW}

9 + edt3x 8 edsae3x {x k6 k W}

8/12/2019 TRIGONOMETRIC Ratio & Identities Theory_H

4/14

HI_JU

"hik`sjfuhirpjys`es.`k" ?

b8fofs.fbarj6 Yyufsi os bpQl U`lks db _ja _r`ldkdhatr`e Buket`dks 7(`) 6bk Bfa prqBff'f asi lU re ](i, c)Bfa prqBff'f asi lfsxfI cnby, i ; =, c ; = vr% nJfj b8fofs.fbarj6 Yyufsi

of afu /fufSao lfsxfI(``) 6bk b}rj6 prqBff'f asi lU re ](i, c)b}rj6 prqBff'f asi lfsxfI cnby, i > =, c ; = vr% s`k, edsaeos afu

/fufSao vfUg 'fs"f T.ffSao lfisxsI(```) 6bk rmrj6 prqBff'f asi lU re ](i, c)rmrj6 prqBff'f asi lfsxfI cnby, i > =, c > = vr% tik, edt os afu /fufSao

vfUg 'fs"f T.ffSao lfssixsI(v) 6bk prqWBf prqBff'f asi lU re ](i, c)prqBfW prqBff'f asi lfsxfI cnby, i ; =, c > =vr% eds, sae os afu

/fufSao vfUg 'fs"f T.ffSao lfssixsIoqK ald w.fW ofs.ffsi os b8fofs.fbarj6 Yyufsi os afu ujsps kj xcW nfg.fj asi bk6s x6s lUi &

.O.K595

9=tik

=3

9

3

9

3

59eds

9?

?

3

5

?

5

3

9

?

3

3

9

?

9=

?

=s`k

35?2=

K.O. 8v`bgJffb"fredsae x, sae xrBff edt xos afu a'f% s`k x, eds xrBff tik xos D6qSa lUI

nAe) ofs.ffsi os b8fofs.fbarj6 vuq`fr _r`ldkdhatr`e Sit`ds db inn`ao iklnas6bk ofscW ofs.f lUX re

3

,,

3

5, 3cS6fbk nAe) ofs.f olyfrs lUiI

( os b8fofs.fbarj6 vuq`fr 7afuf ,o ofs.f afuo bNBfbr asi Bfa prqBff'f asi lUI afuf cnoj vibra Jfqtf ,o dmfX btnof osQkz DrBff b8f_6f rlUX ofs] (x, y) `g ofRrj lUIafuf vibra Jfqtf of dmf os nfBf ofs.f `g oRf{f beQkq ] (x, y)lUIve HD]8 HD] (niV6fSao) vfUg ],di ] os xv{f `g nafu `z{fs` H lUiI DH] DH] x 8 xrBff y 8 y

y

x

]

]

D

r

H

]DH 8

HD] 8

(x , y)

(x , y )

s`k () 8r

y8

r

y8 s`k.

eds () 8r

x8

r

x8 eds

tik () 8

x

y8

x

y8 tik

edt ( ) 8

y

x8

y

x8 edt.

sae ( 8x

r8

x

r8 sae.

edsae () 8 yr

8y

r8 edsae

nafur% 6bk vQ6 prqBff'ffsi asi lfs re `g bk6s x, `bg.ffa bn) bo, tf nors lUiI

8/12/2019 TRIGONOMETRIC Ratio & Identities Theory_H

5/14

HI_JU

"hik`sjfuhirpjys`es.`k" 1

os b8fofs.fbarj6 vuq`frafuf ,o ofs.f Bfa prqBff'f asi afuo bNBfbr asis lUI afuf cnoj vibra Jfqtf ,o dmfX btnof osQkz DrBff b8f_6f rlUX ofs] (x, y) `g ofRrj lUIafuf vibra Jfqtf of dmf os nfBf ofs.f `g oRfu beQkq ] (x, y)lUIafuf HrBff Ha'f% ]rBff ]os x-v{f `g `z{fs` lUpwbo DH] DH], x 8 x, y 8 y

s`k () 8r

y8

r

y8 s`k

eds () 8r

x8

r

x8 eds

tik () 8

x

y8

x

y8 tik

edt () 8

y

x8

y

x8 edt

sae () 8x

r 8x

r 8 sae

edsae () 8 yr

8y

r8 edsae

(3

os b8fofs.fbarj6 vuq fr %

nafur% ujps bk, x, `bg.ffa bn) bo, tf nors lUiI

s`k

3 8 eds , tik

3 8 edt edsae

3 8 sae

eds

3 8 s`k, edt

3 8 tik, sae

3 8 edsae

(3

+ os b8fofs.fbarj6 vuq fr %

nafur% ujps bk, x, `bg.ffa bn) bo, tf nors lUiI

s`k

3

8 eds, tik

3

8 edt, edsae

3

8 sae

eds

3

8 s`k, edt

3

8 tik, sae

3

8 edsae

+ os b8fofs.fbarj6 vuq fr %nafur% ujps bk, x, `bg.ffa bn) bo, tf nors lUiI

s`k ( +) 8 s`k, tik ( +) 8 tik edsae ( +) 8 edsaeeds ( +) 8 eds, edt ( +) 8 edt, sae ( +) 8 sae

(3

5 os b8fofs.fbarj6 vuq fr %

nafur% ujps bk, x, `bg.ffa bn) bo, tf nors lUiI

8/12/2019 TRIGONOMETRIC Ratio & Identities Theory_H

6/14

HI_JU

"hik`sjfuhirpjys`es.`k" 2

s`k

3

58 eds, tik

3

58 edt, edsae

3

58 sae

eds

3

58 s`k, edt

3

58 tik, sae

3

58 edsae,

(3

5+

os b8fofs.fbarj6 vuq fr %nafur% ujps bk, x, `bg.ffa bn) bo, tf nors lUiI

s`k

3

58 eds, eds

3

58 s`k, tik

3

58 edt,

edt

3

58 tik, sae

3

58 edsae, edsae

3

58 sae

ujps kj xcW nfg.fj asi bgOr NBffufsi ofs Nd6i Jfbg,I

.O.K595

9=tik

=3

9

3

9

3

59eds

93

5

3

9

3

9=s`k

32

995

13

55

?2