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1M th
1.
cos2+sin2=1,1+cot2=cosec2 1+tan2= sec2
2. sin()=sin,cosec()=cosec , tan()=tan,cot()=cot
cos()=cossec()=sec
3. 3, 2 , ,2 2
,
F CF F
F( ) F( ) , F(2 ) F( )
3F( ) COF( ) , F( ) COF( )
2 2
4. A B (1) sin(AB)=sinAcosBcosAsinB sin(A+B)=sinAcosB+cosAsinB
cos(AB)=cosAcosB+sinAsinB cos(A+B)=cosAcosBsinAsinB
tan(AB)=tan A tan B
1 tan A tan B
tan(A+B)=
tanA tan B
1 tan A tanB
(2)
asinAx+bcosAx= 2 2a b sin(Ax+)= 2 2a b cos(Ax) tan= ba
(3)
sin2A=2sinAcosA=2
2 tanA
1 tan A
cos2A=cos2Asin2A=12sin2A=2cos2A1=
2
2
1 tan A
1 tan A
2
2 tanAtan 2A
1 tan A
Y
X
2M th
(4)
A 1 cosA A 1 cosAsin cos
2 2 2 2
A 1 cos Atan
2 1 cosA1 cos A sinA
sinA 1 cos A
(5) 3 3
3
2
sin 3A 3 sinA 4 sin A cos 3A 4 cos A 3cosA
3 tan A tan Atan 3A
1 3 tan A
(6) (7)
2 sin A cos B sin(A B) sin(A B)
2 cos A sin B sin(A B) sin(A B)
2 cos A cos B cos(A B) cos(A B)
2 sin A sin B cos(A B) cos(A B)
1 1sinA sinB 2 sin (A B) cos (A B)
2 2
1 1sinA sinB 2cos (A B) sin (A B)
2 2
1 1cosA cosB 2 cos (A B) cos (A B)
2 2
1 1cosA cosB 2 sin (A B) sin (A B)
2 2
5.
[0 , 2) sinx=a x=n+(1)
n nI
cosx=a x=2n nI
tanx=a x=n+ nI
6. ()
y=arcsinx
[1,1]
,
2 2
y=arctanx =(,),
2 2
y=arccosecx (,1][1,)
, 0 0,
2 2
y=arcosx [1,1] [0,]
y=arccotx = (,) (0,)
y=arcsecx
(,1][1,)
0, ,
2 2
3M th
1f (a) x f(x)=a
1. sin(arcsinx)= x; 1x1 arcsin(sinx)=x;x ,2 2
cos(arcosx) = x; 1x1 arcos(cosx)=x;x[0,]
tan(arctanx)= x; x arctan(tanx)=x;x ,2 2
2. arcsin(x)=arcsinx; 1x1 arccos(x)=arccosx;1x1
arctan(x)=arctanx;x arccot(x)=arccotx;x
arccosec(x)=arccosecx;x1,x1 arcsec(x)=arcsecx;x1,x1
3. arcsinx+arccosx=2
; 1x1 arcstanx+arccotx=
2
;x
4. arctanx+arctany=x y
arctan1 xy
; arctan x arctan y
2 2
arctanx+arctany=+x y
arctan1 xy
; arctan x arctan y
2
7.
2 2 2
2 2 2
2 2 2
a b c 2bc cosA
b a c 2ac cosB
c a b 2ab cosC
2 2 2
2 2 2
2 2 2
b c acos A
2bc
a c bcos B
2ac
a b ccos C
2ab
a bcosC c cosB
b = c cos A a cosC
c = a cos B b cosA
a b csin A sin B sinC
=2r
r A,B C
1ABC ab sinC2
1 ac sin B2
1 bc sin A2
ABC s(s a)(s b)(s c) a b cs2
bc
aCB
A
4M th
1. secx+tanx=A A0 sinx
1.2A 2.2
2
A 1
A 1
3.
2
A
A 1 4.
2
2A
A 1
2. 1 tan x 1 A cos x sin x1 tan x cos2x
A
1.1 2.2 3.4 4.6
3. 5cos sin3
sin2
1. 413
2.9
13 3.
4
9 4.
13
9
5M th
4. o o
o o
sin 30 cos30
sin10 cos10
1.1 2.1 3.2 4.2
5. x1 cot201 cot25
x [2]
6. x sin x cos x a sin x cos x b
sin4x
1. 3 31 (a b ab )2
2. 3 31 (ab a b)2
3. 3 3ab a b 4. 3 3a b ab
6M th
7. 1xfxx 1
x0 x1
02 2f(sec )
1. 2sin 2. 2cos 3. 2tan 4. 2cot
8. cos 36 cos 72sin 36 tan18 cos 36
9. osin15 ocos15 2x ax b 0 4a b
1. 1 2. 1 3. 2 4. 1+3 2
7M th
10.
44 44
n 1 n 144 44
n 1 n 1
cosn sin n
sin n cos n
11. n
1(sin1 )(sin 3 )(sin 5 ) (sin 89 )2
4n
8M th
12. a 5(sin a cos a) 2 sin a cos a 0.04
3 3125(sin a cos a) 75 sin a cos a
13. o o o o2 2 2 2log 1 tan1 log 1 tan2 log 1 tan 3 ... log 1 tan 44
9M th
14. A xcos x cos4
A (0, 4 )
15. o 2 o o
o 2 o
cos ec10 3 cos 70 sin 40A ,B
sec10 1 0 cos 50
2 o
o 2 o
cos 20 0C
sin 80 cos 10
det[A(B+C)]
10M th
16. R f : R R
0 ; x 1
f(x) x 1; x 1
x 1
A x R (f f)(x) cot75 1.A ( 3, 2) 2.A ( 4, 3) 3.A (2,3) 4.A (3,4)
17. 180o
11M th
18. tan20 4 sin20
sin 20 sin 40 sin 80
19. a,bR atanb
4 4
2 2
cos sin sin 2
a b ab(a b )
3 2
3a b
b 2a
12M th
20. 0x2
2 2A x log ( 3 cos x) 1 2 log (sin x)
B sec 3x cos2x x A B
21. o o0 45
tanA (sin )
cotB (sin )
sinC (cot )
cosD (cot )
1.A
13M th
22. 2 o o o o2 sin 60 (tan5 tan 85 ) 12 sin70
23.
. 3 1cos cos cos5 5 2
. 7 3tan tan cos ec16 8 8
1.. . 2.. . 3.. . 4.. .
14M th
24.
1. o ocos 75 (2 3)cos15 2. o o ocos10 sin 40 cos 20
3. A tan 3A cos 2A cos 4AcotA cos 2A cos 4A
4. A B sin2A cos2B 2sin(A B)cos(A B)
25.
a b 4 4cos sin 3sin 4 cos
a+b
15M th
26. , [ , 0] 2sin sin3
2cos cos3
1.6
2.3
3. 2
3
4. 4
3
5. 5
3
27. sin9x 6sin7x 17sin5x 12sin3xsin8x 5sin6x 12sin4x
x
16M th
28. 1A {0 x 2 | (cos2x cos 4x cos 6x) 1}3
A
29. arcsinx=2arccosx
1.1 2.2 3.3 4.4
30. 1x1 arccos x arcsin x2552
sin( )2552
1.2x 2.12 2x 3.2 2x 1 4.2x
17M th
31. arcsin(5x)+arcsin(x)=2
tan(arcsinx)
1. 15
2.1
3 3.
1
3 4.
1
2
32. 2 3(sin cos )2
04 arccos(tan 3 ) [0]
33. atanb
2 2 2 2
a acos arcsin sin arccos 1a b a b
sin
18M th
34. x arcsin x4
2sin( arccos(x ))
15
1. 1(0, )2
2.1 1( , )2 2
3.1 3
( , )22
4.3
( ,1)2
35.
71 1tan arc cot arc cot arctan5 3 9
5 12sin arcsin arcsin13 13
19M th
36. cot(arc cot 7 arc cot13 arc cot21 arc cot 31)
1. 114
2. 134
3. 92
4. 252
37. 3 5 8c arcsin arc cot arctan5 3 19
A 1 1arc cot arc cot c2x 3x
A
1. 14
2.1
4 3.
1
6 4.
1
6
20M th
38. A
2arccos(x) arccos(x 3) arccos( 1 x )
B
arccos(x) arcsin(x) arcsin(1 x)
P(S) S
P(AB)
39. 1 2arc sec x arcsin 2arccos17 5
cot arc sec x2
1. 139
2. 13
9 3. 13
16 4. 13
16
21M th
40. 2 1 1sec 2arctan arctan3 7
41. 1x1
1. sin(arccos x) cos(arcsin x) 2.arcsin x arcsin( x) 0
3.arccos x arccos( x)
4.arcsin x arccos x2
5. 1arctan x arctan , x 0x 2
42. 2 1sec (2 tan 2)
22M th
43. ABC D BC BAD CAD
BD 2CD
sin BsinC
1. 12
2.1 3.3
2 4.2
44. ABC A 60 BC= 6 AC=1
cos(2B)
1. 14
2.1
2 3.
3
2 4.
3
4
45. ABC D BC AB=4
AC=3 AD= 52
BC
1.3 2.4 3.5 4.6
23M th
46. ABC AB 2
3 3BC AC 2BC 2AC cotC [1]
1. 1
3 2. 1
2 3.1 4. 3
47. ABC a,b c A B C
1 1 1cosA cosB cosCa b c
1. 2 2 2a b c
2abc 2.
2(a b c)
abc 3.
2(a b c)
2abc 4.
2 2 2a b c
abc
24M th
48. ABC
o oABC 30 BAC 135 AD AE BAC 3
ECBC
1. 13
2. 3 3. 1
2 4. 2
49. ABC 3sin A5
5cos B13
cosC
1. 1665
2. 1665
3. 4865
4. 3365
A
B CD E
25M th
50. ABC A,B C a,b c
2 2 2a b 31c 3 tanC cotA cotB
51. ABC A,BC a,bc
(sinAsinB+sinC)(sinA+sinB+sinC)=3sinAsinC
2 23 cos e c B 3 sec B
26M th
52. ABC a,b c A B
C C o60 b=5 ac=2
ABC
1.25 2.29 3.37 4.45
53. ABC
a,b c A B C
1 1 3a c b c a b c
sinC
1. 22
2. 12
3. 32
4.1
27M th
54. a b A B ABC
2b=3a B 2A cosA
55. ABC A,B C a,b c
2 2 2a b 2555c cotCcotA cotB
28M th
1. sin 5 sin10 sin15 ... sin 355
2. cos12 cos 24 cos 36 ... cos168
3. arctan sin10 sin 20 sin 30 ... sin170
4.
44 44
k 1 k 1
44 44
k 1 k 1
cos k sin k
sin k cos k
***
5. tan18 (sin 36 sin 72 )
29M th
6. 1sin 20 sin 40 sin 80 cot102
7. sin18 sin 36 sin 54 sin 72cot9 1
8. 2 2 2
1 1 1
cos 10 sin 20 sin 40
***
9. sin9x 6sin7x 17sin5x 12sin3xsin8x 5sin6x 12sin4x
x
30M th
10. 3(1 sin )(1 cos )2
(1 sin )(1 cos )
11. n
1sin1 sin 3 sin 5 ... sin 892
4n
12. tan70 a tan10 b tan20 c tan 40 a+b+c
13. A,BR sinA+sinB=1 cosA+cosB= 12
sin2A+sin2B
31M th
14. 6 6sin cos
15. xR sin x cos x tan x cos ecx sexc cot x 8 sin2x
16. tana tanb 2x 17x 19 0
2 2sin (a b) 17 sin(a b)cos(a b) 19 cos (a b)
17. ABC oACB 120 ABC
AC AC D E BD=BE
BCE BE G CG : BD
32M th
18. ABC o A B 90 BC+CA=2(AB) cosC
19. ABC A B
cos2A+3cos2B=2 cosA 2 cos B 0 cosC
20. ABC A,B C a,b c
2 2 2a b 2555c cotCcotA cotB
33M th
1. ABC sin A : sin B : sinC 6 : 5 : 4
cosA : cosB : cosC [2:9:12]
2. O(0,0) A(1,0) B C 1
O C AB C
X X D CD 0.2 AB
[0.4]
3. ABC
(sinA+sinB+sinC)(sinA+sinBsinC)=3sinAsinB
tanC [ 3 ]
34M th
4. x,y z
2 2
2 2
2 2
x xy y 9
y yz z 16
z xz x 25
xy+yz+zx [8 3 ]
5. T(x) cos x cos 3x cos 5x ... cos 2555x x [0, ]
T(x)=0 [1277.5]
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