SSC Math

Aa¨vqwfwËK K‡›U›U
Aa¨vq-9: w·KvYwgwZK AbycvZ
cÖ‡qvRbxq Z_¨:
 Trigonometry
Trigon metry

P P

O N O N

PON OP ON, OPN OP, PN,

PN ON

alpha  beta  gamma  theta  phi  omega 


XOA OA P A
P
P OX PM POM
∆POM PM, OM OP
XOA

O
M x

XOA POM PM OM OP XOA =  

PM
=  (sine) sin
OP
 OM
=  (cosine) cos
OP

PM
=  (tangent) tan
OM

OM
=  (cotangent) cot
PM

OP
=  (secant) sec
OM

OP
=  (cosecant) cosec.
PM

()


 = XOA
A
P
PM OP
sin = OP cosec =
PM
OM OP
cos = sec =  x
OP OM
O M
PM OM
tan = cot =
OM PM

PM OP PM OM
1 sin cosec = . =1 3 tan cot =  =1
OP PM OM PM
1 1 1 1
 sin = Ges cosec =  tan = Ges cot =
cosec sin cot tan
OM OP PM
2 cos sec =  =1
OP OM PM OP
4 tan = = OP
OM OM
 
 cos = Ges sec = OP
sec cos
sin cos
 tan = cot =
cos sin


1 sin2 + cos2 = 1 2. 1 + tan2 = sec2 3 1 + cot2 = cosec2
sin2 = 1 – cos2 sec2 – tan2 = 1 cosec2 – cot2 = 1
cos2 = 1 – sin2
 sin cos
sin2 + cos2 = 1

 =XOA
PM ⊥ OX
POM

2 2 2

∆OPM
OP, PM OM
 OP = PM + OM
2 2 2

2 2 2 A
OP PM OM 2 P
2 = 2 + 2 OP
OP OP OP
1=  +  
PM 2 OM 2

 OP   OP 
1 = (sin) + (cos)
2 2

O x
M
1 = sin  + cos   Sin
2 2
cos = ]

 sin2 + cos2 = 1
P A
 sec tan sec2 − tan2 =1
 =XOA
PM ⊥ OX

 POM O x
M

2 2 2

∆POM
OP, PM OM
 OP2 = PM2 + OM2
OP2 − PM2 = OM2
2 2 2
OP PM OM
2 –
2
2 = 2 [ OM ]
OM OM OM
2 2
 OP  –  PM  =1  = sec = tan]
OM OM
 sec2 – tan2 = 1
A
 cosec cot cosec2 – cot2 = 1 P

 = XOA
PM ⊥ OX
 POM O  x
M
2 2 2

∆POM
OP, = PM OM
 OP = PM + OM
2 2 2

OP – OM = PM
2 2 2

2 2 2
OP OM PM 2
2 – 2 = 2 [ PM
PM PM PM
  OP  – OM = 1 
2 2
cosec = cot]
PM  PM 
 cosec2 – cot2 = 1

0 30 45 60 90

0 30 45 60 90

1 1 3
sine 0
2 2 2 1
3 1 1
cosine 1 2
0
2 2
1
tangent 0 1 3
3
1
cotangent 3 1 3 0

2
secant 1 3 2 2

2
cosecant 2 2 3 1


(i) 0, 1, 2, 3 4 4 sin0, sin30, sin45, sin60
1 1
sin90 sin 30 = =
4 2
(ii) 4, 3, 2, 1 0 4 cos 0 cos 30 cos 45 cos
60  cos 90
2 1 1
cos 45 = 4
= 2
=
2
(iii) 0, 1, 3 9 3 tan 0 tan 30 tan 45 tan
60 tan 90
(iv) 9, 3, 1 0 3 cot30, cot45, cot60, cot90
cot0
  x − y = 30 ................ (ii)
m„Rbkxj cÖkœ: (i) I (ii) bs mgxKiY †hvM K‡i cvB,
2x = 90  x = 45
cÖk
œ 1 [Xv. †ev. 17] Avevi, (i) bs †_‡K (ii) bs mgxKiY we‡qvM K‡i cvB,
A 2y = 30  y = 15
myZivs x = 45 Ges y = 15 (Ans.)
x+y
cÖk
œ 2 [w`. †ev. 17]
a
A = cos + sin Ges B = cos − sin `yBwU w·KvYwgwZK ivwk|
x−y
C K.  = 45 n‡j A Ges B Gi gvb wbY©q Ki| 2
B b L. A = 2 (A − sin) n‡j, cÖgvY Ki †h, B = 2 (A − cos) 4
K. AC Gi ˆ`N©¨ wbY©q Ki| 2 M. A = 1 n‡j,  Gi gvb wbY©q Ki †hLv‡b 0    90. 4
b a 2 bs cÖ‡kœi mgvavb
1− 2 K †`Iqv Av‡Q, A = cos + sin Ges B = cos − sin
a2 + b2 a + b2
L. cÖgvY Ki †h, a
+
b
= 2 cosecA. 4  = 45 n‡j, A = cos45 + sin 45 Ges B = cos45 − sin 45
1− 2 1 1 1 1
−
2 2 2
a +b a +b = + =
2 2 2 2
M. a = 1 Ges b = 3 n‡j x I y Gi gvb wbY©q Ki| 4
1+1 2  B=0
1 bs cÖ‡kœi mgvavb = = =
A 2 2

K 2
x+y A= 2 Ges B = 0 (Ans.)
a L †`Iqv Av‡Q, A = cos + sin Ges B = cos − sin

x−y GLb A = 2 (A − sin) n‡j
B b C cos + sin = 2 (cos + sin − sin)
wPÎvbymv‡i, AB = a, BC = b Ges ABC = 90 ev, cos + sin = 2 cos
cx_v‡Mviv‡mi Dccv`¨ Abymv‡i, ev, sin = 2 cos − cos
AC2 = AB2 + BC2 ev, AC2 = a2 + b2  AC = a2 + b2 (Ans.) ev, sin = ( 2 − 1) cos

L †`Iqv Av‡Q, AB = a, BC = b ev, ( 2 + 1) sin = ( 2 + 1) ( 2 − 1) cos
Ges AC = a2 + b2 [ÔKÕ †_‡K cÖvß] ev, ( 2 + 1) sin = {( 2 )2 − 1} cos
b
1− 2
a BC AB ev, ( 2 + 1) sin = (2 − 1) cos
1−
a2 + b2 a + b2 AC AC ev, 2 sin + sin = cos
evgc¶ =
a
+
b
=
AB
+
BC
[ ÔKÕ
1− 2 1− ev, 2 sin = cos − sin
a +b 2 2
a +b 2 AC AC
ev, cos − sin = 2 sin
Gi wPÎ n‡Z]
sinA 1 − cosA sin2A + (1 − cosA)2
ev, cos − sin = 2 (cos + sin− cos)
=
1 − cosA
+
sinA
=
(1 − cos A) sinA  B = 2 (A − cos) (cÖgvwYZ)
sin A + 1 − 2cosA + cos2A
2 M †`Iqv Av‡Q,
=
(1 − cosA) sinA A = cos + sin
1 + 1 − 2cosA A = 1 n‡j cos + sin = 1
= [ sin2A + cos2A = 1]
(1 − cosA) sinA ev, cos = 1 − sin
2(1 − cosA) 1 ev, cos2 = (1 − sin)2
= = 2. = 2 cosecA = Wvbc¶
(1 − cosA) sinA sinA
ev, cos2 = 1 − 2sin + sin2
b a
1− 2 ev, 1 − sin2 = 1 − 2 sin + sin2
a2 + b2 a + b2
 + = 2cosecA (cÖgvwYZ) ev, 1 − 2sin + sin2 – 1 + sin2 = 0
a b
1− 2 ev, 2 sin2 − 2sin = 0
a + b2 a2 + b2
ev, 2 sin (sin − 1) = 0

M †`Iqv Av‡Q, AB = a = 1
ev, sin (sin − 1) = 0
BC = b = 3 nq, sin = 0 A_ev, sin − 1 = 0
GLb, mg‡KvYx ABC G sin = sin0 ev, sin = 1
BC   = 0
tan BAC = ev, sin = sin90   = 90
AB
3   = 0 ev 90 (Ans.)
ev, tan (x + y) = 1 A
cÖk
œ 3 [Kz. †ev. 17]
ev, tan (x + y) = tan60
 x + y = 60 ............. (i) 1
AB
Avevi, tan ACB = BC
1 B  C
ev, tan (x − y) = 3
3 K.  Gi gvb wbY©q K‡iv| 2
ev, tan (x − y) = tan30
 1 1 = a2 + b2 – a2 = b2 = b
L. DÏxc‡Ki Av‡jv‡K cÖgvY K‡iv †h, 1 + sin2 + 1 + cosec2 = 1.
AB
4 myZivs, tan = BC
cosB – sinB 3–1 a
M. hw` cosB + sinB = nq, Z‡e †`LvI †h, B = . 4  tan = (Ans.)
3+1 b
3 bs cÖ‡kœi mgvavb 
L ÔKÕ †_‡K cvB,

K wPÎ n‡Z cvB, a
A tan =
b
AC
tan = sin a
BC ev, =
1 1 cos b
ev, tan = a sin a2 a
3 ev, = 2 [b Øviv ¸Y K‡i]
ev, tan = tan30 B  C b cos b
3 a sin + b cos a2 + b2
  = 30 ev, = [†hvRb-we‡qvRb K‡i]

L ÔKÕ †_‡K cvB,  = 30 a sin – b cos a2 – b2
a sin − b cos a – b
2 2
1 1
evgc¶ = 1 + sin2 + 1 + cosec2  =
a sin + b cos a2 + b2
(Ans.)

1 1 
M MwYZ cvV¨eB‡qi Abykxjbx 9.1 Gi D`vniY 9 Gi Abyiƒc|
= +
1 + (sin30)2 1 + (cosec 30)2 cÖk
 œ 5 [wm. †ev. 17]
1 1
= + ABC mg‡KvYx wÎfz‡R B = 1 mg‡KvY Ges tanA = 1
1 2 1 + (2)2
1+
2 () K. AC Gi ˆ`N©¨ wbY©q K‡iv| 2
L. DÏxc‡Ki Av‡jv‡K (sec2A + cot2C + sin2A) Gi gvb wbY©q K‡iv| 4
1 1 1 1
= + =
1 1+4 4+1 5
+ 1 – sin2A 2tan2A
1+ M. DÏxc‡Ki Av‡jv‡K cÖgvY K‡iv †h, 1 + sin2A + 3sin2A = 1 4
4 4
4 1 4+1 5 5 bs cÖ‡kœi mgvavb
= + = = = 1 = Wvbc¶ K †`Iqv Av‡Q, B = 1 mg‡KvY Ges tanA =A1

5 5 5 5
1 1 Avgiv Rvwb,
 + = 1 (cÖgvwYZ)
1 + sin2 1 + cosec2 j¤^ BC

M †`Iqv Av‡Q, tanA =
f‚wg = AB
cosB − sinB 3−1 BC
= ev, 1 = AB
cosB + sinB 3+1
B C
cosB − sinB + cosB + sinB 3 −1 + 3 + 1  AB = BC
ev, cosB − sinB − cosB − sinB = [†hvRb ABC n‡Z,
3−1− 3−1
we‡qvRb K‡i] AC2 = AB2 + BC2 = AB2 + AB2 = 2AB2
 AC = 2 AB (Ans.)
2cosB 2 3
ev, − 2sinB = − 2 
L †`Iqv Av‡Q, tanA = 1
cosB ev, tanA = tan45  A = 45
ev, sinB = 3 Avevi, A + B + C = 180
ev, cotB = 3 ev, C = 180 − A − B = 180 − 45 − 90 = 45
ev, cotB = cot30  sec2A + cot2C + sin2A = sec245 + cot245 + sin245
= ( 2)2 + (1)2 +  
 B = 30 1 2
myZivs B =  [ÔKÕ n‡Z cvB] (†`Lv‡bv n‡jv)  2
1 4+2+1 7
=2+1+ = = (Ans.)
cÖk
œ 4 [P. †ev. 17] 2 2 2
A 1 − sin A 2tan A
2 2

M evgc¶ = 1 + sin2A + 3sin2A
1 − sin245 2tan245
= + [ A = 45]
1 + sin 45 3sin(2  45)
2

1− 
B C 1 2 1
 2 2.(1)2 1 − 2 2
AB = a, AC = a2 + b2 Ges C =  n‡j, = + = +
1 31
1+ 
1 2 3.sin90
1+
K. wPÎ n‡Z tan Gi w·KvYwgwZK AbycvZ wbY©q K‡iv| 2  2 2
a sin – b cos 2−1 1
L. tan Gi gv‡bi Dci wfwË K‡i a sin + b cos Gi gvb wbY©q K‡iv|
2 2 2 2 1 2 2 1 2
4 = + = + =  + = +
2+1 3 3 3 2 3 3 3 3
M. hw` tanA + sinA = m, tanA – sinA = n nq Zvn‡j cÖgvY K‡iv †h, 2 2
m2 – n2 = 4 mn. 4 1+2 3
= = = 1 = Wvbc¶
3 3
4 bs cÖ‡kœi mgvavb
A 1 − sin2A 2tan2A

K GLv‡b, AB = a, AC = a2 + b2  + = 1 (cÖgvwYZ)
1 + sin2A 3sin2A
 BC = AC2 – AB2
a
= ( a2 + b2)2 – a2

B C
cÖk
œ 6 [h. †ev. 17] L. †`LvI †h, x = 45 Ges y = 15. 4
1 M. B + 15 Gi w·KvYwgwZK AbycvZ¸‡jv †ei K‡iv| 4
ABC G B = 90 Ges tan =
3 7 bs cÖ‡kœi mgvavb
K. AC evûi ˆ`N©¨ wbY©q K‡iv| 2 
K wc_v‡Mviv‡mi m~Îvbymv‡i cvB,
cosec2 – sec2 1 AB2 = AC2 + BC2 = a2 + (a 3)2 = a2 + 3a2 = 4a2
L. cÖgvY K‡iv †h, cosec2 + sec2 = 2. 4  AB = 2a
M. A = x – y =  Ges C = x + y n‡j x I y Gi gvb wbY©q K‡iv| AB Gi ˆ`N©¨ 2a GKK
4 a 3

L wPÎ †_‡K, tan A = a
6 bs cÖ‡kœi mgvavb
K ABC-G B = 90
 C ev, tan (x + y) = 3 ev, tan (x + y) = tan 60
 AC2 = AB2 + BC2  x + y = 60................(i)
a
 AC = AB2 + BC2 GKK (Ans.) Avevi, tan B =
1 a 3

L tan = B A 1
3 ev, tan (x − y) = ev, tan (x − y) = tan 30
1 3
 tan  =
2
 x − y = 30................(ii)
3
1 4 GLb (i) I (ii) bs †hvM K‡i cvB,
sec2 = 1 + tan2 = 1 + = x + y + x − y = 60 + 30
3 3
1 90
tan2 =  cot2 = 3 ev, 2x = 90 ev, x = 2  x = 45
3
cosec2 = 1 + cot2 = 1 + 3 = 4 x Gi gvb (i) bs G ewm‡q cvB,
4 12 − 4 45 + y = 60 ev, y = 60 − 45  y = 15
4−
cosec2 − sec2 3 3  x = 45 Ges y = 15 (†`Lv‡bv n‡jv)
GLb, evgc¶ = cosec2 + sec2
= =
4+
4 12 + 4 
M ÔLÕ †_‡K cvB, x − y = 30
3 3 wPÎ †_‡K B = x − y = 30
8 3 1
= 
3 16 2
= = Wvbc¶ GLb B + 15 = 30 + 15 = 45
cosec2 − sec2 1 45 †Kv‡Yi w·KvYwgwZK AbycvZ¸‡jv †ei Ki‡Z n‡e|
 cosec2 + sec2 = 2 (cÖgvwYZ) AZci:

M †`Iqv Av‡Q, A = x − y =  Ges C = x + y MwYZ cvV¨ eB‡qi Abykxjb 9.2 Gi Aby‡”Q` 9.6 Gi 45 †Kv‡Yi
BC w·KvYwgwZK AbycvZ `ªóe¨| c„ôv-160
ABC G tanA =
AB
œ 8 [iv.
cÖk †ev. 16] Y
BC
ev, tan = AB A
ev, = AB  tan = 
1 BC 1
3  3
BC 1 y
ev, AB =
3
ev, tanA =
1 
3 O X
x B
1
ev, tan (x − y) =
3 K. cot Gi gvb wbY©q Ki| 2
ev, tan (x − y) = tan 30 L. DÏxc‡Ki Av‡jv‡K R¨vwgwZK c×wZ‡Z †`LvI †h,
 x − y = 30..............(i) sin2 + cos2 = 1| 4
AB
Avevi, tan C = BC sinA 1 – cosA
ev, tan (x + y) = 3 = tan 60
M. DÏxc‡Ki Av‡jv‡K (1 – cosA
+
sinA ) Gi gvb wbY©q Ki,
hLb x = 3, y = 4| 4
 x + y = 60..............(ii)
(i) I (ii) bs †hvM K‡i cvB, 8 bs cÖ‡kœi mgvavb
2x = 90  x = 45 
K wPÎ †_‡K cvB, Y
Avevi, (ii) bs n‡Z (i) bs we‡qvM K‡i cvB, A
f‚wg = OB = x
2y = 30  y = 15 j¤^ = AB = y y
myZivs x = 45 Ges y = 15 (Ans.) Avgiv Rvwb, 
cÖk
œ 7 [e. †ev. 17] A f‚wg OB B
O x X
cot =
j¤^ = AB
x+y x
a  cot  = . (Ans.)
y

L ÔKÕ Gi wPÎvbymv‡i cvB,
x–y
B C f‚wg = OB = x
a 3 j¤^ = AB = y
wP‡Î ABC GKwU wÎfzR Ges AOB =  GKwU m~²‡KvY
K. AB Gi ˆ`N©¨ wbY©q K‡iv| 2 cx_v‡Mviv‡mi Dccv`¨ Abymv‡i cvB,
 (AwZf~R)2 = (f‚wg)2 + (j¤^)2 GLb, cos = AC =
BC 1
ev, OA2 = OB2 + AB2 2
ev, OA2 = x2 + y2 AB 1
Ges cos = AC =
 OA = x2 + y2 2
Avgiv Rvwb, 1 1 1+1 2
j¤^ AB y  cos + cos = + = = = 2 (Ans.)
sin = 2 2 2 2
AwZfyR = OA = x2 + y2 
M ÔKÕ Gi wPÎ †_‡K cvB,
f‚wg OB x AB 1 AB 1
cos =
AwZfyR = OA = x2 + y2 sin =
AC
=
2
Ges cos = AC =
2
evgc¶ = sin2 + cos2
=
y
2 +  x 2
GLb, sin2 + cos2 = AC (AB) + ( ) 2 AB
AC
2

 x2 + y2  x2 + y2
=  +  1 2
1 2
= 2
y2
+ 2
x2 x2 + y2
= 2 =1  2  2
x + y x + y x + y2
2 2
1 1 1+1 2
= Wvbc¶ = + = = =1
2 2 2 2
 sin2 + cos2 = 1 (†`Lv‡bv n‡jv)
 sin2 + cos2 = 1 (cÖgvwYZ)

M ÔLÕ †_‡K cvB,
OB = x œ 10 [Kz.
cÖk †ev. 16]
AB = y tanA + sinA = m Ges tanA − sinA = n
OA = x2 + y2 K. cÖgvY Ki †h, tan2A. sin2A = mn. 2
†`Iqv Av‡Q, x = 3 Ges y = 4 L. †`LvI †h, m2 − n2 = 4 mn. 4
ÔKÕ Gi wPÎ †_‡K cvB, M. cÖgvY Ki †h, secA = mn. cosec2A. 4
OB x 3 3 3
10 bs cÖ‡kœi mgvavb
sinA =
OA
= = = = 
K †`Iqv Av‡Q,
x2 + y2 32 + 42 25 5
tanA + sinA = m
AB y 4 4 4 tanA – sinA = n
cosA = = 2 = 2 = =
OA x +y 2 3 + 42 25 5 GLb, tan2A. sin2A
sinA 1 − cosA = tan2A.(1 – cos2A) [ sin2A + cos2A = 1]
cÖ`Ë ivwk = 1 − cosA + sinA = tan2A – tan2A.cos2A
sin2A
3 4 3 1 = tan2A – . cos2A
1− cos2A
5 5 5 5
= + = + = tan2A – sin2A
4 3 1 3 = (tanA + sinA) (tanA – sinA)
1−
5 5 5 5 = mn [gvb ewm‡q]
3 5 1 5  tan2A.sin2A = mn (cÖgvwYZ)
=  + 
5 1 5 3

L MwYZ cvV¨eB‡qi Abykxjbx 9.1 Gi D`vniY 9 Gi Abyiƒc|
1 9+1
=3+ = 
M ÔLÕ †_‡K cvB,
3 3
m2 – n2 = 4 mn
10
=
3
(Ans.) ev, (tanA + sinA)2 – (tanA – sinA)2 = 4 mn [m, n Gi gvb
ewm‡q]
cÖk
 œ 9 [w`. †ev. 16] ev, 4tanA sinA = 4 mn [ (a + b)2 – (a – b)2 = 4ab]
wb‡Pi wPÎwU j¶¨ Kit ev, tanA.sinA = mn [4 Øviv fvM K‡i]
A
sinA

ev, cosA.sinA = mn
sin2A
1 ev, cosA = mn
 1 mn
B C ev, cosA = sin2A
1
1
K. AwZf‚R Gi cwigvY KZ? 2 ev, secA = mn. sin2A
L. cos + cos Gi gvb wbY©q Ki| 4  secA = mn cosec2A (cÖgvwYZ)
M. wP‡Îi Av‡jv‡K cÖgvY Ki †h, sin2 + cos2 = 1. 4
9 bs cÖ‡kœi mgvavb cÖk
œ 11 [wm. †ev. 16]
A

K cx_v‡Mviv‡mi Dccv`¨ Abymv‡i,

AC = AB2 + BC2
= 12 + 12
= 1+1 = 2
30
AwZfzR 2 GKK (Ans.) B C
3

L ÔKÕ †_‡K cvB, BC = 3 †m.wg., B = GK mg‡KvY, ACB = 30.
AwZfzR AC = 2 K. AB I AC evûi ˆ`N©¨ wbY©q Ki| 2
 1
L. DÏxc‡Ki Av‡jv‡K cÖgvY Ki †h, 2 − sin2A + 2 + tan2A = 1.
1
4 ev, (cos – 1) (2cos – 1) = 0
 cos – 1 = 0 A_ev, 2cos – 1 = 0
M. DÏxcK Abymv‡i  †Kv‡Yi mv‡c‡¶ hw` 2. AC (BC) + 3.AC
2AB
−3=0 ev, cos = 1 ev, 2cos = 1
nq, Z‡e †`LvI †h,  = 60| 4 ev, cos = cos0 ev, cos = 2
1
11 bs cÖ‡kœi mgvavb

K A   = 0 ev, cos = cos60   = 60

  = 60 (†`Lv‡bv n‡jv)
œ 12 [h.
cÖk †ev. 16]
30
B C tan + sin = mGes tan – sin = n.
3
K. DÏxc‡Ki Av‡jv‡K †`LvI †h, m + n = 2 sec.sin. 2
†m.wg.
†`Iqv Av‡Q, BC = 3 †m.wg., B = GK mg‡KvY I ACB = 30 L. cÖgvY Ki †h, m2 – n2 = 4 mn. 4
BC m 2+ 3
cos ACB = M. n = n‡j,  Gi gvb wbY©q Ki, †hLv‡b 0 <  < 90. 4
AC 2– 3
3 12 bs cÖ‡kœi mgvavb
ev, cos30 = AC

K †`Iqv Av‡Q,
3 3
ev, 2 = AC tan + sin = m......... (i) Ges tan − sin = n ........ (ii)
1 1 (i) I (ii) †hvM K‡i cvB,
ev, 2 = AC tan + sin + tan − sin = m + n
 AC = 2 †m.wg. (Ans.) ev, 2tan = m + n
AB ev, m + n = 2tan
Avevi, sinACB = AC sin
AB ev, m + n = 2.cos
ev, sin30 = 2 1
1 AB ev, m + n = 2.cos.sin
ev, 2 = 2
 m + n = 2sec.sin (†`Lv‡bv n‡jv)
 AB = 1 †m.wg. (Ans.)

L MwYZ cvV¨eB‡qi Abykxjbx 9.1 Gi D`vniY 9 Gi Abyiƒc|

L ÔKÕ Gi wPÎ †_‡K cvB,
BC

M †`Iqv Av‡Q,
sinA = .......... (i) m 2+ 3
AC =
3 n 2− 3
 sinA = [ BC = 3 I AC = 2]
2 tan + sin 2 + 3
BC ev, tan − sin = [cÖ`Ë]
Avevi, tanA = AB ........... (ii) 2− 3
tan + sin + tan − sin 2 + 3 + 2 − 3
3 ev, tan + sin − tan + sin = [†hvRb-we‡qvRb
tanA = [ BC = 3 I AB = 1] 2+ 3−2+ 3
1
K‡i]
 tanA = 3
1 1 1 1 2 tan 4
GLb, 2 – sin2A + 2 + tan2A = + ev, 2 sin =
2 3
2– 
3 2 2 + ( 3)2
2 tan
ev, sin =
2
1 1 1 1 1 1 3
= + = + = + sin 1 2
3 2+3 8–3 5 5 5 ev, cos . sin =
2–
4 4 4 3
4 1 4+1 5 1 2
= + =
5 5 5
= =1
5
ev, cos =
3
1 1
 + = 1 (cÖgvwYZ) 3
2 – sin2A 2 + tan2A ev, cos = 2
BC AB ev, cos = cos 30

M wPÎvbyhvqx, sin = AC Ges cos = AC
  = 30 (Ans.)
†`Iqv Av‡Q, 2 AC (BC) + 3.AB
2
AC
–3=0
cÖk
œ 13 [e. †ev. 16]
 2(sin)2 + 3.cos – 3 = 0
ABC-G B = 90, A = x − y, C = x + y, AB = 3, BC = 1.
ev, 2sin2 + 3cos – 3 = 0
ev, 2(1 – cos2) + 3cos – 3 = 0 K. AC Gi ˆ`N©¨ wbY©q Ki| 2
ev, 2 – 2cos2 + 3cos – 3 = 0 cosec2A − sec2A
L. DÏxc‡Ki Av‡jv‡K cos2A − sin2A
Gi gvb wbY©q Ki| 4
ev, –2cos2 + 3cos – 1 = 0
M. x I y Gi gvb wbY©q Ki| 4
ev, 2cos2 – 3cos + 1 = 0
ev, 2cos2 – 2cos – cos + 1 = 0
ev, 2cos (cos – 1) – 1(cos – 1) = 0
 13 bs cÖ‡kœi mgvavb 14 bs cÖ‡kœi mgvavb

K A 
K A

3
x−y

x+y
B C
1 C B
†`Iqv Av‡Q, ABC G B = 90, AB = 3, BC = 1 ABC mg‡KvYx wÎfz‡R C = GK mg‡KvY
wc_v‡Mviv‡mi Dccv`¨ Abymv‡i cvB, cx_v‡Mviv‡mi Dccv`¨ Abyhvqx,
AC2 = AB2 + BC2 = ( 3)2 + 12 = 3 + 1 = 4 AB2 = AC2 + BC2
 AC = 2  AB = AC2 + BC2 (Ans.)
†h‡nZz ˆ`N©¨ FbvÍK n‡Z cv‡i bv|
 AC Gi ˆ`N©¨ 2 GKK (Ans.)

L †`Iqv Av‡Q, tanB = 3
ev, tanB = tan60
L †`Iqv Av‡Q, ABC -G AB = 3, BC = 1
  B = 60
ÔKÕ †_‡K cvB, AC = 2 †h‡nZy C = 90  A = 30
BC
 sinA = cotA + tanB cot30 + tan60 3+ 3 2 3
AC evgc¶ = cotB + tanA = cot60 + tan30 = 1 = =3
1 1 1 2
sinA = ev, sinA = 2 +
2 3 3 3
 cosecA = 2 Wvbc¶ = cotA . tanB = cot30.tan60
AB = 3. 3 =3
Avevi, cosA = AC
= evgc¶
3 1 2 cotA + tanB
 cosA = ev, cosA =  = cotA.tanB (cÖgvwYZ)
2 3 cotB + tanA
2
 secA = M †`Iqv Av‡Q, B = p + q ............. (i)

3 A = p − q ............. (ii)
22 –  
2 2 (i) bs Ges (ii) bs †hvM K‡i cvB,
cosec2A − sec2A  3 B + A = p + q + p – q
cÖ`Ë ivwk = cos2A − sin2A =
 32 – 1 2 ev, 60 + 30 = 2p
2 2 () ev, 2p = 90
4 12 – 4 8 ev, p = 45
4–
3 3 3 8 4 16
= = = =  = (Ans.) (i) bs †_‡K (ii) bs we‡qvM K‡i cvB,
3 1 3–1 2 3 2 3 B − A = p + q − p + q
–
4 4 4 4 ev, 60 − 30 = 2q

M ABC-G AB = 3, BC = 1 ev, 30 = 2q
BC
tanA =  p = 45 Ges q = 15 (Ans.)
AB
1
 tan (x − y) = [ A = x – y] cÖk
œ 15 [w`. †ev. 15]
3
ev, tan (x − y) = tan 30 †Kv‡bv mg‡KvYx wÎfz‡Ri AwZfzR 1 + p Ges  †Kv‡Yi mwbœwnZ evû
 x − y = 30 ......... (i) 2p|
AB K. Z_¨¸‡jv R¨vwgwZK wP‡Î Dc¯’vcb K‡i Aci evûi ˆ`N©¨ wbY©q Ki|
Avevi, tanC = BC
2
3 L. sec2 + tan2 Gi gvb wbY©q Ki| 4
ev, tan (x + y) = 1 [ C = x + y]
1 + cosec2 1
ev, tan (x + y) = tan 60 M. cÖgvY Ki †h, 1 − cosec2 = − p 4
 x + y = 60 ......... (ii) 15 bs cÖ‡kœi mgvavb
mgxKiY (i) I (ii) †hvM K‡i cvB,
K awi, ABC mg‡KvYx

2x = 90  x = 45 A
x Gi gvb (ii) bs G ewm‡q cvB,
wÎfz‡Ri AwZfzR
45 + y = 60  y = 15 AC = 1 + P ,
 x = 45 I y = 15 (Ans.)  †Kv‡Yi mwbœwnZ
evû BC = 2P . 1+P
cÖk
œ 14 [Xv. †ev. 15]
GLb, wc_v‡Mviv‡mi
ABC mg‡KvYx wÎfz‡R C mg‡KvY, tan B = 3.
K. AB Gi gvb KZ? 2
Dccv`¨ Abymv‡i cvB, 
AC2 = AB2 + BC2 C
cotA + tan B 2 2 B 2P
L. DÏxc‡Ki Av‡jv‡K cÖgvY Ki †h, cot B + tan A = cotA.tanB.4 ev, ( 1 + P) = AB2 + ( 2P)
M. B = p + q Ges A = p − q n‡j, p I q Gi gvb wbY©q Ki|4 ev, 1 + P = AB2 + 2P
ev, AB2 = 1 + P − 2P
ev, AB2 = 1 − P
 AB = 1 − P (Ans.)

L ÔKÕ n‡Z cvB,  A = 30 (Ans.)
ABC mg‡KvYx wÎfz‡Ri AC = 1 + P , BC = 2P 
M 4sin2 − (2 + 2 3 )sin + 3 = 0
Ges AB = 1 − P Ges  = ACB ev, 4sin2 − 2sin − 2 3 sin + 3 = 0
AC
GLb, sec = BC =
1+P ev, 2sin(2sin − 1) − 3 (2sin − 1) = 0
2P ev, (2sin − 1) (2sin − 3 ) = 0
AB 1−P nq, 2sin − 1 = 0 A_ev, 2sin − 3 = 0
tan = =
BC 2P ev, 2sin = 1 ev, 2sin = 3
1 + P2  1 − P2
 sec2 + tan2 =  + 1 3
 2P   2P  ev, sin =
2
ev, sin = 2
1+P 1−P 1+P+1−P 2 ev, sin = sin30 ev, sin = sin60
= + = =
2P 2P 2P 2P ev,  = 30
1
= (Ans.)   = A [ÔLÕ n‡Z] ev,  = 2  30
P
  = 2A [ÔLÕ n‡Z]
M ÔKÕ n‡Z cvB, ABC mg‡KvYx wÎfz‡Ri AC = 1 + P , BC = 2P
 (†`Lv‡bv n‡jv)
AB = 1 − P Ges  = ACB œ 17 [P.
cÖk
 †ev. 15]
AC 1+P
 cosec = = L
AB 1−P

1+
1 + P2 1+P 13 GKK
1 + cosec2  1 − P
1+
1−P 12 GKK
evgc¶ =
1 − cosec2
= =
1−
1 + P 2 1−
1+P 
 1 − P 1−P M N
1−P+1+P K. cot Gi gvb wbY©q Ki| 2
1−P 2 (1 − P) −1 L. DÏxc‡Ki Av‡jv‡K cÖgvY Ki †h, tan2 − sin2 = tan2. sin24
= =  =
1 − P − 1 − P (1 − P) −2P P M. R¨vwgwZK c×wZ‡Z cÖgvY Ki †h, sin2 + cos2 = 1 4
1−P 17 bs cÖ‡kœi mgvavb
= Wvbc¶ 
K †`Iqv Av‡Q,
1 + cosec2 −1 L
 = (cÖgvwYZ) LM = 12 GKK
1 − cosec2 P
LN = 13 GKK
œ 16 [Kz.
cÖk †ev. 15]  MN = LN2 − LM2 GKK
12 13 GKK
cosA + sinA 3+1 = 132 − 122 GKK
= , B = 60. GKK
cosA − sinA 3−1 = 169 − 144 GKK
K. cosec2B + cot2B Gi gvb wbY©q Ki| 2 = 25 GKK

L. A Gi gvb wbY©q Ki| 4 = 5 GKK
M N
M. 4sin2 − (2 + 2 3) sin + 3 = 0 mgxKiYwU mgvavb K‡i MN 5
 cot = = (Ans.)
LM 12
†`LvI †h,  = 2A A_ev  = A| 4
16 bs cÖ‡kœi mgvavb 
L DÏxc‡Ki wPÎvbymv‡i cvB,
LM = 12 GKK, LN = 13 GKK
K B = 60
 ÔKÕ n‡Z cvB, MN = 5 GKK
GLb cosec2B + cot2B LM 12
= (cosec60)2 + (cot60)2  tan = =
MN 5
=  + 
2 2 1 2 LM 12
 3  3 sin =
LN 13
=
4 1
= +  evgc¶ = tan2 − sin2
3 3 12 2 12 2
=
5
3
(Ans.)
= ( ) ( )
5
−
13
144 144 24336 − 3600 20736
= − = =

L †`Iqv Av‡Q, 25 169 4225 4225
12 2 12 2 144 144 20736
cosA + sinA
cosA − sinA
=
3+1
3−1
2 2
( )( )
Wvbc¶ = tan .sin  = 5 . 13 = 25 .169 = 4225
 tan2 − sin2 = tan2.sin2 (cÖgvwYZ)
cosA + sinA − cosA + sinA 3+1− 3+1
ev, cosA + sinA + cosA − sinA
=
3+1+ 3−1

M MwYZ cvV¨eB‡qi Abykxjbx 9.1 Gi Aby‡”Q` 9.5(i) `ªóe¨|
[we‡qvRb−†hvRb K‡i]
2sinA 2
ev, =
2cosA 2 3
sinA 1
ev, cosA
=
œ 18 [h.
cÖk
 †ev. 15]
3
ev, tanA = tan30
p = 1 + sinA Ges q = 1 − sinA n‡j ⎯ =
2.tan 45
K. pq Gi gvb KZ? 2 1 − (tan 45)2
2.1
p =
L. cÖgvY Ki †h, q = secA + tanA. 4 1 − (1)2
2
q = = AmsÁvwqZ
M. cÖgvY Ki †h, (secA − tanA)2 = p. 4 0
2tanA
18 bs cÖ‡kœi mgvavb  tan 2A = (†`Lv‡bv n‡jv)
1 − tan2A
K †`Iqv Av‡Q, p = 1 + sinA Ges q = 1 − sinA
 1
 pq = (1 + sinA) (1 − sinA) = 1 − sin2A = cos2A (Ans.) M 3cot2 (B + 45) − 2 cosec2 (B + 45) + 5 sin2(B + 30) − 4
p 1 + sinA cos2(B + 45)

L evgc¶ = =
1 − sinA
1
q = 3cot2 (15 + 45) − cosec2 (15 + 45) +
2
(1 + sinA) (1 + sinA) 5sin2 (15 + 30) − 4cos2 (15 + 45) [ B = 15]
=
(1 + sinA) (1 − sinA)
[je I ni‡K (1 + sinA) Øviv ¸Y
1
K‡i] = 3cot2 60 − cosec2 60 + 5 sin2 45 − 4cos260
2
(1 + sinA)2 (1 + sinA)2
= 3   −   + 5.   − 4
1 2 1 2 2 1 2 1 2
=
1 − sin A
2 =
cos2A  3 2  3  2 ()2
1 + sinA 1 sinA 1 1 4 1 1
=
cosA
= +
cosA cosA =3 −  +5 −4
3 2 3 2 4
= secA + tanA 2 5
= Wvbc¶ =1− + −1
3 2
p 5 2 15 − 4 11
 = secA + tanA (cÖgvwYZ) = − = = (Ans.)
q 2 3 6 6
M evgc¶ = (secA − tanA)2
 cÖk
œ 20 cos A + cos4A = 1
2

1 − sinA2 cos2A
= K. †`LvI †h, 1 + cos2A = (1 + cosA) (1 − cosA)
1 sinA 2
= (cosA −
cosA)  cosA 
L. cÖgvY Ki †h, cot A − cot A = 1
4 2
2
4
(1 − sinA)2 (1 − sinA)2
= = M. †`LvI †h, tan4A + tan2A = 1 Ges sin2A + sec2A = 2 4
cos2A 1 − sin2A
(1 − sinA) (1 − sinA) 1 − sinA q 20 bs cÖ‡kœi mgvavb
= = = 
(1 + sinA) (1 − sinA) 1 + sinA p K †`Iqv Av‡Q,
= Wvbc¶ cos2A + cos4A = 1
q ev, cos4A = 1 − cos2A
 (secA − tanA)2 = (cÖgvwYZ) ev, cos4A = sin2A [ sin2A + cos2A = 1]
p
cÖk
œ 19 2cos (A + B) = 1 = 2sin (A − B)  cos2A = sinA
cos2A sinA
K. DÏxcK Abymv‡i A + B I A − B msKwjZ `yBwU mgxKiY MVb Ki| GLb, 1 + cos2A = 1 + sinA
2 sinA (1 – sinA)
2tanA =
L. A I B Gi gvb wbY©q Ki| Av‡iv †`LvI †h, tan2A = 1 − tan2A (1 + sinA) (1 − sinA)
[ni I je‡K (1 − sinA) Øviv ¸Y K‡i]
4 sinA (1 − sinA)
1 =
M. 3cot2 (B + 45) − 2 cosec2 (B + 45) + 5 sin2 (B + 30) − 4 cos2 1 − sin2A
sinA (1 − sinA)
(B + 45) Gi gvb wbY©q Ki| 4 =
cos2A
19 bs cÖ‡kœi mgvavb sinA (1 − sinA)

K †`Iqv Av‡Q, 2cos (A + B) = 1 =
sinA
ev, cos (A + B) = 2
1 = 1 − sinA = 1 − cos2A
= (1 + cosA) (1 − cosA)
ev, cos (A + B) = cos 60 cos2A
 A + B = 60 ... ... ... ... (i)  = (1 + cosA) (1 − cosA) (†`Lv‡bv n‡jv)
1 + cos2A
Ges 2 sin (A − B) = 1 
L †`Iqv Av‡Q,
1 cos2A + cos4A = 1
ev, sin (A − B) = 2
ev, cos4A = 1 − cos2A
ev, sin (A − B) = sin 30 ev, cos4A = sin2A
 A − B = 30 ... ... ... (ii) cos4A sin2A

L (i) I (ii) bs †hvM K‡i cvB, ev, sin4A = sin4A
2A = 90  A = 45 1
Avevi, (i) n‡Z (ii) we‡qvM K‡i cvB, ev, cot4A = sin2A
2B = 30  B = 15 ev, cot4A = cosec2A
evgc¶ = tan 2A ev, cot4A = 1 + cot2A
= tan (2  45) [ A = 45]  cot4A − cot2A = 1 (cÖgvwYZ)
= tan 90

M ÔLÕ †_‡K cvB,
sin 90 1
= = = AmsÁvwqZ cot4A − cot2A = 1
cos 90 0
1 1
2tanA ev, tan4A − tan2A = 1
Wvbc¶ = 1 − tan2A
 1− tan2A
ev, tan4A = 1
ev, tan4A = 1 − tan2A
 tan4A + tan2A = 1 (†`Lv‡bv n‡jv)
sin2A + sec2A
1
= sin2A +
cos2A
1
= sin2A +
sinA
sin3A + 1
=
sinA
(sinA + 1) (sin2A − sinA + 1)
=
sinA
(sinA + 1) (sin2A − sinA + 1)
= [ÔKÕ n‡Z sinA = cos2A]
cos2A
(sinA + 1) (sin2A − sinA + 1)
= [ sin2A + cos2A = 1]
1 − sin2A
sin2A − sinA + 1
=
1 − sinA
sin2A − sinA + 1
=
1 − cos2A
sin2A − sinA + 1
=
sin2A
1 1
=1− +
sinA sin2A
1 1
=1− +
cos2A sin2A
1 − sec2A + cosec2A
= 1 − sce2A + (1 + cot2A) [  cosec2A − cot2A = 1]
= 2 + cot2A − sec2A
cos2A
= 2 + 2 − sec2A
sin A
cos2A
=2+ − sec2A
cos4A
= 2 + sec2A − sec2A
=2
 sin2A + sec2A = 2 (†`Lv‡bv n‡jv)