Skip to main content

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)

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

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

Labels:

Comments

Post a Comment

Popular posts from this blog

Method of differntiation - Notes, Concept and All Important Formula

METHODS OF DIFFERENTIATION 1. DERIVATIVE OF f(x) FROM THE FIRST PRINCIPLE : Obtaining the derivative using the definition \(\displaystyle\displaystyle \lim_{\delta x \rightarrow 0} \dfrac{\delta y}{\delta x}= \displaystyle\displaystyle \lim_{\delta x \rightarrow 0} \dfrac{f(x+\delta x)-f(x)}{\delta x}=f^{\prime}(x)=\dfrac{d y}{d x}\) is called calculating derivative using first principle or ab initio or delta method. All Chapter Notes, Concept and Important Formula 2. FUNDAMENTAL THEOREMS : If \(f\) and \(g\) are derivable function of \(x\) , then, (a) \(\dfrac{\mathrm{d}}{\mathrm{dx}}(\mathrm{f} \pm \mathrm{g})=\dfrac{\mathrm{df}}{\mathrm{dx}} \pm \dfrac{\mathrm{d} \mathrm{g}}{\mathrm{d} \mathrm{x}}\) , known as SUM RULE (b) \(\dfrac{\mathrm{d}}{\mathrm{dx}}(\mathrm{cf})=\mathrm{c} \dfrac{\mathrm{df}}{\mathrm{dx}}\) , where \(\mathrm{c}\) is any constant (c) \(\dfrac{\mathrm{d}}{\mathrm{dx}}(\mathrm{fg})=\mathrm{f} \dfrac{\mathrm{dg}}{\mathrm{dx}}+\mathrm{g} \dfrac{\mathrm{df}...
Post a Comment
Read more

Hyperbola - Notes, Concept and All Important Formula

HYPERBOLA The Hyperbola is a conic whose eccentricity is greater than unity \((e>1) .\) 1. STANDARD EQUATION & DEFINITION(S): Standard equation of the hyperbola is \(\dfrac{\mathbf{x}^{2}}{\mathbf{a}^{2}}-\dfrac{\mathbf{y}^{2}}{\mathbf{b}^{2}}=\mathbf{1},\) where \(b^{2}=a^{2}\left(e^{2}-1\right)\) or \(a^{2} e^{2}=a^{2}+b^{2}\)    i.e.    \(e^{2}=1+\dfrac{b^{2}}{a^{2}}\) \(=1+\left(\dfrac{\text { Conjugate Axis }}{\text { Transverse Axis }}\right)^{2}\) (a) Foci : \(\mathrm{S} \equiv(\mathrm{a} e, 0) \quad \& \quad \mathrm{~S}^{\prime} \equiv(-\mathrm{a} e, 0) .\) (b) Equations of directrices: \(\mathrm{x}=\dfrac{\mathrm{a}}{e}\quad \) & \(\quad \mathrm{x}=-\dfrac{\mathrm{a}}{e}\) (c) Vertices: \(A \equiv(a, 0)\quad \) & \(\quad A^{\prime} \equiv(-a, 0)\) (d) Latus rectum: (i) Equation: \(\mathrm{x}=\pm \mathrm{ae}\) (ii) Length:  \(\begin{aligned} &=\dfrac{2 b^{2}}{a}=\dfrac{(\text { Conjugate Axis })^{2}}{(\text { Transverse Axis ...
Post a Comment
Read more

What are Function and how its work on Calculus?

What are Function ? Table Of Contents Introduction with beautiful example Here's a plant, and what you see here is it's  shadow. Can you list the things that the length of the shadow is dependent on. One, it's dependent on the position of the source of light. Anything else that you can think of. If the height of the plant grows then the shadows length will also change, right. So the length of the shadow is dependent on the position of the source of light, and the height of the plant too. So we can say that the length of the shadow is a function of the following two things. The output is dependent on these two things, which could be considered as the inputs. That's a very simple way to understand functions. Could you think of more inputs, this output is dependent on here, tell us yours answers in the comment section below.  How do Function work in calculus ? That's what we'll see in this topics. Previously, we saw an idea to find the instant...
1 comment
Read more