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ef$ekeâesCeefceleer
3
5,
1. pej COSA = lej sin A Je tan A ÛÙee efkeâceleer keâe{e.
Gòej : sin2A+cos2A = 1... (ef$ekeâesCeefceleerÙe efvelÙe meceevelee)
sin2A+(35)2 = 1
9
sin2A+ 25 = 1
9
∴ sin2A=1- 25
16
∴ sin2A= 25
∴ sinA = 45 (Jeie&cegUs IesTve)
415
sinA
- ∴ tan A = 315 ∴ tan A = 34
tan A = cosA
2. meeshes ¤he Åee : sinθ (coseθ- sinθ)
Gòej : sinθ (cosecθ-sinθ)
1
)
= sinθ ( sin1 θ - sinθ)... (∴ cosecx = sinx
1
= sinθ x sinθ - sinθ x sinθ
= 1 - sin2θ
= cos2θ
3. sin220+sin270 Ûeer efkebâcele keâe{e.
Gòej : sin220+sin270
= sin220+cos2 (90-70) ... (∴sinθ = cos (90-θ))
= sin2 20+cos220
= 1 ... (∴ sin2θ+cos2θ=1)
4. pej sinA=cosB Demesue Je A DeeefCe B ns ueIegkeâesve Demeleerue, lej oeKeJee.
A+B=900.
efmeælee : sinA = cos B ... (efoues Deens) ... (1)
hejbleg cos(90-θ) = sinθ .... (keâesefškeâesveeÛes ef$ekeâesCeefceleerÙe iegCeesòej)
cnCepesÛe cos(90-A) = sinA
∴ cos (90-A)=cos B... (sin A Ûeer efkebâcele (i) ceOÙes "sJetve)
∴ 90-A = B
∴ A + B = 900
5. pej tan 13 A = co+5 A Demesue Je 5 A DeeefCe 13A ns ueIegkeâesve Demeleerue,
lej A Ûeer efkebâcele keâe{e.
Gòej : tan13A = cot5A ... (efoues Deens) ... (i)
hejbleg, tanΘ = cot (90-Θ) ... (keâesefškeâesveeÛes ef$ekeâesCeefceleerÙe iegCeesòej)
∴ tan13A = cot (90-13 A) ... (2)
cot (90-13A) = cot5A ... [(1) Je (2) Je¤ve]
∴ 90-13A = 5A
∴ 90 = 13A+5A ∴18A = 90 ∴ A=50
6. oeKeJee : 1-tan4x=2sec2x-sec4x
Gòej : 1+tan2x = sec2x ... (ef$ekeâesCeefceleerÙe efvelÙe meceevelee)
ÀFa´FIÊY : [email protected]
AB
tan∠ADB = tan45 = BD
80
∴1= x
∴ x = 80 ... (1)
Je keâeškeâesve UACB ceOÙes,
tan∠ACB = tan30 = AB
_
BC ∴z
80
∴ √1_3 = x+y
∴ x = y = 80 √3
∴tan2x = sec2x-1... (1)
1-tan4x
[eJeer yeepet =
= 1-(tan2x)2
= 1- (sec2x-1)... [(1)Je¤ve]
= 1- (sec4x-2sec2x+1)
= 1-sec4x+2sec2x-1
= 2sec2x-sec4x
= GpeJeer yeepett.
7. peefceveerJejerue Skeâe efyebotheemetve ceveesNÙeeÛÙee JejÛÙee šeskeâeÛee GVele keâesve 300 ceeheeÛee
Deens. lÙee efyebotheemetve ceveesNÙeekeâ[s 30 ceeršj Ûeeuele iesuÙeeme GVele keâesve 600 ceeheeÛee
neslees, lej ceveesNÙeeÛeer GbÛeer efkeâleer?
Gòej : meesyeleÛÙee Deeke=âleerle CD ceveesje Demetve A DeeefCe B ner
efvejer#ekeâeÛeer efvejer#eCe keâjCÙeeÛeer ef"keâeCes Deensle.
∠CAD=300, ∠CBD=600
AB=30 ceeršj, mecepee CD = x ceeršj Je 3D = y ceeršj
keâeškeâesve U CDB ceOÙes,
tan∠CBD = tan60 = CD
BD
9.
∴
8.
1_
√_3
=
x
y+30
y+30
_
√3
∴x=
y+30
_
√3
∴ √3y =
∴ [(1) Je¤ve]
∴ 3y = _y+30 ∴ 2y=30 _∴ y=15
Je x= √3 x115 ∴_ x=15√3
ceveesNÙeeÛeer GbÛeer · 15√3 ceeršj.
80 ceeršj GbÛeerÛÙee oerheie=neÛÙee SkeâeÛe yeepetme DemeuesuÙee oesve penepeebkeâ[s heeefnues
Demelee DeJevele keâesve Deveg›eâces 300 Je 450 ceeheeÛes nesleele. lej lÙee oesve penepeebceOeerue
Deblej keâe{e. leer oesve penepes Je oerheie=neÛes leU ns SkeâeÛe js<esle Deensle Demes mecepee.
Gòej : meesyeleÛÙee Deeke=âleerle AB ns oerheie=n Demetve AH ner ef#eeflepemeceeblej heeleUer Deens.
oerhemlebYeeÛÙee SkeâeÛe yeepetuee C Je D Ùee ef"keâeCeer
penepes Deensle.
∠HAC = 300 Je ∠HAD=450
∠ACB = ∠HAC
}.... (JÙegl›eâce keâesve)
Je ∠ADB = ∠HAD
∴ ∠ACB = 300 Je ∠ADB = 450
mecepee BD = x ceeršj Je CD = y ceeršj
keâeškeâesve UADB ceOÙes,
n-1
_
pej sin A = n+1
Demesue lej oeKeJee : secA = n+1
2√n
2
2
efmeælee : sin A+cos A = 1
n-1 2
) +cos2A=1
(n+1
n-1 2
∴ cos2A = 1 - ( n+1
)
n-1 2
∴ cos2A = 11 - ( n+1
) = (n+1)2 -(n-1)2
2
(n+1)
n2+2n+1-n2+2n-1
4n
=
2
(n+1)2
_ (n+1)
2√n
∴ cosA = (n+1)
n+1
_
1 =
∴ cosA
2√n
n+1
∴ secA = 2√_n ... (∴ cos1θ = secθ)
x
pej cosec(90-A)
= 1, Demesue lej x Ûeer efkebâcele keâe{e.
sin (90 - A)
tan (90-A)
1
cosec
(90-A)
x
:
- tan (90-A) =
sin (90-A)
secA
x
∴ cosA - cotA = 1 (∴ cosec(90-θ) = secθ, sin(90-θ) = cosθ
∴ cos2A =
_
_
∴ √3 = xy ∴ x = √3y ... (1)
keâeškeâesve U CDA ceOÙes
CD
tan ∠CAD = tan30 = AD
_
∴ (1) Je¤ve 80
_ + y = 80 √3_
∴ y = 80 √3 - 80 = 80 _( √3 - 1)
penepeebceOeerue Deblej = 80 ( √3-1) ceeršj.
10.
Gòej
x
1
∴ sec A x cosA - cosA
sinA
1
1
x cosA ∴ cosA
1
∴ cos
- 1 =
2A
x.sinA
cosA
x.sinA
cosA
Je tan(90-θ) - cot)
= 1 ...(∴ cotθ = cos0 )
= 1 ... ( ∴ secθ =
sin0
1
cos0)
A
x.sinA
∴ 1-cos
cos2A =
cosA
sin2A
sinA
∴ cos2A = x x cosA
sinA
∴ x = cosA
∴ x = tanA
1
Demesue lej x Ûeer efkebâcele keâe{e.
mJeeOÙeeÙe : 1. pej x . sin (90- θ) . cot (90-θ) = sec(90θ)
12
2. cotθ = 5 lej tan2θ - sin2θ = sin2θ .tan2θ ÙeeÛee he[leeUe IÙee.
2
CXôFF¨FF d½F¿F¹F : Science and Technology (Part II)
SXFªFZaýi QØFFÂF¹F ¦FûÀFF½Fe,
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