Inverse Trigonometry (NCERT) Class 12 Notes | EduRev

Class 12 : Inverse Trigonometry (NCERT) Class 12 Notes | EduRev

 Page 1


Inverse Trigonometry – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 0
A. INTRODUCTION
sin
–1 
x, cos
–1 
x, tan
–1 
x etc. represents angles or numbers whose values of sine, cosine and  tangent is ‘x’,
provided that the value in numerical form is smallest. These can be written as arc sin x, arc cos x etc. If two
angles whose modulus is equal, in which one is positive and other is negative then we take positive sign.
B. DOMAIN & PRINCIPLE VALUE RANGE OF INVERSE TRIGONOMETRIC FUNCTIONS
} 0 {
2
,
2
y ) , 1 [ ] 1 , (– x x ec cos y . 6
2
] , 0 [ y ) , 1 [ ] 1 , (– x x sec y . 5
) , 0 ( y R x x cot y . 4
2
,
2
y R x x tan y . 3
] , 0 [ y ] 1 , 1 [ x x cos y . 2
2
,
2
y ] 1 , 1 [ x x sin y . 1
) PVR ( range value inciple Pr Domain Function . No . S
1
1
1
1
1
1
?
?
?
?
?
?
? ? ?
? ? ? ? ? ? ? ?
?
?
?
?
?
? ?
? ? ? ? ? ? ? ? ?
? ? ? ?
?
?
?
?
?
? ? ?
? ? ? ?
? ? ? ? ?
?
?
?
?
?
? ? ?
? ? ? ? ?
?
?
?
?
?
?
C. GRAPH OF INVERSE TRIGONOMETRIC FUNCTIONS
(a) f : [– ?/2, ?/2] ? [–1, 1] f
–1
 : [–1, 1] ? [– ?/2, ?/2]
f(x) = sin x f
–1
(x) = sin
–1
x
0 1
–1
2 / ?
– 2 / ?
x
y
y=arcsinx
y=x
y=sinx
y=sinx
y=arcsinx
y=x
2 / ?
1
– 2 / ?
–1
0 1
–1
x
y
y=arcsinx
y=arcsinx
2 / ?
– 2 / ?
(b) f : [0, ?] ? [–1, 1] f 
–1
 : [–1, 1] ? [0, ?]
f(x) = cos x f 
–1
(x) = cos
–1
 x
     
0 1 –1
2 / ?
x
y
y=arc cosx
y=x
y=cosx
y=x
2 / ?
1
–1
?
?
0 1 –1
x
y
2 / ?
1
?
INVERSE TRIGONOMETRY
INVERSE TRIGONOMETRY
Page 2


Inverse Trigonometry – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 0
A. INTRODUCTION
sin
–1 
x, cos
–1 
x, tan
–1 
x etc. represents angles or numbers whose values of sine, cosine and  tangent is ‘x’,
provided that the value in numerical form is smallest. These can be written as arc sin x, arc cos x etc. If two
angles whose modulus is equal, in which one is positive and other is negative then we take positive sign.
B. DOMAIN & PRINCIPLE VALUE RANGE OF INVERSE TRIGONOMETRIC FUNCTIONS
} 0 {
2
,
2
y ) , 1 [ ] 1 , (– x x ec cos y . 6
2
] , 0 [ y ) , 1 [ ] 1 , (– x x sec y . 5
) , 0 ( y R x x cot y . 4
2
,
2
y R x x tan y . 3
] , 0 [ y ] 1 , 1 [ x x cos y . 2
2
,
2
y ] 1 , 1 [ x x sin y . 1
) PVR ( range value inciple Pr Domain Function . No . S
1
1
1
1
1
1
?
?
?
?
?
?
? ? ?
? ? ? ? ? ? ? ?
?
?
?
?
?
? ?
? ? ? ? ? ? ? ? ?
? ? ? ?
?
?
?
?
?
? ? ?
? ? ? ?
? ? ? ? ?
?
?
?
?
?
? ? ?
? ? ? ? ?
?
?
?
?
?
?
C. GRAPH OF INVERSE TRIGONOMETRIC FUNCTIONS
(a) f : [– ?/2, ?/2] ? [–1, 1] f
–1
 : [–1, 1] ? [– ?/2, ?/2]
f(x) = sin x f
–1
(x) = sin
–1
x
0 1
–1
2 / ?
– 2 / ?
x
y
y=arcsinx
y=x
y=sinx
y=sinx
y=arcsinx
y=x
2 / ?
1
– 2 / ?
–1
0 1
–1
x
y
y=arcsinx
y=arcsinx
2 / ?
– 2 / ?
(b) f : [0, ?] ? [–1, 1] f 
–1
 : [–1, 1] ? [0, ?]
f(x) = cos x f 
–1
(x) = cos
–1
 x
     
0 1 –1
2 / ?
x
y
y=arc cosx
y=x
y=cosx
y=x
2 / ?
1
–1
?
?
0 1 –1
x
y
2 / ?
1
?
INVERSE TRIGONOMETRY
INVERSE TRIGONOMETRY
Inverse Trigonometry – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 2
(c) f : (– ?/2, ?/2) ? R f 
–1
 : R ? (– ?/2, ?/2)
f(x) = tan x f
–1
 (x) = tan
–1
 x
0
2 / ? – 2 / ?
x
y
y=tanx
y=x
y=arc tanx
y=arc tanx
y=tanx
y=x
–
?
– 2 / ?
?
?
2 / ?
0
x
y
y=arc tanx
y=arc tanx
– 2 / ?
2 / ?
(d) f : (0, ?) ? R f 
–1
 : R ? (0, ?)
f(x) = cot x f
–1
 (x) = cot
–1
 x
0 2 / ?
x
y
y=x
y=arc cotx
y=arc cotx
y=cotx
y=x
–
?
– 2 / ?
?
?
2 / ?
0
x
y
y=arc cotx
?
?
2 / ?
y=arc cotx
(e) f : [0, ?/2) ? ( ?/2, ?] ? (– ?, –1] ? [1, ?)
f(x) = sec x
f
–1
 : (– ?, –1] ? [1, ?) ? [0, ?/2) ? [ ?/2, ?]
0 1 –1
x
y
2 / ?
?
f
–1
 (x) = sec
–1
 x
(f) f : [– ?/2, 0) ? (0, ?/2] ? (– ?, –1] ? [1, ?)
f(x) = cosec x
0 1 –1
x
y
2 / ?
– 2 / ?
f
–1
 : (– ?, –1] ? [1, ?) ? [– ?/2, 0) ? (0, ?/2]
f
–1
 (x) = cosec
–1
 x
Basis on the above discussion we get following results :
(i) All inverse trigonometric functions shows angle.
(ii) If x ? 0 then all six trigonometric functions sin
–1
 x, cos
–1
 x, tan
–1
 x, sec
–1
 x, cosec
–1
 x, cot
–1
 x
shows acute angle.
(iii) If x < 0 then sin
–1
 x, tan
–1
 x and cosec
–1
 x, shows angle between – ?/2 to 0 (IV quadrant)
(iv) If x < 0 then cos
–1
 x, cot
–1
 x and sec
–1
 x shows obtuse angle (II quadrant)
(v) III quadrant never used in inverse trigonometric functions.
Page 3


Inverse Trigonometry – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 0
A. INTRODUCTION
sin
–1 
x, cos
–1 
x, tan
–1 
x etc. represents angles or numbers whose values of sine, cosine and  tangent is ‘x’,
provided that the value in numerical form is smallest. These can be written as arc sin x, arc cos x etc. If two
angles whose modulus is equal, in which one is positive and other is negative then we take positive sign.
B. DOMAIN & PRINCIPLE VALUE RANGE OF INVERSE TRIGONOMETRIC FUNCTIONS
} 0 {
2
,
2
y ) , 1 [ ] 1 , (– x x ec cos y . 6
2
] , 0 [ y ) , 1 [ ] 1 , (– x x sec y . 5
) , 0 ( y R x x cot y . 4
2
,
2
y R x x tan y . 3
] , 0 [ y ] 1 , 1 [ x x cos y . 2
2
,
2
y ] 1 , 1 [ x x sin y . 1
) PVR ( range value inciple Pr Domain Function . No . S
1
1
1
1
1
1
?
?
?
?
?
?
? ? ?
? ? ? ? ? ? ? ?
?
?
?
?
?
? ?
? ? ? ? ? ? ? ? ?
? ? ? ?
?
?
?
?
?
? ? ?
? ? ? ?
? ? ? ? ?
?
?
?
?
?
? ? ?
? ? ? ? ?
?
?
?
?
?
?
C. GRAPH OF INVERSE TRIGONOMETRIC FUNCTIONS
(a) f : [– ?/2, ?/2] ? [–1, 1] f
–1
 : [–1, 1] ? [– ?/2, ?/2]
f(x) = sin x f
–1
(x) = sin
–1
x
0 1
–1
2 / ?
– 2 / ?
x
y
y=arcsinx
y=x
y=sinx
y=sinx
y=arcsinx
y=x
2 / ?
1
– 2 / ?
–1
0 1
–1
x
y
y=arcsinx
y=arcsinx
2 / ?
– 2 / ?
(b) f : [0, ?] ? [–1, 1] f 
–1
 : [–1, 1] ? [0, ?]
f(x) = cos x f 
–1
(x) = cos
–1
 x
     
0 1 –1
2 / ?
x
y
y=arc cosx
y=x
y=cosx
y=x
2 / ?
1
–1
?
?
0 1 –1
x
y
2 / ?
1
?
INVERSE TRIGONOMETRY
INVERSE TRIGONOMETRY
Inverse Trigonometry – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 2
(c) f : (– ?/2, ?/2) ? R f 
–1
 : R ? (– ?/2, ?/2)
f(x) = tan x f
–1
 (x) = tan
–1
 x
0
2 / ? – 2 / ?
x
y
y=tanx
y=x
y=arc tanx
y=arc tanx
y=tanx
y=x
–
?
– 2 / ?
?
?
2 / ?
0
x
y
y=arc tanx
y=arc tanx
– 2 / ?
2 / ?
(d) f : (0, ?) ? R f 
–1
 : R ? (0, ?)
f(x) = cot x f
–1
 (x) = cot
–1
 x
0 2 / ?
x
y
y=x
y=arc cotx
y=arc cotx
y=cotx
y=x
–
?
– 2 / ?
?
?
2 / ?
0
x
y
y=arc cotx
?
?
2 / ?
y=arc cotx
(e) f : [0, ?/2) ? ( ?/2, ?] ? (– ?, –1] ? [1, ?)
f(x) = sec x
f
–1
 : (– ?, –1] ? [1, ?) ? [0, ?/2) ? [ ?/2, ?]
0 1 –1
x
y
2 / ?
?
f
–1
 (x) = sec
–1
 x
(f) f : [– ?/2, 0) ? (0, ?/2] ? (– ?, –1] ? [1, ?)
f(x) = cosec x
0 1 –1
x
y
2 / ?
– 2 / ?
f
–1
 : (– ?, –1] ? [1, ?) ? [– ?/2, 0) ? (0, ?/2]
f
–1
 (x) = cosec
–1
 x
Basis on the above discussion we get following results :
(i) All inverse trigonometric functions shows angle.
(ii) If x ? 0 then all six trigonometric functions sin
–1
 x, cos
–1
 x, tan
–1
 x, sec
–1
 x, cosec
–1
 x, cot
–1
 x
shows acute angle.
(iii) If x < 0 then sin
–1
 x, tan
–1
 x and cosec
–1
 x, shows angle between – ?/2 to 0 (IV quadrant)
(iv) If x < 0 then cos
–1
 x, cot
–1
 x and sec
–1
 x shows obtuse angle (II quadrant)
(v) III quadrant never used in inverse trigonometric functions.
Inverse Trigonometry – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 2
Ex.1 Find the domain of the following functions.
(i) sin
–1
 ?n x (ii) cos
–1
 [x] (iii) sin
–1
 {x}
Sol. (i) f(x) = sin
–1
 ?n x  ?  –1 ? ?n x ? 1  ? ? ?
e
1
 ? x ? e
(ii) f(x) = cos
–1
 [x]  ?  –1 ? [x] ? 1  ? ? ?[x] = –1, 0, 1  ?  x ? [–1, 0) ? [0, 1) ? [1, 2) ? ?x ? [–1, 2)
(iii) f(x) = sin
–1
 {x}  ? ? –1 ? {x} ? 1  ?  x ? R
Ex.2 tan
–1
 (1) + cos
–1
 
?
?
?
?
?
?
?
2
1
 + sin
–1
 
?
?
?
?
?
?
?
2
1
 is equal to
Sol. tan
–1
 (1) + cos
–1
 
?
?
?
?
?
?
?
2
1
 + sin
–1
 
?
?
?
?
?
?
?
2
1
 = 
4
3
2 4 6 3
2
4
?
?
?
?
?
?
?
?
?
?
?
D. PROPERTIES OF INVERSE TRIGONOMETRIC FUNCTIONS
P-1 (i) y = sin (sin
–1
 x) = x (ii) y = cos (cos
–1
 x) = x
x ? [–1, 1], y ? [–1, 1] x ? [–1, 1], y ? [–1, 1]
45º
0
y = x
1
x
y
–1
–1
1
45º
0
y = x
1
x
y
–1
–1
1
(iii) y = tan (tan
–1
 x) = x (iv) y = cot (cot
–1
 x) = x
x ? R, y ? R x ? R, y ? R
45º
0
y = x
x
y
45º
0
y = x
x
y
(v) y = cosec (cosec
–1
 x) = x (vi) y = sec (sec
–1
 x) = x
| x | ? 1, | y | ? 1 | x | ? 1 ; | y | ? 1
0
y = x
1
x
y
–1
–1
1
y = x
0
y = x
1
x
y
–1
–1
1
y = x
Page 4


Inverse Trigonometry – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 0
A. INTRODUCTION
sin
–1 
x, cos
–1 
x, tan
–1 
x etc. represents angles or numbers whose values of sine, cosine and  tangent is ‘x’,
provided that the value in numerical form is smallest. These can be written as arc sin x, arc cos x etc. If two
angles whose modulus is equal, in which one is positive and other is negative then we take positive sign.
B. DOMAIN & PRINCIPLE VALUE RANGE OF INVERSE TRIGONOMETRIC FUNCTIONS
} 0 {
2
,
2
y ) , 1 [ ] 1 , (– x x ec cos y . 6
2
] , 0 [ y ) , 1 [ ] 1 , (– x x sec y . 5
) , 0 ( y R x x cot y . 4
2
,
2
y R x x tan y . 3
] , 0 [ y ] 1 , 1 [ x x cos y . 2
2
,
2
y ] 1 , 1 [ x x sin y . 1
) PVR ( range value inciple Pr Domain Function . No . S
1
1
1
1
1
1
?
?
?
?
?
?
? ? ?
? ? ? ? ? ? ? ?
?
?
?
?
?
? ?
? ? ? ? ? ? ? ? ?
? ? ? ?
?
?
?
?
?
? ? ?
? ? ? ?
? ? ? ? ?
?
?
?
?
?
? ? ?
? ? ? ? ?
?
?
?
?
?
?
C. GRAPH OF INVERSE TRIGONOMETRIC FUNCTIONS
(a) f : [– ?/2, ?/2] ? [–1, 1] f
–1
 : [–1, 1] ? [– ?/2, ?/2]
f(x) = sin x f
–1
(x) = sin
–1
x
0 1
–1
2 / ?
– 2 / ?
x
y
y=arcsinx
y=x
y=sinx
y=sinx
y=arcsinx
y=x
2 / ?
1
– 2 / ?
–1
0 1
–1
x
y
y=arcsinx
y=arcsinx
2 / ?
– 2 / ?
(b) f : [0, ?] ? [–1, 1] f 
–1
 : [–1, 1] ? [0, ?]
f(x) = cos x f 
–1
(x) = cos
–1
 x
     
0 1 –1
2 / ?
x
y
y=arc cosx
y=x
y=cosx
y=x
2 / ?
1
–1
?
?
0 1 –1
x
y
2 / ?
1
?
INVERSE TRIGONOMETRY
INVERSE TRIGONOMETRY
Inverse Trigonometry – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 2
(c) f : (– ?/2, ?/2) ? R f 
–1
 : R ? (– ?/2, ?/2)
f(x) = tan x f
–1
 (x) = tan
–1
 x
0
2 / ? – 2 / ?
x
y
y=tanx
y=x
y=arc tanx
y=arc tanx
y=tanx
y=x
–
?
– 2 / ?
?
?
2 / ?
0
x
y
y=arc tanx
y=arc tanx
– 2 / ?
2 / ?
(d) f : (0, ?) ? R f 
–1
 : R ? (0, ?)
f(x) = cot x f
–1
 (x) = cot
–1
 x
0 2 / ?
x
y
y=x
y=arc cotx
y=arc cotx
y=cotx
y=x
–
?
– 2 / ?
?
?
2 / ?
0
x
y
y=arc cotx
?
?
2 / ?
y=arc cotx
(e) f : [0, ?/2) ? ( ?/2, ?] ? (– ?, –1] ? [1, ?)
f(x) = sec x
f
–1
 : (– ?, –1] ? [1, ?) ? [0, ?/2) ? [ ?/2, ?]
0 1 –1
x
y
2 / ?
?
f
–1
 (x) = sec
–1
 x
(f) f : [– ?/2, 0) ? (0, ?/2] ? (– ?, –1] ? [1, ?)
f(x) = cosec x
0 1 –1
x
y
2 / ?
– 2 / ?
f
–1
 : (– ?, –1] ? [1, ?) ? [– ?/2, 0) ? (0, ?/2]
f
–1
 (x) = cosec
–1
 x
Basis on the above discussion we get following results :
(i) All inverse trigonometric functions shows angle.
(ii) If x ? 0 then all six trigonometric functions sin
–1
 x, cos
–1
 x, tan
–1
 x, sec
–1
 x, cosec
–1
 x, cot
–1
 x
shows acute angle.
(iii) If x < 0 then sin
–1
 x, tan
–1
 x and cosec
–1
 x, shows angle between – ?/2 to 0 (IV quadrant)
(iv) If x < 0 then cos
–1
 x, cot
–1
 x and sec
–1
 x shows obtuse angle (II quadrant)
(v) III quadrant never used in inverse trigonometric functions.
Inverse Trigonometry – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 2
Ex.1 Find the domain of the following functions.
(i) sin
–1
 ?n x (ii) cos
–1
 [x] (iii) sin
–1
 {x}
Sol. (i) f(x) = sin
–1
 ?n x  ?  –1 ? ?n x ? 1  ? ? ?
e
1
 ? x ? e
(ii) f(x) = cos
–1
 [x]  ?  –1 ? [x] ? 1  ? ? ?[x] = –1, 0, 1  ?  x ? [–1, 0) ? [0, 1) ? [1, 2) ? ?x ? [–1, 2)
(iii) f(x) = sin
–1
 {x}  ? ? –1 ? {x} ? 1  ?  x ? R
Ex.2 tan
–1
 (1) + cos
–1
 
?
?
?
?
?
?
?
2
1
 + sin
–1
 
?
?
?
?
?
?
?
2
1
 is equal to
Sol. tan
–1
 (1) + cos
–1
 
?
?
?
?
?
?
?
2
1
 + sin
–1
 
?
?
?
?
?
?
?
2
1
 = 
4
3
2 4 6 3
2
4
?
?
?
?
?
?
?
?
?
?
?
D. PROPERTIES OF INVERSE TRIGONOMETRIC FUNCTIONS
P-1 (i) y = sin (sin
–1
 x) = x (ii) y = cos (cos
–1
 x) = x
x ? [–1, 1], y ? [–1, 1] x ? [–1, 1], y ? [–1, 1]
45º
0
y = x
1
x
y
–1
–1
1
45º
0
y = x
1
x
y
–1
–1
1
(iii) y = tan (tan
–1
 x) = x (iv) y = cot (cot
–1
 x) = x
x ? R, y ? R x ? R, y ? R
45º
0
y = x
x
y
45º
0
y = x
x
y
(v) y = cosec (cosec
–1
 x) = x (vi) y = sec (sec
–1
 x) = x
| x | ? 1, | y | ? 1 | x | ? 1 ; | y | ? 1
0
y = x
1
x
y
–1
–1
1
y = x
0
y = x
1
x
y
–1
–1
1
y = x
Inverse Trigonometry – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 4
(vii) y = sin
–1
 (sin x), x ? R, y ? ?
?
?
?
?
? ? ?
?
2
,
2
, is periodic function with period 2 ? and it is an odd function
sin
–1
 (sin x) = 
?
?
?
?
?
?
?
?
?
? ? ?
?
? ?
?
? ?
?
?
?
? ? ? ? ? ? ? ?
x
2
, x
2
x
2
, x
2
x , x
     
/2
y
0
?
?
? ?
–  /2
?
y=–( +x) ?
2
3 ?
?
? ?2
x
2
?
?
2
?
2
3 ?
y=2 +x ?
y=x
? 2
y=x–2 ?
y=  –x ?
(viii) y = cos
–1
 (cos x), x ? R, y ? [0, ?], is periodic function with period 2 ? and it is an even function
cos
–1
 (cos x) = 
x , x 0
x , 0 x
2 x , x 2
x 2 , 2 x 3
? ? ? ? ? ?
? ? ? ?
?
? ? ? ? ? ?
?
? ? ? ? ? ?
?
y=x+2 ?
y
?
?
2 ?
0 ? ?
–3 ?/2 –2 ?
– ?/2 ?/2 3 ?/2
y=–x
y=x
y=2 –x ?
2
?
(ix) y = tan
–1
 (tan x), x ? R – ?
(2n 1) , n I
2
? ? ?
? ?
? ?
? ?
;  y ? 
,
2 2
? ? ? ?
?
? ?
? ?
 is periodic function with period ? and it
is an odd function
     tan
–1
 (tan x) = 
3
x ; x
2 2
x ; x
2 2
3
x ; x
2 2
? ? ?
? ? ? ? ? ?
?
?
? ? ?
? ? ?
?
?
? ?
?
? ? ? ?
?
?
   
? ?2
? 2
0
y=x
y=x+ ?
2
3 ?
?
? ?
2
?
?
2
? ?
y=x– ?
2
3 ?
2
?
2
?
?
x
y
(x) y = cot
–1
 (cot x), x ? R – {n ?, n ? I}, y ? [0, ?], is periodic function with period ? and it is nei-
ther an even nor odd function
cot
–1
 (cot x) = 
?
?
?
?
?
? ? ? ? ? ?
? ? ?
? ? ? ? ? ?
2 x ; x
x 0 ; x
0 x ; x
? ?2 ? 2 0
y=x
y=x+ ?
? ? ?
?
y=x– ?
x
y
y=x+2 ?
(xi) y = cosec
–1
 (cosec x), x ? R – {n ?, n ? I}, y ? 
?
?
?
?
?
? ?
? ?
?
?
?
?
? ?
?
2
, 0 0 ,
2
 is periodic function with period
2 ? and it is an odd function
Page 5


Inverse Trigonometry – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 0
A. INTRODUCTION
sin
–1 
x, cos
–1 
x, tan
–1 
x etc. represents angles or numbers whose values of sine, cosine and  tangent is ‘x’,
provided that the value in numerical form is smallest. These can be written as arc sin x, arc cos x etc. If two
angles whose modulus is equal, in which one is positive and other is negative then we take positive sign.
B. DOMAIN & PRINCIPLE VALUE RANGE OF INVERSE TRIGONOMETRIC FUNCTIONS
} 0 {
2
,
2
y ) , 1 [ ] 1 , (– x x ec cos y . 6
2
] , 0 [ y ) , 1 [ ] 1 , (– x x sec y . 5
) , 0 ( y R x x cot y . 4
2
,
2
y R x x tan y . 3
] , 0 [ y ] 1 , 1 [ x x cos y . 2
2
,
2
y ] 1 , 1 [ x x sin y . 1
) PVR ( range value inciple Pr Domain Function . No . S
1
1
1
1
1
1
?
?
?
?
?
?
? ? ?
? ? ? ? ? ? ? ?
?
?
?
?
?
? ?
? ? ? ? ? ? ? ? ?
? ? ? ?
?
?
?
?
?
? ? ?
? ? ? ?
? ? ? ? ?
?
?
?
?
?
? ? ?
? ? ? ? ?
?
?
?
?
?
?
C. GRAPH OF INVERSE TRIGONOMETRIC FUNCTIONS
(a) f : [– ?/2, ?/2] ? [–1, 1] f
–1
 : [–1, 1] ? [– ?/2, ?/2]
f(x) = sin x f
–1
(x) = sin
–1
x
0 1
–1
2 / ?
– 2 / ?
x
y
y=arcsinx
y=x
y=sinx
y=sinx
y=arcsinx
y=x
2 / ?
1
– 2 / ?
–1
0 1
–1
x
y
y=arcsinx
y=arcsinx
2 / ?
– 2 / ?
(b) f : [0, ?] ? [–1, 1] f 
–1
 : [–1, 1] ? [0, ?]
f(x) = cos x f 
–1
(x) = cos
–1
 x
     
0 1 –1
2 / ?
x
y
y=arc cosx
y=x
y=cosx
y=x
2 / ?
1
–1
?
?
0 1 –1
x
y
2 / ?
1
?
INVERSE TRIGONOMETRY
INVERSE TRIGONOMETRY
Inverse Trigonometry – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 2
(c) f : (– ?/2, ?/2) ? R f 
–1
 : R ? (– ?/2, ?/2)
f(x) = tan x f
–1
 (x) = tan
–1
 x
0
2 / ? – 2 / ?
x
y
y=tanx
y=x
y=arc tanx
y=arc tanx
y=tanx
y=x
–
?
– 2 / ?
?
?
2 / ?
0
x
y
y=arc tanx
y=arc tanx
– 2 / ?
2 / ?
(d) f : (0, ?) ? R f 
–1
 : R ? (0, ?)
f(x) = cot x f
–1
 (x) = cot
–1
 x
0 2 / ?
x
y
y=x
y=arc cotx
y=arc cotx
y=cotx
y=x
–
?
– 2 / ?
?
?
2 / ?
0
x
y
y=arc cotx
?
?
2 / ?
y=arc cotx
(e) f : [0, ?/2) ? ( ?/2, ?] ? (– ?, –1] ? [1, ?)
f(x) = sec x
f
–1
 : (– ?, –1] ? [1, ?) ? [0, ?/2) ? [ ?/2, ?]
0 1 –1
x
y
2 / ?
?
f
–1
 (x) = sec
–1
 x
(f) f : [– ?/2, 0) ? (0, ?/2] ? (– ?, –1] ? [1, ?)
f(x) = cosec x
0 1 –1
x
y
2 / ?
– 2 / ?
f
–1
 : (– ?, –1] ? [1, ?) ? [– ?/2, 0) ? (0, ?/2]
f
–1
 (x) = cosec
–1
 x
Basis on the above discussion we get following results :
(i) All inverse trigonometric functions shows angle.
(ii) If x ? 0 then all six trigonometric functions sin
–1
 x, cos
–1
 x, tan
–1
 x, sec
–1
 x, cosec
–1
 x, cot
–1
 x
shows acute angle.
(iii) If x < 0 then sin
–1
 x, tan
–1
 x and cosec
–1
 x, shows angle between – ?/2 to 0 (IV quadrant)
(iv) If x < 0 then cos
–1
 x, cot
–1
 x and sec
–1
 x shows obtuse angle (II quadrant)
(v) III quadrant never used in inverse trigonometric functions.
Inverse Trigonometry – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 2
Ex.1 Find the domain of the following functions.
(i) sin
–1
 ?n x (ii) cos
–1
 [x] (iii) sin
–1
 {x}
Sol. (i) f(x) = sin
–1
 ?n x  ?  –1 ? ?n x ? 1  ? ? ?
e
1
 ? x ? e
(ii) f(x) = cos
–1
 [x]  ?  –1 ? [x] ? 1  ? ? ?[x] = –1, 0, 1  ?  x ? [–1, 0) ? [0, 1) ? [1, 2) ? ?x ? [–1, 2)
(iii) f(x) = sin
–1
 {x}  ? ? –1 ? {x} ? 1  ?  x ? R
Ex.2 tan
–1
 (1) + cos
–1
 
?
?
?
?
?
?
?
2
1
 + sin
–1
 
?
?
?
?
?
?
?
2
1
 is equal to
Sol. tan
–1
 (1) + cos
–1
 
?
?
?
?
?
?
?
2
1
 + sin
–1
 
?
?
?
?
?
?
?
2
1
 = 
4
3
2 4 6 3
2
4
?
?
?
?
?
?
?
?
?
?
?
D. PROPERTIES OF INVERSE TRIGONOMETRIC FUNCTIONS
P-1 (i) y = sin (sin
–1
 x) = x (ii) y = cos (cos
–1
 x) = x
x ? [–1, 1], y ? [–1, 1] x ? [–1, 1], y ? [–1, 1]
45º
0
y = x
1
x
y
–1
–1
1
45º
0
y = x
1
x
y
–1
–1
1
(iii) y = tan (tan
–1
 x) = x (iv) y = cot (cot
–1
 x) = x
x ? R, y ? R x ? R, y ? R
45º
0
y = x
x
y
45º
0
y = x
x
y
(v) y = cosec (cosec
–1
 x) = x (vi) y = sec (sec
–1
 x) = x
| x | ? 1, | y | ? 1 | x | ? 1 ; | y | ? 1
0
y = x
1
x
y
–1
–1
1
y = x
0
y = x
1
x
y
–1
–1
1
y = x
Inverse Trigonometry – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 4
(vii) y = sin
–1
 (sin x), x ? R, y ? ?
?
?
?
?
? ? ?
?
2
,
2
, is periodic function with period 2 ? and it is an odd function
sin
–1
 (sin x) = 
?
?
?
?
?
?
?
?
?
? ? ?
?
? ?
?
? ?
?
?
?
? ? ? ? ? ? ? ?
x
2
, x
2
x
2
, x
2
x , x
     
/2
y
0
?
?
? ?
–  /2
?
y=–( +x) ?
2
3 ?
?
? ?2
x
2
?
?
2
?
2
3 ?
y=2 +x ?
y=x
? 2
y=x–2 ?
y=  –x ?
(viii) y = cos
–1
 (cos x), x ? R, y ? [0, ?], is periodic function with period 2 ? and it is an even function
cos
–1
 (cos x) = 
x , x 0
x , 0 x
2 x , x 2
x 2 , 2 x 3
? ? ? ? ? ?
? ? ? ?
?
? ? ? ? ? ?
?
? ? ? ? ? ?
?
y=x+2 ?
y
?
?
2 ?
0 ? ?
–3 ?/2 –2 ?
– ?/2 ?/2 3 ?/2
y=–x
y=x
y=2 –x ?
2
?
(ix) y = tan
–1
 (tan x), x ? R – ?
(2n 1) , n I
2
? ? ?
? ?
? ?
? ?
;  y ? 
,
2 2
? ? ? ?
?
? ?
? ?
 is periodic function with period ? and it
is an odd function
     tan
–1
 (tan x) = 
3
x ; x
2 2
x ; x
2 2
3
x ; x
2 2
? ? ?
? ? ? ? ? ?
?
?
? ? ?
? ? ?
?
?
? ?
?
? ? ? ?
?
?
   
? ?2
? 2
0
y=x
y=x+ ?
2
3 ?
?
? ?
2
?
?
2
? ?
y=x– ?
2
3 ?
2
?
2
?
?
x
y
(x) y = cot
–1
 (cot x), x ? R – {n ?, n ? I}, y ? [0, ?], is periodic function with period ? and it is nei-
ther an even nor odd function
cot
–1
 (cot x) = 
?
?
?
?
?
? ? ? ? ? ?
? ? ?
? ? ? ? ? ?
2 x ; x
x 0 ; x
0 x ; x
? ?2 ? 2 0
y=x
y=x+ ?
? ? ?
?
y=x– ?
x
y
y=x+2 ?
(xi) y = cosec
–1
 (cosec x), x ? R – {n ?, n ? I}, y ? 
?
?
?
?
?
? ?
? ?
?
?
?
?
? ?
?
2
, 0 0 ,
2
 is periodic function with period
2 ? and it is an odd function
Inverse Trigonometry – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 4
/2
y
0
?
?
? ?
–  /2
?
y=–( +x) ?
2
3 ?
?
? ?2
x
2
?
?
2
?
2
3 ?
y=2 +x ?
y=x
? 2
y=x–2 ?
y=  –x ?
(xii) y = sec
–1
 (sec x), x ? R – ?
?
?
?
?
?
?
?
?
? I n ,
2
) 1 n 2 (
, y ? 
0, ,
2 2
? ? ? ? ? ?
? ?
? ? ? ?
? ? ? ?
 is periodic function with period 2 ?
and it is an even function
       
y=x+2 ?
y
?
?
2 ?
0 ? ?
–3 ?/2 –2 ?
– ?/2 ?/2 3 ?/2
y=–x
y=x
y=2 –x ?
2
?
Ex.3 Evaluate following
(i) sin(cos
–1
 3/5) (ii) cos(tan
–1
 3/4) (iii) sin 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
2
1
sin
2
1
Sol. (i) Let cos
–1
 3/5 = ? then cos ? = 3/5  ?  sin ? = 4/5 ? ?? ? ?  sin(cos
–1
 3/5) = sin ? = 4/5
(ii) Let tan
–1
 3/4 = ? then tan ? = 3/4  ?  cos ? = 4/5 ? ? ? ? ?  cos(tan
–1
 3/4) = cos ? = 4/5
(iii) sin 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
2
1
sin
2
1
 = sin ?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
? ?
?
6 2
 = sin 
3
2 ?
 = 
2
3
Ex.4 Define the function,  f(x) = cos
 ?1
 (cos x) ? sin
 ?1
 (sin x)  in  [0, 2
 
?]  and  find the area bounded by the
graph of the function and the  x
 
?
 
axis.
Sol. cos
 ?1
 (cos x)  =  
x
x
x
x 2
0
2 ?
?
? ? ?
? ?
? ?
?
?
?
; sin
 ?1
 (sin x)  =  
x
x
x
x
x
x
?
?
?
?
?
?
?
?
?
? ?
? ?
? ?
?
?
?
?
?
?
2
0
2
2
2
3
2
3
2
Hence    f
 
(x) =  
? ?
? ?
? ?
? ?
0 0
2
4 2 2
2
2
3
2
3
2
i f x
x i f x
i f x
x i f x
?
? ?
?
? ?
?
?
?
?
?
?
?
?
,
,
,
,
?
?
?
?
? ?
? ?
? ?
Area =  
3
2 2
? ?
?
?
?
?
?
?
? ? 
?
2
=  ?
2
Ex.5 Let  y = sin
–1
(sin 8) – tan
–1
(tan 10) + cos
–1
(cos 12) – sec
–1
(sec 9) + cot
–1
(cot 6) – cosec
–1
(cosec 7).
If  y  simplifies to  a ? + b  then find ( a – b).
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