Ch. 2.

Inverse Trigonometric Functions

1. The domain and range of the trigonometric functions are as follows:

FUNCTION DOMAIN RANGE
y  sin x R  1,1
y  cos x R  1,1
y  tan x R  n : n  I  R
R   1,1
y  cos ecx R  n : n  I  or
 ,1  1,  
R   1,1
y  sec x   
R   2n  1 : n  I  or
 2   ,1  1,  
y  cot x   
R   2n  1 : n  I  R
 2 
2. The domain and range of the inverse trigonometric functions are as follows:
DOMAIN RANGE

3. Graphs of Inverse Trigonometric Functions (Principal Branch)

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 IMPORTANT FORMULAE FOR INVERSE TRIGONOMETRIC FUNCTIONS

𝝅 𝝅
4. 𝒔𝒊𝒏−𝟏 (𝒔𝒊𝒏𝒙) = 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 [− 𝟐 , 𝟐 ]
5. 𝒄𝒐𝒔−𝟏 (𝒄𝒐𝒔𝒙) = 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 [𝟎, 𝝅 ]
−𝟏 (𝒕𝒂𝒏𝒙) 𝝅 𝝅
6. 𝒕𝒂𝒏 = 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 (− 𝟐 , 𝟐 )
𝝅 𝝅
7. 𝒄𝒐𝒔𝒆𝒄−𝟏 (𝒄𝒐𝒔𝒆𝒄𝒙) = 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 [− 𝟐 , 𝟐 ] , 𝒙 ≠ 𝟎
𝝅
8. 𝒔𝒆𝒄−𝟏 (𝒔𝒆𝒄𝒙) = 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 [𝟎, 𝝅 ], 𝒙 ≠ 𝟐
9. 𝒄𝒐𝒕−𝟏 (𝒄𝒐𝒕𝒙) = 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 (𝟎, 𝝅)
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10. 𝒔𝒊𝒏(𝒔𝒊𝒏−𝟏 𝒙) = 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 [−𝟏, 𝟏 ]
−𝟏
11. 𝒄𝒐𝒔(𝒄𝒐𝒔 𝒙) = 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 [−𝟏, 𝟏 ]
−𝟏
12. 𝒕𝒂𝒏(𝒕𝒂𝒏 𝒙) = 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 𝑹
−𝟏
13. 𝒄𝒐𝒔𝒆𝒄(𝒄𝒐𝒔𝒆𝒄 𝒙) = 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 (− ∞, −𝟏] ∪ [𝟏, ∞)
−𝟏
14. 𝒔𝒆𝒄(𝒔𝒆𝒄 𝒙) = 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 (− ∞, −𝟏] ∪ [𝟏, ∞)
−𝟏
15. 𝒄𝒐𝒕(𝒄𝒐𝒕 𝒙) = 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 𝑹
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16. 𝒔𝒊𝒏−𝟏 (−𝒙) = − 𝒔𝒊𝒏−𝟏 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 [−𝟏, 𝟏]
17. 𝒄𝒐𝒔−𝟏 (−𝒙) = 𝝅 − 𝒄𝒐𝒔−𝟏 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 [−𝟏, 𝟏]
18. 𝒕𝒂𝒏−𝟏 (−𝒙) = − 𝒕𝒂𝒏−𝟏 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 𝑹
19. 𝒄𝒐𝒔𝒆𝒄−𝟏 (−𝒙) = − 𝒄𝒐𝒔𝒆𝒄−𝟏 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 (− ∞, −𝟏] ∪ [𝟏, ∞)
20. 𝒔𝒆𝒄−𝟏 (−𝒙) = 𝝅 − 𝒔𝒆𝒄−𝟏 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 (− ∞, −𝟏] ∪ [𝟏, ∞)
−𝟏 (−𝒙) −𝟏
21. 𝒄𝒐𝒕 = 𝝅 − 𝒄𝒐𝒕 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 𝑹
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 𝟏
22. 𝒄𝒐𝒔−𝟏 (𝒙) = 𝒔𝒆𝒄−𝟏 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 (− ∞, −𝟏] ∪ [𝟏, ∞)
𝟏
23. 𝒔𝒊𝒏−𝟏 ( ) = 𝒄𝒐𝒔𝒆𝒄−𝟏 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 (− ∞, −𝟏] ∪ [𝟏, ∞)
𝒙

𝟏 𝒄𝒐𝒕−𝟏 𝒙 , 𝒇𝒐𝒓 𝒙 > 𝟎

24. 𝒕𝒂𝒏−𝟏 (𝒙) = {
− 𝜋 + 𝑐𝑜𝑡 −1 𝑥 , 𝑓𝑜𝑟 𝑥 < 0
𝟏 𝒕𝒂𝒏−𝟏 𝒙 , 𝒇𝒐𝒓 𝒙 > 𝟎
25. 𝒄𝒐𝒕−𝟏 (𝒙) = {
𝜋 + 𝑡𝑎𝑛−1 𝑥 , 𝑓𝑜𝑟 𝑥 < 0
𝟏
26. 𝒄𝒐𝒔𝒆𝒄−𝟏 ( ) = 𝒔𝒊𝒏−𝟏 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 [−𝟏, 𝟏]
𝒙
𝟏
27. 𝒔𝒆𝒄−𝟏 (𝒙) = 𝒄𝒐𝒔−𝟏 𝒙, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 [−𝟏, 𝟏]

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𝝅
28. 𝒔𝒊𝒏−𝟏 𝒙 + 𝒄𝒐𝒔−𝟏 𝒙 = 𝟐
, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 [−𝟏, 𝟏]
𝝅
29. 𝒕𝒂𝒏−𝟏 𝒙 + 𝒄𝒐𝒕−𝟏 𝒙 = 𝟐
, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 𝑹
−𝟏 −𝟏 𝝅
30. 𝒔𝒆𝒄 𝒙 + 𝒄𝒐𝒔𝒆𝒄 𝒙= 𝟐
, 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝝐 (− ∞, −𝟏] ∪ [𝟏, ∞)

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𝒙+𝒚
𝒕𝒂𝒏−𝟏 (𝟏 − 𝒙𝒚) , 𝒊𝒇 𝒙𝒚 < 𝟏
𝑥+𝑦
31. 𝒕𝒂𝒏−𝟏 𝒙 + 𝒕𝒂𝒏−𝟏 𝒚 = 𝜋 + 𝑡𝑎𝑛−1 ( ), 𝑖𝑓 𝑥 > 0, 𝑦 > 0 𝑎𝑛𝑑 𝑥𝑦 > 1
1 − 𝑥𝑦
𝑥+𝑦
{− 𝜋 + 𝑡𝑎𝑛−1 ( ), 𝑖𝑓 𝑥 < 0, 𝑦 < 0 𝑎𝑛𝑑 𝑥𝑦 > 1
1 − 𝑥𝑦

𝒙−𝒚
𝒕𝒂𝒏−𝟏 (𝟏 + 𝒙𝒚) , 𝒊𝒇 𝒙𝒚 > −𝟏
𝑥−𝑦
32. 𝒕𝒂𝒏−𝟏 𝒙 − 𝒕𝒂𝒏−𝟏 𝒚 = 𝜋 + 𝑡𝑎𝑛−1 ( ), 𝑖𝑓 𝑥 > 0, 𝑦 < 0 𝑎𝑛𝑑 𝑥𝑦 < − 1
1 + 𝑥𝑦
𝑥−𝑦
{− 𝜋 + 𝑡𝑎𝑛−1 (1 + 𝑥𝑦) , 𝑖𝑓 𝑥 < 0, 𝑦 > 0 𝑎𝑛𝑑 𝑥𝑦 < − 1

𝑥 + 𝑦 + 𝑧 − 𝑥𝑦𝑧
33. 𝑡𝑎𝑛−1 𝑥 + 𝑡𝑎𝑛−1 𝑦 + 𝑡𝑎𝑛−1 𝑧 = 𝑡𝑎𝑛−1 (1 − 𝑥𝑦 − 𝑦𝑧 − 𝑧𝑥)

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𝒔𝒊𝒏−𝟏 [𝒙√𝟏 − 𝒚𝟐 + 𝒚√𝟏 − 𝒙𝟐 ] , 𝒊𝒇 − 𝟏 ≤ 𝒙, 𝒚 ≤ 𝟏 𝒂𝒏𝒅 𝒙𝟐 + 𝒚𝟐 ≤ 𝟏

34. 𝒔𝒊𝒏−𝟏 𝒙 + 𝒔𝒊𝒏−𝟏 𝒚 = 𝜋 − 𝑠𝑖𝑛−1 [𝑥√1 − 𝑦 2 + 𝑦√1 − 𝑥 2 ] , 𝑖𝑓 0 ≤ 𝑥, 𝑦 ≤ 1 𝑎𝑛𝑑 𝑥 2 + 𝑦 2 > 1
−1 2 2 2 2
{− 𝜋 − 𝑠𝑖𝑛 [𝑥√1 − 𝑦 + 𝑦√1 − 𝑥 ] , 𝑖𝑓 − 1 ≤ 𝑥, 𝑦 ≤ 0 𝑎𝑛𝑑 𝑥 + 𝑦 > 1

35. 𝒔𝒊𝒏−𝟏 𝒙 − 𝒔𝒊𝒏−𝟏 𝒚 =

𝒔𝒊𝒏−𝟏 [𝒙√𝟏 − 𝒚𝟐 − 𝒚√𝟏 − 𝒙𝟐 ] , 𝒊𝒇 − 𝟏 ≤ 𝒙, 𝒚 ≤ 𝟏 𝒂𝒏𝒅 𝒙𝟐 + 𝒚𝟐 ≤ 𝟏
𝜋 − 𝑠𝑖𝑛−1 [𝑥√1 − 𝑦 2 − 𝑦√1 − 𝑥 2 ] , 𝑖𝑓 0 < 𝑥 ≤ 1; −1 ≤ 𝑦 ≤ 0 𝑎𝑛𝑑 𝑥 2 + 𝑦 2 > 1
−1 2 2 2 2
{− 𝜋 − 𝑠𝑖𝑛 [𝑥√1 − 𝑦 − 𝑦√1 − 𝑥 ] , 𝑖𝑓 − 1 ≤ 𝑥 < 0; 0 < 𝑦 ≤ 1 𝑎𝑛𝑑 𝑥 + 𝑦 > 1
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𝒄𝒐𝒔−𝟏 [𝒙𝒚 − √𝟏 − 𝒙𝟐 √𝟏 − 𝒚𝟐 ] , 𝒊𝒇 − 𝟏 ≤ 𝒙, 𝒚 ≤ 𝟏 𝒂𝒏𝒅 𝒙 + 𝒚 ≥ 𝟎

−𝟏 −𝟏
36. 𝒄𝒐𝒔 𝒙 + 𝒄𝒐𝒔 𝒚={
2𝜋 − 𝑐𝑜𝑠 −1 [𝑥𝑦 − √1 − 𝑥 2 √1 − 𝑦 2 ], 𝑖𝑓 − 1 ≤ 𝑥, 𝑦 ≤ 1 𝑎𝑛𝑑 𝑥 + 𝑦 ≤ 0

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 𝒄𝒐𝒔−𝟏 [𝒙𝒚 + √𝟏 − 𝒙𝟐 √𝟏 − 𝒚𝟐 ] , 𝒊𝒇 − 𝟏 ≤ 𝒙, 𝒚 ≤ 𝟏 𝒂𝒏𝒅 𝒙 − 𝒚 ≤ 𝟎
−𝟏 −𝟏
37. 𝒄𝒐𝒔 𝒙 − 𝒄𝒐𝒔 𝒚={
−𝑐𝑜𝑠 −1 [𝑥𝑦 + √1 − 𝑥 2 √1 − 𝑦 2 ], 𝑖𝑓 − 1 ≤ 𝑦 ≤ 0; 0 < 𝑥 ≤ 1 𝑎𝑛𝑑 𝑥 − 𝑦 ≥ 0

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38.
 1  x2 
sin 1
 x   cos 1
 1 x 2
  x 
1
 tan 
 1 x 
2
1
  cot 
 x



 1  x2 
cos 1
 x   sin 1
 1 x 2
  x 
1
 cot 
 1 x 
2
1
  tan 
 x



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39.

 
2sin 1  x   sin 1 2 x 1  x 2  cos 1 1  2 x 2  , 
1
2
x
1
2
2 cos 1  x   sin 1  2 x 1  x   cos
2 1
2x 2
 1 ,
1
2
 x 1

 2x 
2 tan 1  x   sin 1  , x  1
1 x 
2

 1  x2 
 cos 1  2 
, x0
1 x 
 2x 
 tan 1  2 
, 1  x  1
1 x 
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40.
3sin 1  x   sin 1  3 x  4 x 3 
3cox 1  x   cos 1  4 x 3  3 x 
 3x  x3 
3 tan 1  x   tan 1  2 
 1  3x 
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41. List of substitutions:

S. No. Form substitution
1 √𝒙𝟐 − 𝒂𝟐 x = acosecθ or x = asecθ
2 √𝒙𝟐 + 𝒂𝟐 x = atanθ or x = acotθ
3 √𝒂𝟐 − 𝒙𝟐 x = acosθ or x = asinθ
4 √𝒂 − 𝒙 & √𝒂 + 𝒙 x = acos2θ or x = acosθ

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 Trigonometric Formulae
I Sum and Difference of angle formulae:

a) Sin( x + y) = sinx cosy + cosx siny

b) Sin(x – y) = sinx cosy – cosx siny
c) Cos(x + y) = cosx cosy – sinx siny
d) Cos(x – y ) = cosx cosy + sin x siny
tan x  tan y
e) tan(x + y) =
1  tan x tan y
tan x  tan y
f) tan(x – y) =
1  tan x tan y
cot x cot y  1
g) cot (x + y) =
cot y  cot x
cot x cot y  1
h) cot (x – y) =
cot y  cot x

II Compound angle Formulae

2 tan x
a) sin 2x = 2sinx cosx =
1  tan 2 x
1  tan 2 x
b) cos 2x = cos2 x  sin 2 x = 1 – 2sin2x = 2 cos2 x  1 =
1  tan 2 x
2 tan x
c) tan 2x =
1  tan 2 x
d) 1 – cos 2x = 2 sin2x
e) 1 + Cos 2x = 2 cos2x
x
f) 1 – cos x = 2 sin2 2
x
g) 1 + cos x = 2 cos2 2

x
2 tan
x x 2
h) Sinx = 2 sin cos 
2 2 x
1  tan 2
2
x
1  tan 2
x 2 x  2 x  2 x  2
i) Cosx = 2 cos2   1  1  2 sin    cos    sin   
2 2 2 2 1  tan 2
x
2
j) Sin 3x = 3 sin x – 4sin3x
k) Cos 3x = 4cos3x – 3 cos x
3 tan x  tan 3 x
l) tan 3x =
1  3 tan 2 x

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III Transformation Formulae (converting product of trig. Functions into sum/difference)

a) 2 sin x cosy = sin (x + y) + sin (x – y)

b) 2 cosx sin y = sin(x + y) – sin (x – y)
c) 2 cosx cosy = cos(x + y) + cos (x – y)
d) – 2 sin x siny = cos(x + y) – cos (x – y) or cos(x – y) – cos (x + y) = 2 sin x siny

IV. Transformation Formulae (Converting sum/difference of trig. Functions into product)

xy xy
a) sin x + sin y = 2 sin cos
2 2
xy xy
b) sin x – sin y = 2 cos sin
2 2
xy xy
c) cos x + cos y = 2 cos cos
2 2
xy xy
d) cos x – cos y = – 2 sin sin
2 2

V. Solution of Trigonometric Equations:

1. a) sinx = 0 x = n ; n Z


b) cos x = 0 x = (2n  1) ; n Z
2
c) tan x = 0 x = n ; n Z

2. a) sin x = sin y x = n  (1) n y; n Z

b) cos x = cos y x = 2n  y; n Z

c) tan x = tan y x = n  y; n Z

VI Sine and Cosine rules (Not required for exam):

a) Sine rule: If a, b, c are sides opposite to angles A, B and C of ΔABC respectively, then
a b c
  k
sin A sin B sin C
b) Cosine rule:
b2  c2  a2
1. a  b  c  2bc cos A
2 2 2
OR cosA =
2bc
c2  a2  b2
2. b 2  c 2  a 2  2ca cos B OR cosB =
2ca
a2  b2  c2
3. c 2  a 2  b 2  2ab cosC OR cosC =
2ab

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