Inverse Trigonometric Function _ Class notes __ Manzil Legends-JEE (4).pdf

Short notes for itf which are quick for jee main

1. &

a: MANZIL 747

ONE SHOT
Inverse Trigonometry
Functions

2. La)
Cite
Vv

The ITF, denoted by sin”! x or arc sin x, cos”! x, tan”! x etc. denote the angles.

ae RR)

Siq AS!
€

&

3. la But trigonometric functions such as f(x) = tan x, f(x) = sin x etc. are not

bijective. Then how does these inverse exists?

4. &

a
{el | Domain, Range & Graph of ITF

Vv 3 = Sin la.

= SIMA, DA Be
E ) XE 33)

5. La)

| comments |

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(1) sin”! xis monotonical increasing in its domain.
(2) It is a bounded function.
er It is an odd function.

(4) It is aperiodic.

(5) It is continuous.

6. O = cos 19)

| | Je cos In)
| Y= ze, Ve DO. coso= |
4 ,

7. “a
IG cones (o

(1) cos’! xis monotonic decreasing in its domain.

(2) It is a bounded function.
(3) It is aperiodic.
(4) It is neither even nor odd.

(5) It is continuous.
A

8. | f= tm) Pa 2)

sin E A ES) os @ | =

en

10. La)
| comments |
v

(Taking image of tan x in y = x)

(1)
(2)
(3)
(4)
(5)

tan”! x is monotonic increasing.

It is an odd function.
It is a continuous function.
It is aperiodic.

It is bounded function.

y=arc tar

12. La)

| comments |

Vv

(1) _ Itis monotonic decreasing function.
(2) It is a bounded function.

(3) It is aperiodic. arc coix

(4) Itis continuous everywhere.

y=arc cotx

(9 It is neither even nor odd.

13. La]

Gi Fred
D 4

(1) It is increasing in (-, -1] and

Ss

then again it is increasing [1, ©)
(2) Itis aperiodic.
{3) Itis neither even nor odd.
(4) It is continuous, wherever it is
defined for |x| > 1.

(5) Itis a bounded function.

14. La)
AL comments
Vv

(1)

(2)
(3)
(4)
(5)

It is decreasing in (-, -1] and
then again it is decreasing [1, ©)
It is aperiodic function.

Itis a bounded function.

Itis an odd function.

It is continuous, wherever it is

defined.

15. |x| <1
|x| <1

xeR
E
Ix|>1 10,1] = [5] or [0,3) u Gr]
nn

-5,5]- 0

Ixl21 2:2

xeR |

(0, 1)

16. From the above discussion following IMPORTANT points can be concluded.

A All the inverse trigonometric functions represent an angle.

(2)

(3)

If x > 0, then all six inverse trigonometric functions viz sin”! x, cos”! x, tan
Sun IN

“1x, sec“ x, cosec! x, cot"? x. Civer ong le, a Ast quede

If x < 0, then sin”! x, tan! x and cosec”! x represent an angle from -x/2
IN Nes =

to 0 (IV! quadrant).

17. a
{|| cos) e pS seé\az E

(4) If x < 0, then cos”! x, cot-! x and sec”! x le an obtuse _
— Le La ew —,

(Id quadrant)

(5) — HI quadrant is never used in the range of any inverse trigonometric

—

function.

18. @

(2) +c0s7! (-3) ON) + cot-1( (-3 ad
IN 0
a (ne raw)
7 1
lA
a ab
a 8 à Y ea !

N.

EP
Bact 6)

&

19. Us ise

(20) = 255 eof

DS

an

E)

20. ke en 1 D: + me = a — 0

COSRX + a
"

Ita RnB

Venta t tip

ha NE ne.
+. ety

Ty =

=o.

&

21. &

i 1 Aa)
sın (tan cos cot ( +) zu,

22. &

tan (2tan!2 2) = =

23. Find domain & range of the following functions : =

(a) fx) = cost [x]
ane: a i e,
4 <@< ay’ Peet E), e 2 ) e)

Tisak) [Paren 3,0
Fes EE)

24. a ER =y= sin"! be

o< <a

&

25. (d) fix) = cos"! (sgn x)

&

26. (a) (a) If cos”! x + cos”! y + cos”! z = 31, then compute the value of
x2004 4 y2004 4 72004 4 6

12003 42003 472003

27. (b) If (sin! x + sin”! y) (sin-! w + -1 z) = x2, then compute the value of =

(x+y) (w+z)

30. A

-1 1/77 E
The number of real solutions of tan VxG +1) + sin 1x2 + x +1 = Fis:

tel RD... Na

B: X+az 0: nn. xt
ds Rau:

© Infinite

31. © Ko Ko Ko |

If «= 3sin”* 8) and ß = 3cos”* 0) where the inverse trigonometric
functions take only the principal values, then the correct option(s) is/are :
[2005 Adv.]

cosß>0
sinB <0
cos (a + B) > 0

cosa<0

PN

32. The number of real roots of the equation

-1 1/2 = ice ins-
tan-1/x(x +1) + sin her x+1=7 is: [JEE Mains-2021]
APIO FS
pee

&

33. a

The domain of the function cosec” 4: is:

À
[JEE Mains-2021] “7

mE...

lo) von atkeyp or atl <-|
[-7,0) v 4,0) e a dí Be
> e IV

D 3% CE S Ne

ERICH
. _ y 0
DB 50 +.

34. A . any (3x+x-1 AN
The domain of the function f(x) = sin”! ( Fan: ) cos 1 (23) is
[JEE Mains-2022]

D p3
sofa
D [uo
B bos

&

35. La)
lll Properties of Inverse Trigonometric Function
Vv

0)
(ii)
(iii)
(iv)
(v)
(vi)

sin-!(-x) = -sin-1x ;
tan-!(-x) = -tan-ix;
cos!(-x)=n-cos'x;
cot-!(-x) = 11 - cot!x;
sec”1(-x) = x - sec’'x;

cosec-1(-x) = x - cosec-1x;

Ix] <1
xeR
Ix] <1
xeR
Ix]>1

Ix]>1

36. x e [-1, 1], y e [-1, 1], y is aperiodic
Sinfsi'(4)) =
in 3 (& )) %.
DA

(ii) y = cos(cos”*x) = x

x e [-1, 1], y e [-1, 1], y is aperiodic

37. (iii) y = tan(tan-!x) =x
ue
xe Ry eR, y is aperiodic

——
a

(4e (3) = 1

(iv) y = cot(cot-!x) = x
cae
xe Ry e R,yis aperiodic

38. (v) y= cosec(cosec-1x) = x

|x| > 1, ly] > 1, y is aperiodic

(vi) y = sec(sec-!x) = x

|x| > 1, ly| > 1, y is aperiodic

39. [Y= we}

„el a).

sl ae FE
Sin (e

wa

en

41. 0)

&

y = sin”*(sin x), x, e Rye [- = =] periodic with period 2x and it is an odd
function.

Note : To draw the graph, plot only between x e [0, 1] and draw rest of the
graph using periodicity and odd function property.

7
—T—X, ASS

a 7 x
sin”*(sinx) = % RFA

42. (ii) y = cos-!(cos x), x, e R, y € [0, x] periodic with period 2x and it is an even
function.
Note : To draw the graph, plot only between x e [0, x] and draw rest of the
graph using periodicity and even function property.

=x, —Tr<x<o0

x, 0<x<rT yelasy-r relo,X].

cos”*(cosx) =

43. \@

Y= sit (sian) |] sin (sma) =
Sin (sh) = 1-3
EE AA à
AS

44. | gin !(sinz2) = -22 4 A N {ke mol!

sin!(siaa3) = 23-8% = a...
Urs ame

45. 3

cos (cost) +

E ess (esu) = HA
i , css) :

46. co (0533) = 2348

ZN RES BK e

lo

x

\@

47. \@

Sy a
sm) = su) sel (sig = a
sint{_sina Y _ _ giql( sina
ee) | 1) N 4)

= SI, de 2 sn sra,

A = she

1

48. (ii) y = tan”*(tan x), x, e R- {(2n = 1)3n € 1} ye (5,5) periodic with period
mand it is an odd function.
Note : To draw the graph, plot only between x e (5,5) and draw rest of the
graph using periodicity.

xtn, —E<xr<-!
2 2

n 7
tan”*(tanx) ={ x, SDE

49. (iv) y = cot (cot x), x, e R- {nn}, y e (0, x) periodic with period x and neither
even nor odd function
Note : To draw the graph, plot only between x e (0, x ) and draw rest of the
graph using periodicity.

Ste, H<x<0
cot (cotxy={ x 0O<x<r
XT, n<x<2n

50. (v) y = cosec”!(cosec x), x, e R- {na,ne I}, ye [-2,0) U (0,2] periodic with

period 27 and it is an odd function.

51. (vi) 5 y=sec’!(sec x), y is periodic with period 2x.

n

xer-{(2n-1%nei},ye [0,2) u x]

2

52. Solve : cos” cos (= =) + cot cot (=)

&

53. MO torear (oS) + sin"? (nt de = er
(ns À oe éal ( u

cos (+ 3) al) E
on

Sin (5) A.

OB

54. Find the integral solution of the inequality
3x? + 8x < 2sin”*(sin 4) - cos“!(cos 4)

55. P-4
() cosec"! x = sin“!

(ii)

; |x] > 1 and sin"!x = cosec-1à |x} <1x#0

sec! x= ee) |x| > 1 and cos”tx = sect, a <1x#0
“11 zl

(iii) cotTix= \. MER ee E COsec a Sin at) = 2
&® «E (= fe) _A
© , 3 6 sec A(g)= ie 10

>? cot (8) = 52 Pace a

+! (-4 = i à

N:

56. Ps ®

DW simtxpcosi=zho, |x| <1 © «ini la = À
] Be reia) à.
Gi) tamixpcorj=h , xeR a “7
n
(ii) cosec”* xfsecd= 5 e |x| 21 % G

le: mel spi &@ vie rests) =
2.
I nl =
yang Oy

=| E |
> + ceo Eo ESE: |

57. Find sin-tcos (E)
10

58. Solve : i

a 4sin-!x + cos'!x=n
Se

Ust. + nu Pa

Ssitaz À
Z.

&

59. Solve : 0 neon n-!x + 3cot"!x = 27

&+n | lat 3 (dei! At E AR.

ES

Rh SAA _ x
2.

5 3G)

&

60. Maximum & Minimum values of (sin”!x)? + ad A

Pz CS Sab (arb).

> u (sit E est) = Es x) — 3 (cor viene Atsia F9

Ê = sx (cos! gotta) (sta! x)

y= 2 max es er x). [ a
Tr,

61. Maximum & Minimum values of (sin-!x)? + (cos-!x)? ) = Range =

PE = jo gs
a 34 + 3a! à

EYE

+

x
4

\~

62. (sita + Ces x us minimum vue

ie ern

en

ei +" a

63. Maximum & Minimum values of sin-1x + cos“!x + tan”!x
SS

64. @

E

The value of (e *) is equal to [JEE Main-2022] ==
4.

| NT ea ES)

ro je.

65. \@

3%

Ct
sono = 70528 +34 _ | —cos3%
a ana &- ar
9= 1%. 1.

66. a

© Ko Ko Ko |

sin“? (sin=) +cos? (cos) + tan? (tan) is equal to [JEE Main -2022] =.

117
12

177
12

317
12

ELA

67. a

© Ko Ko Ko |

The value of tan (2tan- 8) + sin"! 8) isequalto [JEE Main -2021]

—181
69

220
2

—291
76

151
63

68. @
The set of all values of k for which (tan~1x)3 + (cot"!x)? = =kn3,xER,is 7
the interval. [JEE Mains-2022]
<a A = Han! a) (cot A ars ae
Gen! at je 1G ee y) 2)
Co

Lo lea) y = Be 25 (al) RE tr), let ma
lO)

3 Ss
(8) Le a) ya ney un nt
de .

=e ES)

3 +3
3
ra (e hts 7 2)

D |
D EE TK x)

=)

69. &

The set of all values of k for which (tan=!x)? + (cot"!x)? = = kn?,x € R,is

the interval. [JEE Mains- ms

DEEE ie
gt (a +

I is lb tence
G2) = tea)
lo Eu 2) : ; -

32'8

D ki RAI) 2

71. tan”! x+y

tan-!x + tan-1y =| + tan! =,

T

== p

+y

ro Se: "| Say;
EN) Fa (a) 4485 (3) = er

x > 0,y > 0 and xy < 1(acute angle)

y $ > 0, y > 0 and. xy > 1(obtuse angle)

, x*>0y>0andxy=1
—

1

2x3)
= K+ toa.
= T- el

&

72. 1 18
Ds da et ee ee ®
1 ten ¢ = |
nee, Ai

|— tank fang
PES) 5
6. = an B-
SES LG = Ge)
o <XtBCA
9¢ BC
ire
«+=. > v= ton" la,

Faces) = Hand. my
|

73. CRD! RS
wees aa

\@

Û A ten OL

Ia

Gall X+Y |
See ey (tano), a)
a >

ie cay ee
Tess ack.

(ES
JT. Hog <cotp

| =
; aa)
É

KB,
X<T

74. (a) tanTix —tan"ly = tan 2

1+xy

SALE 9$B<%H
2 2

2 Sx-p <a

x>0y>0

&

75. La]

ca
u,

[ah 1+tan-12 + td? 3=x) (Remember)

(ii) cot! 1 +cot! 2 + cot13== = (Remember)
“Y q

z Hi) qe (1 E
rare)

“a=

&

76. la Prove that: tan”*5 — tan-13 + tan-t2 = 2

77. Prove that: sec E) se)

Le
poa _H

B

Secx =

Me

tua o)

E)

a, + sin” a

Sr

T

N
Eb

3.

78. Powhatan (Gr) (0) + an (Ga) = ET
ee

(2) + tel
ik EA o

E an! Er =f A \

(i) fe) (Bee)

&

79. q . (tan741+tan712+tan713
(o) Prove that: sin”! [sin ze] =n-2

cot”t1+cot”12+cot”13

80. If tan-14 + tan-*5 = cot~1(A), then find 2.

81. oe 17 3
la Which is greater cos”! 7, teos 12 or cot"! (-1).

82. V7

lo, tanta + tan 'y + tan”?

(bte), 4
ian ei) where x > 0,y > 0,z > 0 and xy + yz + zx
E a [_x+ytz-xyz ],
a al where x > 0,y > 0,2 >0andxy+yz+zx>1

83. &

sin-!x + sin=!y
u sin*(x/1—y? +yV1=22) ‚ifx>0;y>0andx2+y2<ı)

n-sint(x AY +yvI—x) ifx >0;y >Oandx*+y?>1

OA reos O. cs
Sid nt sity = 8: ks om
x

ee 0 = sir (ange + VTE),
Sinx+p) = sin. eee ESA

Sing cosp + aa = Sin). T-B= sin! (arg HT)
NP te |

84. sin-1x — sin!y = sin*(x/ 1-y?-y

2),

>0;y>0

E)

85. cos~!x + cos~ty = cos”! [xy -v1-x2/1 =y2]
was es

er]
Cos (+B) zesse.

cosx CHR — SNASUB = cos D.
STE
Ne ETES

oo) ETS) kosa» = 0

86. à A cos“!(xy + I=? 1-72) y <y,x,y>0
cos” X — cos” y= :
bw Up —cos“*(xy + VI =x2,/1 =y?) x>y,xy>0

&

87. T—sin 12)

65.

33

Ti COS E =)

= inc =)
5 sin

65.

Ecos”! (2)
2 65

oo gs

The value of sin”! (2) — sin? 8) is equal to: [JEE Main-2019 (April)]

Sia =sinty = si (a Vay TE)

ET - 25)

— su 19 “6

88. The value of sin”! (2) — sin? 8) is equal to: [JEE Main-2019 (April)]

ns (©) id = i
= Ti NG:)(a 9 = NE x 196.

a — cos”! (8) = 6

»
æ
Pu
D

Z—cos*(2=) _ Mer oa) )

89. 15 1 16
ao 3 +sin + Di is equal to:

[JEE Main-2020 (September)]

OO)

mp2 RA [si 12, 3 4/16
» | (ats = an ©)
gr à E 2)
Sin y Is _
D 57/2 | D ü IE NE
Cor

2n - (sin “144 sin

90. n

(x > >, then x is equal to

[2019 Main, 9 Jan I]

&

91. La]

E | Inverse Trigonometry Function
Vv

&

92. Simplification & Transformation of Inverse Functions

by Elementary Substitution & Their Graphs

a —(n + 2tan"*x) if x < —1
1. y=f(x) = sin! (5) = 2tanix if |x| <1
m—2tan-1x if x>1

(where, / = increasing and D = Decreasing)

Domain: R, Range : [- a

93. age (-%%)
NE de) . =

=
Let O= tm x.

ehr ge =
Y= sin! tan)
US

IRRE See (5 =

) TNR tainty XQ] à
e)= tx! Mo AL
EA

(q

94. ©

95. 1+x2

y = f(x) = cos“! (=) yA [A <0

Domain: R,

2tan-1x ifx>0

Range: [0, x)
X=+m8 .
DEl-x x
(3%)

8. ert)

&

96. &

97. E m+2tanix ifx<-1
y = f(x) = tan! A = 2tan"!x if lx] <1
—(m— 2tan™x) if x>1

Domain : R - (-1, 1), Range : (— =)
Katou

99. La)
GIN
Vv

Fi 2x 1-x? 2x
-1y = sin"! = -1 = -1
2tan”*x = sin a cos He = tan 12

for x € [0-1)

100. re es = f(5)+f(10)+f(3)
If f (x) = sin Ta + 2tan™x, then find POTES PRESS