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 ; (-frac{pi}{2} leq y leq frac{pi}{2})  and sin y = x.
• y = cos^-1x where −1 ≤ x ≤ 1 ; 0 ≤ y ≤ π and cos y = x.
• y = tan^-1x where x ∈ R ; (-frac{pi}{2} leq x leq frac{pi}{2}) and tan y = x.
• y = cosec^-1x where x ≤ − 1 or x ≥ 1 ; (-frac{pi}{2} leq y leq frac{pi}{2}) , y ≠ 0 and cosec y = x
• y = sec^-1x where x ≤ −1 or x ≥ 1 ; 0 ≤ y ≤ π ; (mathrm{y} neq frac{pi}{2}) 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. (-frac{pi}{2} leq y leq 0.)

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

  Note very carefully that:
  (sin ^{-1} frac{2 mathrm{x}}{1+mathrm{x}^{2}}=left[ begin{array}{ll}{2 tan ^{-1} mathrm{x}} & {text { if }|mathrm{x}| leq 1} {pi-2 tan ^{-1} mathrm{x}} & {text { if } quad mathrm{x}>1} {-left(pi+2 tan ^{-1} mathrm{x}right)} & {text { if } quad mathrm{x}<-1}end{array}right.)
  (cos ^{-1} frac{1-x^{2}}{1+x^{2}}=left[ begin{array}{ll}{2 tan ^{-1} x} & {text { if } x geq 0} {-2 tan ^{-1} x} & {text { if } x<0}end{array}right.)
  (tan ^{-1} frac{2 mathrm{x}}{1-mathrm{x}^{2}}=left[ begin{array}{ll}{2 tan ^{-1} mathrm{x}} & {text { if } quad|mathrm{x}|<1} {pi+2 tan ^{-1} mathrm{x}} & {text { if } quad mathrm{x}<-1} {-left(pi-2 tan ^{-1} mathrm{x}right)} & {text { if } quad mathrm{x}>1}end{array}right.)

  Remember That:
  (i) sin^-1x + sin^-1y + sin^-1z = (frac {3pi}{2})  ⇒ 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(frac {1}{2}) + tan^-1(frac {1}{3}) = (frac {pi}{2})

Inverse Trigonometric Functions | Some Useful Graphs

1. y = sin^-1x , |x| ≤ 1 , y ∈ (left[-frac {pi}{2}, frac {pi}{2}right])

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

3. y = tan^-1x, x ∈ R , y ∈ (left(-frac {pi}{2}, frac {pi}{2}right))

4. y = cot^-1x, x ∈ R, y ∈ (0 , π)

5. y = sec^-1x, |x| ≥ 1, y ∈ (left[0, frac{pi}{2}right) cupleft(frac{pi}{2}, piright])

6. y = cosec^-1x, |x| ≥ 1, y ∈ (left[-frac{pi}{2}, 0right) cupleft(0, frac{pi}{2}right])

7. (a) y = sin^-1(sin x) , x ∈ R , y ∈ (left[-frac {pi}{2}, frac {pi}{2}right])
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
(x in R-left{(2 n-1) frac{pi}{2} n in Iright}, y inleft(-frac{pi}{2}, frac{pi}{2}right)), 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 ∈ (left[-frac{pi}{2}, 0right) cupleft(0, frac{pi}{2}right])
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 ;
(x in mathrm{R}-left{(2 mathrm{n}-1) frac{pi}{2} mathrm{n} in mathrm{I}right} quad mathrm{y} inleft[0, frac{pi}{2}right) cupleft(frac{pi}{2}, piright])

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

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