CLASS 12 INVERSE TRIGNOMETRIC DPP CHAP 2
CHAPTER 2 - GRAPHS
GRAPHS OF DIFFERENT INVERSE TRIGRNOMETRIC RATIO
1) The inverse of cosine function
is denoted by cos⁻¹ (or arc cosine)
function. Thus
y = cos⁻¹ x if and only if x = cos y
x ∈ [-1, 1] and y ∈ [0, π]
Graph of y = cos⁻¹ x
3) The inverse of tangent function is
denoted by tan⁻¹ (or arc tangent)
function. Thus
y = tan⁻¹ x if and only if x = tan y
x ∈ (-∞, ∞) and y ∈ (-π/2, π/2)
Graph of y = tan⁻¹ x.
4) The inverse of cotangent function
is denoted by cot⁻¹ (or arc
cotangent) function. Thus
y = cot⁻¹ x if and only if x = cot y
x ∈ (-∞, ∞) and y ∈ (0, π)
Graph of y = cot⁻¹ x.
DAILY PRACTICE QUESTION
✤ Very Short (Objective Type) / Short Answer Type
1) Principal value of sin⁻¹(⁻¹/₂) is
(a) π/3 (b) - π/3 (c) 5π/6 (d) -π/6
2) sin⁻¹(sin 2π/3) = 2π/3 , state true or false.
3) tan⁻¹[sin (-π/2)] is equal to
(a) -1 (b) 1 (c) π/2 (d) -π/4
4) sec {tan⁻¹ (y/3)} is equal to
(a) √(9+y²) / 9 (b) √(9+y²) / 3
(c) 3 / √(9+y²) (d) 9 / √(9+y²)
5) Principal branch of tan⁻¹ x is ________
6) Find the value of tan⁻¹√3 - cot⁻¹(-√3).
7) Find the principal value of cos⁻¹(¹/₂) .
8) Find the principal value of sin⁻¹(⁻¹/₂) + cos⁻¹(⁻¹/₂).
9) What is the domain of the function sin⁻¹ x?
10) Find the value of sin [π/3- sin⁻¹(-1/2)].
[NCERT; Delhi 2011]
11) If 〖cos〗^(-1) x/a+ 〖cos〗^(-1) y/b=∝, prove that x^2/a^2 -2xy/ab+y^2/b^2 =〖sin〗^2∝.
12) Show that tan(1/2 sin-1 (¾) ) = (4 - √7) / 3.
13) Prove that sec2(tan-12) + cosec2(cot-13)=15.
14) If a1, a2, a3,……, an is an arithmetic progression with common difference d, then evaluate the following expression
tan[〖tan〗^(-1) (d/(1+a_1 a_2 ))+〖tan〗^(-1) (d/(1+a_2 a_3 ))+〖tan〗^(-1) (d/(1+a_3 a_4 ))+⋯+〖tan〗^(-1) (d/(1+a_(n-1) a_n ))].
15) Prove that 2 tan-1(1/5) + sec-1(5√2/7) + 2 tan-11/8 = π /4
16) Write the following in the simplest form:
tan⁻¹ (cos x / 1 + sin x)
17) Simplify: tan⁻¹ [(a cos x - b sin x) / (b cos x + a sin x)], if a/b tan x > -1.
[NCERT]
18) Solve for x, tan⁻¹ [(1-x)/(1+x)] = ½ tan⁻¹ x, x > 0.
[NCERT; NCERT Exemplar; Foreign 2011]
19) Solve the following equation:
tan⁻¹ [(x+1)/(x-1)] + tan⁻¹ [(x-1)/(x-1)] = tan⁻¹(-7).
20) Write the following function in the simplest form :
Tan-1[√(1+x2)-1/x] , x is not equal to 0.
21) Solve the following equation:
Tan-1(x+1)/(x-1) + tan-1(x-1)/x=tan-1(-7).
22) Find the value of the expression
Sin (2 tan-1(1/3)) + cos(tan-12√2)
23) Find the value of cot ½[cos-1(2x/1+x2)+sin-1(1-y2)/(1+y2)], |x| <1, y>0 and xy<1.
24) Prove that: 〖tan〗^(-1) ((√(1+x^2 )+√(1-x^2 ))/(√(1+x^2 )- √(1-x^2 )))=π/4+1/2 〖cos〗^(-1) x^2; -1<x<1
[Foreign 2017]
11) cos⁻¹(cos 13π/6). [NCERT]
12) Write the principal value of the following: sin⁻¹(sin 4π/5)
13) Using principal value evaluate the following:
cos⁻¹(cos 2π/3) + sin⁻¹(sin 2π/3).
[AI 2011]
14) If sin⁻¹ x + sin⁻¹ y = π, then find the values of x and y.
15) If x < 0, y < 0, such that xy = 1, then find the value of
tan⁻¹ x + tan⁻¹ y.
✤ Long Answer I / Long Answer II Type
Prove that in Exercises 16 to 26:
16) sin⁻¹(12/13) + cos⁻¹(4/5) + tan⁻¹(63/16) = π. [NCERT]
17) cot⁻¹((√1 + sin x + √1 - sin x)/(√1 + sin x - √1 - sin x))= x/2,
x ∈ (0, π/4)
[NCERT; Delhi 2014, 11]
18) tan⁻¹(1/4) + tan⁻¹(2/9) = 1/2 cos⁻¹(3/5) [HOTS]
19) cos⁻¹(12/13) + sin⁻¹(3/5) = sin⁻¹(56/65) [NCERT; Dehradun 2019]
20) tan⁻¹ 2x + tan⁻¹ (4x/(1-4x²)) = tan⁻¹((6x-8x³)/(1-12x²)),
|x| < 1/(2√3) [Foreign 2017]
21) 2 tan⁻¹(1/2) + tan⁻¹(1/7) = tan⁻¹(31/17). [NCERT; AI 2011]
22) sin⁻¹(8/17) + sin⁻¹(3/5) = cos⁻¹(36/85) = tan⁻¹(77/36)
[NCERT; Delhi 2012, 13 (C)]
23) sin⁻¹(1/√5) + cot⁻¹(3) = π/4. [HOTS]
24) cos(sin⁻¹(3/5) + sin⁻¹(5/13)) = 33/65.
25) cos⁻¹(4/5) + cos⁻¹(12/13) = cos⁻¹(33/65). [NCERT]
26) 2 sin⁻¹(3/5) = tan⁻¹(24/7). [NCERT]
Write the following functions in the simplest form (Exercises 27 and
28):
27) tan⁻¹ |3x - x³| / |1 - 3x²|
28) tan⁻¹ √ (1 - cos 3x) / (1 + cos 3x), x < π
29) Solve for x, tan⁻¹(x + 1) + tan⁻¹(x - 1) = tan⁻¹(3/31). [Foreign 2015]
30) Find the value of x, if sin [cot⁻¹(x + 1)] = cos(tan⁻¹ x)
[DoE; Bhubaneswar 2015, Delhi 2015]
31) Prove the following: 2 sin⁻¹(3/5) - tan⁻¹(17/31) = π/4.
[Bhubaneshwar 2015]
32) Solve the following for x:
tan⁻¹((x - 2)/(x - 3)) + tan⁻¹((x + 2)/(x + 3)) = π/4, |x| < 1.
[Patna 2015]
33) Prove the following:
sin [tan⁻¹((1 - x²)/2x) + cos⁻¹((1 - x²)/(1 + x²))] = 1, 0 < x < 1.
[Guwahati 2015]
34) Solve for x, 2 tan⁻¹(sin x) = tan⁻¹(2 sec x), x ≠ π/2
[DoE: Foreign 2012]
35) Solve for x: tan⁻¹(2x / (1 - x²)) + cot⁻¹(1 - x² / 2x) = 2π/3, x > 0.
36) Solve for x: cos⁻¹((x² - 1) / (x² + 1)) + 1/2 tan⁻¹(2x / (1 - x²)) = 2x/3
37) Solve for x: sin⁻¹(2α / (1 + α²)) + sin⁻¹(2β / (1 + β²)) = 2 tan⁻¹x.
[HOTS]
38. Solve for x: sin⁻¹ x + sin⁻¹ 2x = π/3. [HOTS]
INTEGRATED EXERCISE
✤ Very Short (Objective Type) / Short Answer Type
1. The principal value of sin⁻¹(sin 2π/3) is
(a) 2π/3 (b) π/3 (c) -π/6 (d) π/6
2. The value of cos⁻¹(1/2) + 3 sin⁻¹(1/2) is equal to
(a) π/4 (b) π/6 (c) 2π/3 (d) 5π/6
3. The greatest and least values of (sin⁻¹ x)² + (cos⁻¹ x)² are
respectively ______
4. The value of sin(2 sin⁻¹(-6)) is ______
5. Find the principal value of cosec⁻¹(2). [NCERT]
6. Evaluate tan⁻¹[sin(-(π/2))]. [NCERT Exemplar]
7. Write the value of cos⁻¹(-1/2) + 2 sin⁻¹(1/2)
[Foreign 2014]
8. Write one branch of sin⁻¹ x other than the principal branch.
9. Find the principal value of tan⁻¹(-1)
[NCERT]
10. Find the principal value of cos⁻¹(cos (7π/6))
[NCERT; HOTS; Delhi 2011]
11. Find the value of sin(2 sin⁻¹(3/5)).
[Foreign 2013]
12. Find the principal value of tan⁻¹(tan (9π/8)).
[NCERT Exemplar; Foreign 2013]
13. Write the principal value of tan⁻¹(tan (3π/4)).
[NCERT; HOTS; Delhi 2011]
✤ Long Answer I / Long Answer II Type
14. Prove that tan⁻¹(1) + tan⁻¹(2) + tan⁻¹(3) = π
15. Prove the following: (9π/8) - (9/4)sin⁻¹(1/3) = sin⁻¹((2√2)/3)
16. Show that 2 tan⁻¹ [tan (α/2) - tan((π/4) - β/2)]
= tan⁻¹[sin α cos β / (cos α + sin β)] [NCERT Exemplar]
17. Write the following function in the simplest form:
sin⁻¹[x√1 - x - √x√1 - x²].
[DoE; HOTS]
18. Solve the following for x:
cos⁻¹((x² - 1)/(x² + 1)) + tan⁻¹((2x)/(x² - 1)) = (2π/3)
19. Prove that sin⁻¹(63/65) = sin⁻¹(5/13) + cos⁻¹(3/5).
[Foreign 2012]
20. Find the value of the following:
tan⁻¹ (2x/1+x²) + cos⁻¹ ((1-y²)/1+y²) | | b < 1, y > 0 and xy < 1.
[NCERT; D.o.E; Delhi 2013]
21. If cos⁻¹ (x/2) + cos⁻¹ (y/3) = 0, then prove that
9x² - 12xy cos 0 + 4y² = 36 sin² 0. [HOTS]
22. Evaluate cot (√(1+x² + x)).
23. Solve the following equation: sin⁻¹(1 - x) - 2 sin⁻¹ x = π/2
[NCERT; Panchkula 2015]
24. Evaluate tan [ 2 tan⁻¹ (1/5) + π/4] [Ajmer 2015]
25. Solve for x, tan⁻¹(2x) + tan⁻¹(3x) = π/4
[NCERT; Delhi 2019, 2013(C)]
26. Solve for x: tan⁻¹ (√(1+x²) - √1-x²)/ (√(1+x²) + √1-x²) = β.
27. Find the solution of the equation
tan⁻¹ x - cot⁻¹ x = tan⁻¹ (1/√3) [NCERT Exemplar]
28. If tan⁻¹ (1/(1+1.2)) + tan⁻¹ (1/(1+2.3)) + ... +
tan⁻¹ (1/(1+n.(n+1))) = tan⁻¹ 0, then find the value of 0.
[D.o.E; Forrige 2015]
29. Find the principal value of tan⁻¹ (tan 5π/6) [D.o.E]
30. Prove that
tan⁻¹ ((cos x)/(1-sin x)) - cot⁻¹ (√(1+cos x)/(1-cos x)) = π/4,
x ∈ (0, π/2). [D.o.E]
31. Evaluate tan [ 1/2 cos⁻¹ (3/√11) ] [D.o.E]
32. If tan⁻¹ a + tan⁻¹ b + tan⁻¹ c = π, then prove that a + b + c = abc.
ASSESS YOURSELF
1. The equation tan⁻¹x - cot⁻¹x = tan⁻¹(1/√3) has solution as ______
2. If α ≤ 2 sin⁻¹x + cos⁻¹x ≤ β, then α = ____, β = ____
3. If tan⁻¹x = π/10 for some x ∈ R, then the value of cot⁻¹x is
(a) π/5 (b) 2π/5 (c) 3π/5 (d) 4π/5
4. Show that sin⁻¹(√(a - x)/2a) = (1/2)cos⁻¹x/a.
Write the principal values in Exercises 5 to 8:
5. cosec⁻¹(2) 6. cos⁻¹ (-√3/2)
7. tan⁻¹(-√3) 8. Tan⁻¹(tan 3π/4)
Write the value in Exercises 9 to 11:
9. cosec⁻¹(√2) + sec⁻¹(√2)
10. cos⁻¹(cos 2π/3) + sin⁻¹(cos 2π/3)
* * * * * * * * * * * * * * *
