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 ; \(-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}\)  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 = \(\frac {3\pi}{2}\)  ⇒ 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 ∈ \(\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\)

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

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
\(x \in R-\left\{(2 n-1) \frac{\pi}{2} n \in I\right\}, y \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), 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 ∈ \(\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]\)
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 ;
\(x \in \mathrm{R}-\left\{(2 \mathrm{n}-1) \frac{\pi}{2} \mathrm{n} \in \mathrm{I}\right\} \quad \mathrm{y} \in\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\)

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