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QUESTION BANK MATHEMATICS ( INVERSE TRIGNOMETRIC FUNCTIONS)
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THIS PDF CONTAINS QUESTION BANK OF INVERSE TRIGONOMETRIC FUNCTIONS ALONG WITH THE ANSWER KEY AT THE BOTTOM LANGUAGE-ENGLISH
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Inverse Trigonometric FunctionsQUICK RECAP
INVERSE TRIGONOMETRIC FUNCTIONS 1
8 sin −1 = cosec −1x , ∀ x ≥ 1 or x ≤ – 1
x
8 Trigonometric functions are not one-one and
onto over their natural domains and ranges 1
cos −1 = sec −1x , ∀ x ≥ 1 or x ≤ – 1
i.e., R(real numbers). But some restrictions x
on domains and ranges of trigonometric 1
tan −1 = cot −1 x , ∀ x > 0
function ensures the existence of their x
inverses. = – p + cot–1 x, ∀ x < 0
Let y = f(x) = cos x, then its inverse is x = cos–1y 8 sin(sin–1x) = x, ∀ – 1 ≤ x ≤ 1
8 The domains and ranges (principal value cos(cos–1x) = x, ∀ – 1 ≤ x ≤ 1
branches) of inverse trigonometric functions tan(tan–1x) = x, ∀ x ∈ R
are as follows : π
8 sin–1x + cos–1x = ,∀–1≤x≤1
Functions Domain Range 2
π
y = sin–1 x [–1, 1] tan −1 x + cot −1 x = , ∀ x ∈ R
π π 2
− ,
2 2 π
sec −1 x + cosec −1x = , ∀ x ≤ –1 or x ≥ 1
y = cos–1 x [–1, 1] [0, p] 2
y = tan x
–1
R π π
− , 8 sin −1 x = cos −1 1 − x 2 , ∀ 0 ≤ x ≤ 1
2 2
y = cot–1 x R (0, p) sin −1 x = − cos −1 1 − x 2 , ∀ – 1 ≤ x < 0
y = cosec–1 x R–(–1, 1) π π { } cos −1 x = sin −1 1 − x 2 ,∀0≤x≤1
− 2 , 2 − 0 −1
cos x = π − sin −1 2
1− x , ∀ – 1 ≤ x < 0
{}
y = sec–1 x R–(–1, 1) 8 sin–1(–x) = –sin–1x, ∀ – 1 ≤ x ≤ 1
[0,π] − π cos–1(–x) = p – cos–1x , ∀– 1 ≤ x ≤ 1
2
tan–1(–x) = –tan–1x, ∀ x ∈ R
X The value of the inverse trigonometric cot–1(–x) = p – cot–1x, ∀ x ∈ R
functions which lies in its principal value sec–1(–x) = p – sec–1x, ∀ |x| ≥ 1
branch is called the principal value of inverse cosec–1(–x) = – cosec–1x, ∀ |x| ≥ 1
trigonometric function. x+y
8 tan–1x + tan–1y = tan–1 , ∀ xy < 1
PROPERTIES OF INVERSE 1 − xy
( )
TRIGONOMETRIC FUNCTIONS
sin–1x + sin–1y = sin −1 x 1 − y 2 + y 1 − x 2 ,
π π
8 sin–1(sinx) = x, ∀ − ≤x≤ –1 ≤ x, y ≤ 1, x2 + y2 ≤ 1
2 2
cos–1(cosx) = x , ∀ 0 ≤ x ≤ p
π π
(
cos–1x + cos–1y = cos −1 xy − 1 − x 2 1 − y 2 , )
tan–1(tanx) = x, ∀ − < x < –1 ≤ x, y ≤ 1, x + y ≥ 0
2 2 x−y
tan −1 x − tan −1 y = tan −1 , xy > –1
1 + xy
, { 2 2
sin–1x – sin–1y = sin–1 x 1 − y − y 1 − x , }
–1 ≤ x, y ≤ 1 and x2 + y2 ≤ 1
{
cos–1x – cos–1y = cos–1 xy + 1 − x 2 1 − y 2 , }
–1 ≤ x, y ≤ 1 and x ≤ y.
2x
8 2 tan −1 x = sin −1 , ∀–1 ≤ x ≤ 1
1 + x2
1 − x2
2 tan −1 x = cos −1 ,∀x≥0
1 + x 2
2 x , ∀| x | < 1
2 tan −1 x = tan −1
1 − x2
( )
2 sin −1 x = sin −1 2 x 1− x 2 , ∀−
1
2
≤x≤
1
2
1
2 cos −1 x = sin −1 (2 x 1 − x 2 ) , ∀ ≤ x ≤1
2
2 cos–1x = cos–1(2x2 – 1), ∀ 0 ≤ x ≤ 1
1 1
8 3sin–1x = sin–1(3x – 4x3), ∀ − ≤ x ≤
2 2
1
3cos–1x = cos–1(4x3 – 3x), ∀ ≤ x ≤ 1
2
−1 3x − x
3
1 1
3tan x = tan
–1
2 , ∀− <x<
1 − 3x 3 3
, Previous Years’ CBSE
PREVIOUS Board
YEARS MCQS Questions
2.2 Basic Concepts 12. Using principal values, write the value of
3
VSA (1 mark) sin −1 − . (AI 2011C)
2
1 1
1. Write the value of cos −1 − + 2 sin −1 .
2 2 LA 1 (4 marks)
(Foreign 2014)
13. Prove that :
π 12 3 56
2. Write the principal value of tan −1 sin − . cos−1 + sin−1 = sin−1
2 13 5 65
(AI 2014C) (AI 2019, AI 2012)
3. Find the value of the following : 14. Prove that
π 12 4 56
cot − 2 cot −1 3 (AI 2014C) cos −1 + cos −1 = tan −1 . (AI 2013C)
2 13 5 33
4. Write the principal value of 15. Prove that :
4 12 33
1 cos −1 + cos −1 = cos −1
tan −1 (1) + cos −1 − . (Delhi 2013) 5 13 65
2
(AI 2012)
3
5. Write the value of tan −1 2 sin 2 cos −1 .
2
2.3 Properties of Inverse
(AI 2013) Trigonometric Functions
6. Write the principal value of
VSA (1 mark)
−1 3 1
cos + cos −1 − . (Delhi 2013C) 3π
2 2 16. The principal value of tan −1 tan is
5
7. Write the principal value of 2π −2π
(a) (b)
[tan −1 (− 3 ) + tan −1 (1)]. (AI 2013C) 5 5
3π −3π
8. Write the principal value of (c) (d) (2020)
6 6
1 1
cos −1 − 2 sin −1 − . (Delhi 2012) 7 1
17. tan −1 + tan −1 is equal to
2 2
9 8
9. Using principal values, write the value of 65 63
(a) tan −1 (b) tan −1
1 1 72 65
cos −1 + 2 sin −1 . (AI 2012C) π π
2 2 (c) (d) (2020)
4 2
π −1
10. Evaluate : sin − sin −1 . (Delhi 2011) 17 π
3 2 18. Find the value of sin −1 sin − .
8
1 (2020)
11. Write the principal value of sin −1 − .
2 19. Find the value of tan −1 3 − cot −1 (− 3 ) .
(Delhi 2011C) (2018)
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