We know from previous chapter, A function is invertable if it is one-one and onto both and we can make function invertible by restricting the domain and range of the function
Now we know that trigonometric function are many one function,so it is not possible to define the inverse function in thet domain and range
But we can restrict the domain and range of the function to make them invertible.
In this page, we will check out the inverse of all the six trigonometric functions with their domain and range.
Inverse of Sin function
sin function is a many one function with Domain and range as given below
Domain= R
Range =[-1,1]
if we restrict its domain to $[-\frac {\pi}{2},\frac {\pi}{2}]$ , then it becomes one-one and onto with the range [-1,1]
This is also true if we restrict its domain to $[\frac {\pi}{2},\frac {3\pi}{2}]$, $[-\frac {3\pi}{2},\frac {-\pi}{2}]$,.. , then also it becomes one-one and onto with the range [-1,1]. So it is becoming invertible with different branches
The branch $[-\frac {\pi}{2},\frac {\pi}{2}]$ is the called the Principal value branch
We define inverse sin for this branch only
So inverse sin function is defined as
f: [-1,1] -> $[-\frac {\pi}{2},\frac {\pi}{2}]$ $f(x) =sin^{-1}(x)$
Also $sin (sin^{-1} x) =x$ where $x \in [-1,1]$ and $sin^{-1} (sin x) =x$ where $x \in [-\frac {\pi}{2},\frac {\pi}{2}]$
Also $y=sin^{-1} x$ then $x= sin y$
Sometimes the function $sin^{-1} x$ is written as arc sin (x)
Graph of Inverse Sin function
The graph of the sin function in the Domain $[-\frac {\pi}{2},\frac {\pi}{2}]$ is given by
The graph of the inverse sin function can be obtained by interchanging the x and y axis
Inverse of Cos function
cos function is a many one function with Domain and range as given below
Domain= R
Range =[-1,1]
if we restrict its domain to $[0,\pi]$ , then it becomes one-one and onto with the range [-1,1]
This is also true if we restrict its domain to $[\pi,2\pi]$, $[-\pi,0]$,.. , then also it becomes one-one and onto with the range [-1,1]. So it is becoming invertible with different branches
The branch $[0,\pi]$ is the called the Principal value branch
We define inverse cos for this branch only
So inverse cos function is defined as
f: [-1,1] -> $[0,\pi]$ $f(x) =cos^{-1}(x)$
Also $cos (cos^{-1} x) =x$ where $x \in [-1,1]$ and $cos^{-1} (cos x) =x$ where $x \in [0,\pi]$
Also $y=cos^{-1} x$ then $x= cos y$
Sometimes the function $cos^{-1} x$ is written as arc cos (x)
Graph of Inverse cos function
The graph of the cos function in the Domain $[0,\pi]$ is given by
The graph of the inverse cos function can be obtained by interchanging the x and y axis
Inverse of Cosec function
cosec function is a many one function with Domain and range as given below
Domain= [x:$x \in R$ and $x \neq n \pi$, $n \in Z$]
Range =[y:$|y| \geq 1$]
$f(x) =cosec x= \frac {1}{sin x}$
if we restrict its domain to $[-\frac {\pi}{2},\frac {\pi}{2}] - {0} $ , then it becomes one-one and onto with the range R-(-1,1)
This is also true if we restrict its domain to $[\frac {\pi}{2},\frac {3\pi}{2}] - {\pi}$, $[-\frac {3\pi}{2},\frac {-\pi}{2}] - {-\pi} $,.. , then also it becomes one-one and onto with the range R-(-1,1). So it is becoming invertible with different branches
The branch $[-\frac {\pi}{2},\frac {\pi}{2}] -{0}$ is the called the Principal value branch
We define inverse cosec for this branch only
So inverse cos function is defined as
f: R-(-1,1) -> $[-\frac {\pi}{2},\frac {\pi}{2}] - {0} $ $f(x) =cosec^{-1}(x)$
Also $cosec (cosec^{-1} x) =x$ where $x \in R-(-1,1)$ and $cosec^{-1} (cosec x) =x$ where $x \in [-\frac {\pi}{2},\frac {\pi}{2}] -{0}$
Also $y=cosec^{-1} x$ then $x= cosec y$
Sometimes the function $cosec^{-1} x$ is written as arc cosec (x)
Graph of Inverse cosec function
The graph of the cosec function in the Domain $[-\frac {\pi}{2},\frac {\pi}{2}] - {0} $ is given by
The graph of the inverse cosec function can be obtained by interchanging the x and y axis
Inverse of sec function
sec function is a many one function with Domain and range as given below
Domain= [x:$x \in R$ and $x \neq \frac {(2n+1)}{2} \pi$, $n \in Z$]
Range =[y:$|y| \geq 1$]
$f(x) =sec x= \frac {1}{sec x}$
if we restrict its domain to $[0,\pi] - \frac {\pi}{2}$ , then it becomes one-one and onto with the range R-(-1,1)
This is also true if we restrict its domain to $[\pi,2\pi] -\frac {3\pi}{2}$, $[-\pi,0] - \frac {-\pi}{2}$,.. , then also it becomes one-one and onto with the R-(-1,1). So it is becoming invertible with different branches
The branch $[0,\pi] - \frac {\pi}{2}$ is the called the Principal value branch
We define inverse sec for this branch only
So inverse cos function is defined as
f: R-(-1,1) -> $[0,\pi] - \frac {\pi}{2}$, $f(x) =sec^{-1}(x)$
Also $sec (sec^{-1} x) =x$ where $x \in R-(-1,1)$ and $sec^{-1} (sec x) =x$ where $[0,\pi] - \frac {\pi}{2}$
Also $y=sec^{-1} x$ then $x= sec y$
Sometimes the function $sec^{-1} x$ is written as arc sec (x)
Graph of Inverse sec function
The graph of the cos function in the Domain $[0,\pi]$ is given by
The graph of the inverse cos function can be obtained by interchanging the x and y axis
Inverse of tan function
tan function is a many one function with Domain and range as given below
Domain= [x:$x \in R$ and $x \neq \frac {(2n+1}{2} \pi$, $n \in Z$]
Range =R
if we restrict its domain to $(-\frac {\pi}{2},\frac {\pi}{2})$ , then it becomes one-one and onto with the range R
This is also true if we restrict its domain to $(\frac {\pi}{2},\frac {3\pi}{2})$, $(-\frac {3\pi}{2},\frac {-\pi}{2})$,.. , then also it becomes one-one and onto with the range R. So it is becoming invertible with different branches
The branch $(-\frac {\pi}{2},\frac {\pi}{2})$ is the called the Principal value branch
We define inverse tan for this branch only
So inverse sin function is defined as
f: R -> $(-\frac {\pi}{2},\frac {\pi}{2})$ $f(x) =tan^{-1}(x)$
Also $tan (tan^{-1} x) =x$ where $x \in R$ and $tan^{-1} (tan x) =x$ where $x \in (-\frac {\pi}{2},\frac {\pi}{2})$
Also $y=tan^{-1} x$ then $x= tan y$
Sometimes the function $tan^{-1} x$ is written as arc tan (x)
Graph of Inverse tan function
The graph of the tan function in the Domain $(-\frac {\pi}{2},\frac {\pi}{2})$ is given by
The graph of the inverse tan function can be obtained by interchanging the x and y axis
Inverse of cot function
cot function is a many one function with Domain and range as given below
Domain= [x:$x \in R$ and $x \neq n\pi$, $n \in Z$]
Range =R
$cot(x) = \frac {1}{tan x}$
if we restrict its domain to $(0,\pi)$ , then it becomes one-one and onto with the range R
This is also true if we restrict its domain to $(\pi,2\pi)$, $(-\pi,0)$,.. , then also it becomes one-one and onto with the range R. So it is becoming invertible with different branches
The branch $(0,\pi)$ is the called the Principal value branch
We define inverse cot for this branch only
So inverse cot function is defined as
f: R -> $(0,\pi)$ $f(x) =cot^{-1}(x)$
Also $cot (cot^{-1} x) =x$ where $x \in R$ and $cot^{-1} (cot x) =x$ where $x \in (0,\pi)$
Also $y=cot^{-1} x$ then $x= cot y$
Sometimes the function $cot^{-1} x$ is written as arc cot (x)
Graph of Inverse cot function
The graph of the cot function in the Domain $(0,\pi)$ is given by
The graph of the inverse cot function can be obtained by interchanging the x and y axis
