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 ;   and sin y = x.
• y = cos^-1x where −1 ≤ x ≤ 1 ; 0 ≤ y ≤ π and cos y = x.
• y = tan^-1x where x ∈ R ; and tan y = x.
• y = cosec^-1x where x ≤ − 1 or x ≥ 1 ; , y ≠ 0 and cosec y = x
• y = sec^-1x where x ≤ −1 or x ≥ 1 ; 0 ≤ y ≤ π ; 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.

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

  Note very carefully that:
  
  
  

  Remember That:
  (i) sin^-1x + sin^-1y + sin^-1z =   ⇒ 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^-1 + tan^-1 =

Inverse Trigonometric Functions | Some Useful Graphs

1. y = sin^-1x , |x| ≤ 1 , y ∈

2. y = cos^-1x , |x| ≤ 1 , y ∈ [0 , π]

3. y = tan^-1x, x ∈ R , y ∈

4. y = cot^-1x, x ∈ R, y ∈ (0 , π)

5. y = sec^-1x, |x| ≥ 1, y ∈

6. y = cosec^-1x, |x| ≥ 1, y ∈

7. (a) y = sin^-1(sin x) , x ∈ R , y ∈
Periodic with period 2 π

7.(b) y = sin (sin^-1x) ,
= x, x ∈ [− 1 , 1] , y ∈ [− 1 , 1] , y is  a periodic

8. (a) y = cos^-1(cos x), x ∈ R, y ∈ [0, π],
= x periodic with period 2 π

8. (b) y = cos (cos^-1x),
= x,  x ∈ [− 1 , 1] , y ∈ [− 1 , 1], y is a periodic

9. (a) y = tan (tan^-1x) , x ∈ R , y ∈ R , y is a periodic
= x

9. (b)y = tan^-1(tan x) ,
= x
, periodic with period π

10. (a) y = cot^-1(cot x),
= x
x ∈ R − { nπ} , y ∈ (0 , π) , periodic with π

10. (b) y = cot (cot^-1x) ,
= x
x ∈ R , y ∈ R , y is a periodic

11. (a) y = cosec^-1(cosec x),
= x
x ε R − { nπ , n ε I }, y ∈
y is periodic with period 2 π

11. (b) y = cosec (cosec^-1x) ,
= x
|x| ≥ 1, |y| ≥ 1, y is aperiodic

12. (a) y = sec −1 (sec x) ,
= x
y is periodic with period 2π ;

12. (b) y = sec (sec −1 x), |x| ≥ 1 ; |y| ≥ 1], y is a periodic

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