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Inverse Trigonometric function - Notes, Concept and All Important Formula

INVERSE TRIGONOMETRIC FUNCTION

1. DOMAIN, RANGE & GRAPH OFINVERSE TRIGONOMETRIC

FUNCTIONS :

(a)

Graph of arc sin(x) or graph of sin^(-1)x

\(\mathrm{f}^{-1}:[-1,1] \rightarrow[-\pi / 2, \pi / 2]\),

\(\mathrm{f}^{-1}(\mathrm{x})=\sin ^{-1}(\mathrm{x})\)

(b)

Graph of arc cos(x) or graph of cos^(-1)x

\(\mathrm{f}^{-1}:[-1,1] \rightarrow[0, \pi]\),

\(\mathrm{f}^{-1}(\mathrm{x})=\cos ^{-1} \mathrm{x}\)

(c)

Graph of arc tan(x) or graph of tan^(-1)x

\(\mathrm{f}^{-1}: \mathrm{R} \rightarrow(-\pi / 2, \pi / 2)\),

\(\mathrm{f}^{-1}(\mathrm{x})=\tan ^{-1} \mathrm{x}\)

(d)

Graph of arc cot(x) or graph of cot^(-1)x

\(\mathrm{f}^{-1}: \mathrm{R} \rightarrow(0, \pi)\)

\(\mathrm{f}^{-1}(\mathrm{x})=\cot ^{-1} \mathrm{x}\)

(e)

Graph of arc sec(x) or graph of sec^(-1)x

\(\mathrm{f}^{-1}:(-\infty,-1] \cup[1, \infty)\)\(\rightarrow[0, \pi / 2) \cup(\pi / 2, \pi]\),

\(\mathrm{f}^{-1}(\mathrm{x})=\sec ^{-1} \mathrm{x}\)

(f)

Graph of arc cosec(x) or graph of cosec^(-1)x

\(\mathrm{f}^{-1}:(-\infty,-1] \cup[1, \infty)\)\(\rightarrow[-\pi / 2,0) \cup(0, \pi / 2]\)

\(\mathrm{f}^{-1}(\mathrm{x})=\operatorname{cosec}^{-1} \mathrm{x}\)


2. PROPERTIES OF INVERSE CIRCULAR FUNCTIONS:

Property-1 :

(i) \(y=\sin \left(\sin ^{-1} x\right)=x, \quad x \in[-1,1], y \in[-1,1], y\) is aperiodic

(ii) \(y=\cos \left(\cos ^{-1} x\right)\)\(=x \quad, x \in[-1,1],\)\( y\,\, \in \,[-1,1], y\) is aperiodic

(iii) \(y=\tan \left(\tan ^{-1} x\right)=x, x \in R, y \in R, y\) is aperiodic

(iv) \(y=\cot \left(\cot ^{-1} x\right)=x, x \in R, y \in R, y\) is aperiodic

(v) \(y=\operatorname{cosec}\left(\operatorname{cosec}^{-1} x\right)=x,|x| \geq 1,|y| \geq 1, y\) is aperiodic

(vi) \(y=\sec \left(\sec ^{-1} x\right)=x,|x| \geq 1 ;|y| \geq 1, y\) is aperiodic

Property 2 :

(i) \(y=\sin ^{-1}(\sin x), x \in R, y \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] .\) Periodic with period \(2 \pi\).

\(\sin ^{-1}(\sin x)=\left\{\begin{array}{cc}-\pi-x & ,-\frac{3 \pi}{2} \leq x \leq-\frac{\pi}{2} \\ x & , \quad-\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \\ \pi-x & , \quad \frac{\pi}{2} \leq x \leq \frac{3 \pi}{2} \\ x-2 \pi & , \quad \frac{3 \pi}{2} \leq x \leq \frac{5 \pi}{2} \\ 3 \pi-x & , \quad \frac{5 \pi}{2} \leq x \leq \frac{7 \pi}{2} \\ x-4 \pi & , \quad \frac{7 \pi}{2} \leq x \leq \frac{9 \pi}{2}\end{array}\right.\)

Graph of arc sin(sin x)

(ii) \(y=\cos ^{-1}(\cos x), x \in R, y \in[0, \pi]\), periodic with period \(2 \pi\)

\(\cos ^{-1}(\cos x)=\left\{\begin{array}{ccc}-x & , & -\pi \leq x \leq 0 \\ x & , & 0 \leq x \leq \pi \\ 2 \pi-x & , & \pi \leq x \leq 2 \pi \\ x-2 \pi & , & 2 \pi \leq x \leq 3 \pi \\ 4 \pi-x & , & 3 \pi \leq x \leq 4 \pi\end{array}\right.\)

Graph of arc cos(cos x)

(iii) \(y=\tan ^{-1}(\tan x)\)  \(\mathrm{x} \in \mathrm{R}-\left\{(2 \mathrm{n}-1) \frac{\pi}{2}, \mathrm{n} \in \mathrm{I}\right\}; \)\(y \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), periodic with period \(\pi\)

\(\tan ^{-1}(\tan x)=\left\{\begin{array}{cc}x+\pi & ,-\frac{3 \pi}{2}<x<-\frac{\pi}{2} \\ x & , \quad-\frac{\pi}{2}<x<\frac{\pi}{2} \\ x-\pi & , \quad \frac{\pi}{2}<x<\frac{3 \pi}{2} \\ x-2 \pi & , \quad \frac{3 \pi}{2}<x<\frac{5 \pi}{2} \\ x-3 \pi & , \quad \frac{5 \pi}{2}<x<\frac{7 \pi}{2}\end{array}\right.\)

Graph of arc tan(tan x)

(iv) \(\mathrm{y}=\cot ^{-1}(\cot \mathrm{x}), \mathrm{x} \in \mathrm{R}-\{\mathrm{n} \pi\}, \mathrm{y} \in(0, \pi)\), periodic with period \(\pi\)

Graph of arc cot(cot x)

(v) \(y=\operatorname{cosec}^{-1}(\operatorname{cosec} x),\) \( x \in R-\{n \pi, n \in I\} \) \(y \in\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]\), \(y\) is periodic with period \(2 \pi\).

Graph of arc cosec(cosec x)

(vi) \(\mathrm{y}=\sec ^{-1}(\sec \mathrm{x}), \mathrm{y}\) is periodic with period \(2 \pi\)

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

Graph of arc sec(sec x)

Property 3 :

(i) \(\operatorname{cosec}^{-1} \mathrm{x}=\sin ^{-1} \frac{1}{\mathrm{x}} ; \quad \mathrm{x} \leq-1\) or \(\mathrm{x} \geq 1\)

(ii) \(\sec ^{-1} \mathrm{x}=\cos ^{-1} \frac{1}{\mathrm{x}} ; \quad \mathrm{x} \leq-1\) or \(\mathrm{x} \geq 1\)

(iii) \(\cot ^{-1} \mathrm{x}=\tan ^{-1} \frac{1}{\mathrm{x}} ; \mathrm{x}>0\)

\(\qquad \qquad \quad=\pi+\tan ^{-1} \frac{1}{\mathrm{x}} ; \mathrm{x}<0\)

Property 4:

(i) \(\sin ^{-1}(-\mathrm{x})=-\sin ^{-1} \mathrm{x}, \quad-1 \leq \mathrm{x} \leq 1\)

(ii) \(\tan ^{-1}(-\mathrm{x})=-\tan ^{-1} \mathrm{x}, \mathrm{x} \in \mathrm{R}\)

(iii) \(\cos ^{-1}(-\mathrm{x})=\pi-\cos ^{-1} \mathrm{x},-1 \leq \mathrm{x} \leq 1\)

(iv) \(\cot ^{-1}(-\mathrm{x})=\pi-\cot ^{-1} \mathrm{x}, \mathrm{x} \in \mathrm{R}\)

(v) \(\sec ^{-1}(-x)=\pi-\sec ^{-1} x, x \leq-1\) or \(x \geq 1\)

(vi) \(\operatorname{cosec}^{-1}(-\mathrm{x})=-\operatorname{cosec}^{-1} \mathrm{x}, \mathrm{x} \leq-1\) or \(\mathrm{x} \geq 1\)

Property 5 :

(i) \(\sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2},-1 \leq x \leq 1\)

(ii) \(\tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}, \quad x \in R\)

(iii) \(\operatorname{cosec}^{-1} x+\sec ^{-1} x=\frac{\pi}{2},|x| \geq 1\)

Property 6:

(i) \(\tan ^{-1} x+\tan ^{-1} y\)\(=\left\{\begin{array}{l}\tan ^{-1} \frac{x+y}{1-x y}, \text { where } x>0, y>0 \, \&\, x y<1 \\ \pi+\tan ^{-1} \frac{x+y}{1-x y}, \text { where } x>0, y>0\, \&\, x y>1 \\ \frac{\pi}{2}, \text { where } x>0, y>0 \, \&\, x y=1\end{array}\right.\)

(ii) \(\tan ^{-1} x-\tan ^{-1} y=\tan ^{-1} \frac{x-y}{1+x y}\), where \(x>0,\) \( y>0\)

(iii) \(\sin ^{-1} x+\sin ^{-1} y\)\(=\sin ^{-1}\left[x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right]\)
where \(x>0, y>0 \) & \(\left(x^{2}+y^{2}\right)<1\)

Note that : \(\mathrm{x}^{2}+\mathrm{y}^{2}<1 \Rightarrow 0<\sin ^{-1} \mathrm{x}+\sin ^{-1} \mathrm{y}<\frac{\pi}{2}\)

(iv) \(\sin ^{-1} x+\sin ^{-1} y\)\(=\pi-\sin ^{-1}\left[x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right]\),
where \(x>0, y>0 \) & \(x^{2}+y^{2}>1\)

Note that : \(\mathrm{x}^{2}+y^{2}>1 \Rightarrow \frac{\pi}{2}<\sin ^{-1} \mathrm{x}+\sin ^{-1} \mathrm{y}<\pi\)

(v) \(\sin ^{-1} x-\sin ^{-1} y\)\(=\sin ^{-1}\left[x \sqrt{1-y^{2}}-y \sqrt{1-x^{2}}\right]\) where \(x>0, y>0\)

(vi) \(\cos ^{-1} x+\cos ^{-1} y\)\(=\cos ^{-1}\left[x y-\sqrt{1-x^{2}} \sqrt{1-y^{2}}\right]\), where \(x>0, y>0\)

(vii) \(\cos ^{-1} x-\cos ^{-1} y=\)\(\left\{\begin{array}{ll}\cos ^{-1}\left(x y+\sqrt{1-x^{2}} \sqrt{1-y^{2}}\right); &  x<y, \, x, y>0 \\ -\cos ^{-1}\left(x y+\sqrt{1-x^{2}} \sqrt{1-y^{2}}\right) ; & x>y, \, x, y>0\end{array}\right.\)

(viii) \(\tan ^{-1} \mathrm{x}+\tan ^{-1} \mathrm{y}+\tan ^{-1} \mathrm{z}\)\(=\tan ^{-1}\left[\dfrac{\mathrm{x}+\mathrm{y}+\mathrm{z}-\mathrm{xyz}}{1-\mathrm{x} y-\mathrm{yz}-\mathrm{zx}}\right]\)
if \(x>0, y>0, z>0 \) & \(x y+y z+z x<1\)

Note : In the above results \(\mathrm{x} \) & \( \mathrm{y}\) are taken positive. In case if these are given as negative, we first apply \(\mathrm{P}-4\) and then use above results.


3. SIMPLIFIED INVERSE TRIGONOMETRIC FUNCTIONS :

(a) 

Graph of arc sin(2x/(1+x²))

\(y=f(x)=\sin ^{-1}\left(\dfrac{2 x}{1+x^{2}}\right)\)

\(=\left[\begin{array}{ccc}2 \tan ^{-1} x & \text { if } & \mid x \leq 1 \\ \pi-2 \tan ^{-1} x & \text { if } & x>1 \\ -\left(\pi+2 \tan ^{-1} x\right) & \text { if } & x<-1\end{array}\right.\)

(b) 

Graph of arc cos((1-x²)/(1+x²))

\(y=f(x)=\cos ^{-1}\left(\dfrac{1-x^{2}}{1+x^{2}}\right)\)

\(=\left[\begin{array}{ccc}2 \tan ^{-1} x & \text { if } & x \geq 0 \\ -2 \tan ^{-1} x & \text { if } & x<0\end{array}\right.\)

(c) 

Graph of arc tan(2x/(1-x²))

\(y=f(x)=\tan ^{-1} \dfrac{2 x}{1-x^{2}}\)

\(=\left[\begin{array}{llc}2 \tan ^{-1} x & \text { if } & |x|<1 \\ \pi+2 \tan ^{-1} x & \text { if } & x<-1 \\ -\left(\pi-2 \tan ^{-1} x\right) & \text { if } & x>1\end{array}\right.\)

(d)

Graph of arc sin(3x-4x³)

\(y=f(x)=\sin ^{-1}\left(3 x-4 x^{3}\right)\)

\(=\left[\begin{array}{ccc}-\left(\pi+3 \sin ^{-1} x\right) & \text { if } & -1 \leq x \leq-\frac{1}{2} \\ 3 \sin ^{-1} x & \text { if } & -\frac{1}{2} \leq x \leq \frac{1}{2} \\ \pi-3 \sin ^{-1} x & \text { if } & \frac{1}{2} \leq x \leq 1\end{array}\right.\)

(e)
Graph of arc cos(4x³-3x)

 \(y=f(x)=\cos ^{-1}\left(4 x^{3}-3 x\right)\)  

\(=\left\{\begin{array}{ll}3 \cos ^{-1} x-2 \pi & \text { if } & -1 \leq x \leq-\frac{1}{2} \\ 2 \pi-3 \cos ^{-1} x & \text { if } & -\frac{1}{2} \leq x \leq \frac{1}{2} \\ 3 \cos ^{-1} x & \text { if } & \frac{1}{2} \leq x \leq 1\end{array}\right.\)

(f) 

Graph of arc sin(2x√(1-x²))

\(\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right) \)

\(=\left\{\begin{array}{ll}-\left(\pi+2 \sin ^{-1} x\right) & -1 \leq x \leq-\frac{1}{\sqrt{2}} \\ 2 \sin ^{-1} x & -\frac{1}{\sqrt{2}} \leq x \leq \frac{1}{\sqrt{2}} \\ \pi-2 \sin ^{-1} x & \frac{1}{\sqrt{2}} \leq x \leq 1\end{array}\right.\)

(g)

graph of arc cos(2x²-1)

\(\cos ^{-1}\left(2 x^{2}-1\right) \)

\(=\left\{\begin{array}{ll}2\cos ^{-1} x & 0 \leq x \leq 1 \\ 2 \pi-2 \cos ^{-1} x & -1 \leq x \leq 0\end{array}\right.\)

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