Inverse Trigonometric Functions

Inverse Trigonometric Functions (Inverse Trig Functions)

Inverse trig functions: sin^-1x , cos^-1x , tan^-1x etc. denote angles or real numbers whose sine is x , whose cosine is x and whose tangent is x, provided that the answers given are numerically smallest available. These are also written as arc sinx , arc cosx etc . If there are two angles one positive & the other negative having same numerical value, then positive angle should be taken.

Principal Values And Domains Of Inverse Circular Functions

• y = sin^-1x where −1 ≤ x ≤ 1 ; Image may be NSFW.
  Clik here to view.  and sin y = x.
• y = cos^-1x where −1 ≤ x ≤ 1 ; 0 ≤ y ≤ π and cos y = x.
• y = tan^-1x where x ∈ R ; Image may be NSFW.
  Clik here to view. and tan y = x.
• y = cosec^-1x where x ≤ − 1 or x ≥ 1 ; Image may be NSFW.
  Clik here to view. , y ≠ 0 and cosec y = x
• y = sec^-1x where x ≤ −1 or x ≥ 1 ; 0 ≤ y ≤ π ; Image may be NSFW.
  Clik here to view. and sec y = x.
• y = cot^-1x where x ∈ R , 0 < y < π and cot y = x .
  Note:
  (i) 1st quadrant is common to all the inverse functions.
  (ii) 3rd quadrant is not used in inverse functions.
  (iii) 4th quadrant is used in the Clockwise Direction i.e. Image may be NSFW.
  Clik here to view.

Properties Of Inverse Circular Functions | Inverse Trigonometric Functions

• Property 1:
  
  • sin (sin^-1x) = x , −1 ≤ x ≤ 1
  • cos (cos^-1x) = x , −1 ≤ x ≤ 1
  • tan (tan^-1 x) = x , x ∈ R
  • sin^-1(sin x) = x , Image may be NSFW.
    Clik here to view.
  • cos^-1(cos x) = x ; 0 ≤ x ≤ π
  • tan^-1(tan x) = x ; Image may be NSFW.
    Clik here to view.
• Property 2:
  
  • cosec^-1x = sin^-1Image may be NSFW.
    Clik here to view. ; x ≤ −1 , x ≥ 1
  • sec^-1x = cos^-1Image may be NSFW.
    Clik here to view.  ; x ≤ −1 , x ≥ 1
  • cot^-1x = tan^-1Image may be NSFW.
    Clik here to view. ; x > 0 = π + tan^-1Image may be NSFW.
    Clik here to view. ; x < 0
• Property 3:
  
  • sin^-1(−x) = − sin^-1x , −1 ≤ x ≤ 1
  • tan^-1(−x) = − tan^-1x , x ∈ R
  • cos^-1(−x) = π − cos^-1x , −1 ≤ x ≤ 1
  • cot^-1(−x) = π − cot^-1x , x ∈ R
• Property 4:
  
  • sin^-1x + cos^-1x = Image may be NSFW.
    Clik here to view. −1 ≤ x ≤ 1
  • tan^-1x + cot^-1x = Image may be NSFW.
    Clik here to view.  x ∈ R
  • cosec^-1x + sec^-1x = Image may be NSFW.
    Clik here to view. |x|≥1
• Property 5:
  • Image may be NSFW.
    Clik here to view. where x > 0 , y > 0 & xy < 1
    Image may be NSFW.
    Clik here to view. where x > 0 , y > 0 & xy > 1
  • Image may be NSFW.
    Clik here to view. where x > 0 , y > 0
• Property 6:
  
  • Image may be NSFW.
    Clik here to view. where x ≥ 0 ,y≥0 & (x²+y²)≤1
    Note: x²+y²≤ 1 ⇒ 0 ≤ sin^-1x + sin^-1y ≤ Image may be NSFW.
    Clik here to view.
  • Image may be NSFW.
    Clik here to view.  where x≥0,y ≥ 0 & x²+y²>1
    Note: x² + y² >1 ⇒ Image may be NSFW.
    Clik here to view. < sin^-1x + sin^-1y < π
  • Image may be NSFW.
    Clik here to view.  where x > 0 , y > 0
  • Image may be NSFW.
    Clik here to view.  where x ≥ 0 , y ≥ 0
• Property 7:
  If tan^-1x + tan^-1y + tan^-1z = Image may be NSFW.
  Clik here to view.
  Note:
  (i) If tan^-1x + tan^-1y + tan^-1z = π then x + y + z = xyz
  (ii) If tan^-1x + tan^-1y + tan^-1z = Image may be NSFW.
  Clik here to view. then xy + yz + zx = 1
• Property 8:
  Image may be NSFW.
  Clik here to view.

  Note very carefully that:
  Image may be NSFW.
  Clik here to view.
  Image may be NSFW.
  Clik here to view.
  Image may be NSFW.
  Clik here to view.

  Remember That:
  (i) sin^-1x + sin^-1y + sin^-1z = Image may be NSFW.
  Clik here to view.  ⇒ x = y = z = 1
  (ii) cos^-1x + cos^-1y + cos^-1z = 3π ⇒ x = y = z = −1
  (iii) tan^-11 + tan^-12 + tan^-13 = π and tan^-11 + tan^-1Image may be NSFW.
  Clik here to view. + tan^-1Image may be NSFW.
  Clik here to view. = Image may be NSFW.
  Clik here to view.

Inverse Trigonometric Functions | Some Useful Graphs

1. y = sin^-1x , |x| ≤ 1 , y ∈ Image may be NSFW.
Clik here to view.
Image may be NSFW.
Clik here to view.

2. y = cos^-1x , |x| ≤ 1 , y ∈ [0 , π]
Image may be NSFW.
Clik here to view.

3. y = tan^-1x, x ∈ R , y ∈ Image may be NSFW.
Clik here to view.
Image may be NSFW.
Clik here to view.
4. y = cot^-1x, x ∈ R, y ∈ (0 , π)
Image may be NSFW.
Clik here to view.
5. y = sec^-1x, |x| ≥ 1, y ∈ Image may be NSFW.
Clik here to view.
Image may be NSFW.
Clik here to view.
6. y = cosec^-1x, |x| ≥ 1, y ∈ Image may be NSFW.
Clik here to view.
Image may be NSFW.
Clik here to view.
7. (a) y = sin^-1(sin x) , x ∈ R , y ∈ Image may be NSFW.
Clik here to view.
Periodic with period 2 π
Image may be NSFW.
Clik here to view.
7.(b) y = sin (sin^-1x) ,
= x, x ∈ [− 1 , 1] , y ∈ [− 1 , 1] , y is  a periodic
Image may be NSFW.
Clik here to view.

8. (a) y = cos^-1(cos x), x ∈ R, y ∈ [0, π],
= x periodic with period 2 π
Image may be NSFW.
Clik here to view.
8. (b) y = cos (cos^-1x),
= x,  x ∈ [− 1 , 1] , y ∈ [− 1 , 1], y is a periodic
Image may be NSFW.
Clik here to view.
9. (a) y = tan (tan^-1x) , x ∈ R , y ∈ R , y is a periodic
= x
Image may be NSFW.
Clik here to view.
9. (b)y = tan^-1(tan x) ,
= x
Image may be NSFW.
Clik here to view., periodic with period π
Image may be NSFW.
Clik here to view.
10. (a) y = cot^-1(cot x),
= x
x ∈ R − { nπ} , y ∈ (0 , π) , periodic with π
Image may be NSFW.
Clik here to view.

10. (b) y = cot (cot^-1x) ,
= x
x ∈ R , y ∈ R , y is a periodic
Image may be NSFW.
Clik here to view.

11. (a) y = cosec^-1(cosec x),
= x
x ε R − { nπ , n ε I }, y ∈ Image may be NSFW.
Clik here to view.
y is periodic with period 2 π
Image may be NSFW.
Clik here to view.

11. (b) y = cosec (cosec^-1x) ,
= x
|x| ≥ 1, |y| ≥ 1, y is aperiodic
Image may be NSFW.
Clik here to view.

12. (a) y = sec −1 (sec x) ,
= x
y is periodic with period 2π ;
Image may be NSFW.
Clik here to view.
Image may be NSFW.
Clik here to view.

12. (b) y = sec (sec −1 x), |x| ≥ 1 ; |y| ≥ 1], y is a periodic
Image may be NSFW.
Clik here to view.

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