Aóg Aa¨vq

w·KvYwgwZ
cvV m¤úwK©Z MyiæZ¡c~Y© welqvw` 8.1

mvaviYfv‡e w·KvYwgwZ ej‡Z wZbwU †Kv‡Yi cwigvc †evSvq|

e¨envwiK cÖ‡qvR‡b wÎfy‡Ri wZbwU †KvY I wZbwU evûi cwigvc
Ges wÎfy‡Ri mv‡_ m¤úwK©Z wel‡qi Av‡jvPbv †_‡KB
w·KvYwgwZi m~ÎcvZ nq| w·KvYwgwZ‡K `ywU kvLvq wef³ Kiv
nq| kvLv `ywU n‡”QÑ mgZjxq w·KvYwgwZ Ges †MvjKxq
w·KvYwgwZ|
 abvZ¥K I FYvZ¥K †KvY : †Kv‡bv GKwU w¯’i iwk¥i †cÖwÿ‡Z
Aci GKwU N~Y©vqgvb iwk¥‡K Nwoi KuvUvi wecixZ w`‡K
†Nviv‡bvi d‡j †h †KvY Drcbœ nq, Zv‡K abvZ¥K †KvY ejv nq|

Avevi, N~Y©vqgvb iwk¥wU‡K Nwoi KuvUvi w`‡K †Nviv‡bvi d‡j

Drcbœ †KvY‡K FYvZ¥K †KvY ejv nq|
 †KvY cwigv‡ci GKK : †Kv‡Yi cwigvY I gvb eY©bvq mvaviYZ
`yB ai‡bi GKK c×wZ e¨envi Kiv nq| h_v : (1) lvUg~jK GKK
c×wZ Ges (2) e„Ëxq GKK c×wZ|
1. lvUg~jK c×wZ : lvUg~jK c×wZ‡Z mg‡KvY‡K †KvY cwigv‡ci
GKK aiv nq| GK mg‡KvY‡K mgvb 90 fv‡M wef³ Ki‡j cÖwZ fvM‡K
GK wWwMÖ ejv nq| Avevi, GK wWwMÖ‡K 60 fvM K‡i cÖwZ
fvM‡K GK wgwbU Ges GK wgwbU‡K mgvb 60 fvM K‡i cÖwZ
fvM‡K GK †m‡KÛ ejv nq|
60 (†m‡KÛ)
= 1
(wgwbU)
 60 (wgwbU)
= 1
(wWwMÖ)
90
(wWwMÖ) =
1 mg‡KvY|
2.e„Ëxq c×wZ : e„Ëxq c×wZ‡Z GK †iwWqvb †KvY‡K †KvY
cwigv‡ci GKK aiv nq| †Kv‡bv e„‡Ëi e¨vmv‡a©i mgvb Pvc H
e„‡Ëi †K‡›`ª †h m¤§yL †KvY Drcbœ K‡i, Zv‡KB GK †iwWqvb ejv
nq Ges †iwWqvb GKwU aªæe †KvY|
†Kv‡Yi wWwMÖ cwigvc I †iwWqvb cwigv‡ci m¤úK© :
1 †iwWqvb = mg‡KvY| A_©vr 1c = mg‡KvY
 1 mg‡KvY =
 90 =
1 = Ges 1c =
 90 = 1 mg‡KvY = †iwWqvb
A_©vr 180 = 2 mg‡KvY =  †iwWqvb|

cvV m¤úwK©Z MyiæZ¡c~Y© welqvw` 8.2

 m~²‡Kv‡Yi w·KvYwgwZK AbycvZmg~n :

m~²‡Kv‡Yi w·KvYwgwZK AbycvZmg~n wbY©q Kivi Rb¨ Avgiv
GKwU mg‡KvYx wÎfyR OPQ we‡ePbv Kwi| OPQ G OQP
mg‡KvY|
P
POQ Gi mv‡c‡ÿ : OP wÎfy‡Ri AwZfyR, OQ yR
Zf
Aw
f‚wg, PQ j¤^ Ges POQ =  (m~²‡KvY)| OPQ j¤^

mg‡KvYx wÎfy‡R m~²‡KvY  Gi Rb¨ QqwU O f‚wg

Q

w·KvYwgwZK AbycvZ h_vµ‡g wb‡gœv³fv‡e

msÁvwqZ Kiv nq :
sin = cosec = cos = sec = tan = cot
=
 w·KvYwgwZK AbycvZ¸‡jvi cvi¯úwiK m¤úK© :
Y
w·KvYwgwZK AbycvZmg~‡ni A

msÁv †_‡K Avgiv jÿ Kwi †h, P

sin = Y

cosec = = = 
X
O
 sin = Ges cosec =
Abyiƒcfv‡e cos = , sec = = =
A_©vr, cos = Ges sec =
GKBfv‡e, tan = Ges cot =
 mnRfv‡e g‡b ivLvi Rb¨ :
2q PZzf©vM 1g PZzf©vM
sin (+ve) All (+ve)
cosec (+ve)

tan (+ve) cos (+ve)

cot (+ve) sec (+ve)

3q PZzf©vM 4_© PZzf©vM

 ¸iæZ¡c~Y© m~Îvewj :
 sin =  cos =
 tan =  cosec =
 sec =  cot =
 tan =  cot =
 cosec =  sec =
 cot =  cos =
 sin2 + cos2 = 1  cos2 = 1  sin2
 sin2 = 1  cos2  1 + tan2 = sec2
 tan2 = sec2  1  1 + cot2 = cosec2
 cosec2  cot2 = 1  1 = sec2  tan2
 cosec2 = 1 + cot2  cot2 = cosec2  1
 wkÿv_x©‡`i myweav‡_© w·KvYwgwZK
AbycvZmg~‡ni ZvwjKv :
= = =
†KvY 0 = 90
30 45 60
sin 0 1
cos 1 0
AmsÁv
tan 0 1
wqZ
AmsÁv
cot 1 0
wqZ
sec 1 2 AmsÁv
 wqZ
AmsÁv
cosec 2 1
wqZ

cvV m¤úwK©Z MyiæZ¡c~Y© welqvw` 8.3

 †h‡Kv‡bv †Kv‡Yi A_©vr (n   ) †Kv‡Yi w·KvYwgwZK

AbycvZmg~n wbY©‡qi c×wZ (0 <  < ).
wb‡gœv³ c×wZ‡Z †h‡Kv‡bv w·KvYwgwZK †Kv‡Yi
AbycvZ wbY©q Kiv hvq|
avc-1 : cÖ_‡g cÖ`Ë †KvY‡K `yBfv‡M fvM Ki‡Z n‡e| hvi
GKwU Ask ev Gi n ¸wYZK Ges AciwU m~²‡KvY| A_©vr
cÖ`Ë †KvY‡K (n   ) AvKv‡i cÖKvk Ki‡Z n‡e|
avc-2 : n †Rvo msL¨v n‡j Abycv‡Zi aib GKB _vK‡e
A_©vr sine AbycvZ sine _vK‡e cosine AbycvZ cosine _vK‡e
BZ¨vw`|
n we‡Rvo n‡j sine, tangent I secant AbycvZ¸‡jv cosine,
cotangent I cosecant G cwiewZ©Z n‡e| GKBfv‡e Gi wecixZ
cwieZ©b NU‡e|
avc-3 : (n   ) †Kv‡Yi Ae¯’vb †Kvb PZzf©v‡M †mUv
Rvbvi ci H PZzf©v‡M cÖ`Ë Abycv‡Zi †h wPý †mB wPý
avc-2 †_‡K wbiƒwcZ Abycv‡Zi c~‡e© emv‡Z n‡e|
 () †Kv‡Yi w·KvYwgwZK AbycvZ :
sin() =  sin cosec() = cosec
cos() = cos sec() = sec
tan() =  tan cot() =  cot
 (90 + ) †Kv‡Yi w·KvYwgwZK AbycvZ :
 sin(90 + ) = cos cosec (90 + ) = sec
cos(90 + ) = sin sec(90 + ) =  cosec
tan(90 + ) = cot cot(90 + ) =  tan
 (90  ) †Kv‡Yi w·KvYwgwZK AbycvZ :
sin(90  ) = cos cosec(90  ) = sec
cos(90  ) = sin sec(90  ) = cosec
tan(90  ) = cot cot(90  ) = tan
 (180  ) †Kv‡Yi w·KvYwgwZK AbycvZ :
sin(180  ) = sin cosec (180  ) = cosec
cos(180  ) = cos sec (180  ) = sec
tan(180  ) = tan cot (180  ) = cot
 (180 + ) †Kv‡Yi w·KvYwgwZK AbycvZ :
sin(180 + ) = sin cosec(180 + ) = cosec
cos(180 + ) = cos sec(180 + ) = sec
tan(180 + ) = tan cot(180 + ) = cot
 (270  ) †Kv‡Yi w·KvYwgwZK AbycvZ :
sin (270  ) = cos cosec(270  ) = sec
cos (270  ) = sin sec(270  ) = cosec
tan (270  ) = cot  cot(270  ) = tan
 (270 + ) †Kv‡Yi w·KvYwgwZK AbycvZ :
sin(270 + ) = cos cosec(270 + ) = sec
cos(270 + ) = sin sec(270 + ) = cosec
tan(270 + ) = cot cot(270 + ) = tan
 (360  ) †Kv‡Yi w·KvYwgwZK AbycvZ :
sin(360  ) = sin cosec(360  ) = cosec
cos(360  ) = cos sec(360  ) = sec
 tan(360  ) = tan cot(360  ) = cot
 (360 + ) †Kv‡Yi w·KvYwgwZK AbycvZ :
sin(360 + ) = sin cosec(360 + ) = cosec
cos(360 + ) = cos sec(360 + ) = sec
tan(360 + ) = tan cot(360 + ) = cot