Advertisement

Basic Concepts of Trigonometric Functions

description

sine, cosine, tangent, cotangent, secant, cosecant function

notes

In Class XI, we have studied trigonometric functions, which are defined as follows: 

Trigonometric functions Domain Range
sin R [-1,1]
cos R [-1.1]
tan R- {x:x= (2n+1)π/2,  n∈ Z} R
cot R- {x:x= nπ, n∈ Z} R
sec R- {x:x= (2n+1) π/2, n∈ Z} R- (-1,1)
cosec R- {x:x= nπ, n∈ Z} R- (-1,1)

We already know that, if f: x→y
f(x)= y and f is bijective, 
then there exists g: y→x
g(y)= x
g is called `"f"^-1`
So, with the help of this knowledge we will find the inverse of the trigonometric functions
1)Inverse of sin-
Sin R→[-1,1]
sin x= y

`sin: [-π/2, π/2]→[-1,1]`


`sin^(-1): [-1,1]→[-π/2, π/2],` here `[-π/2, π/2]` is the principle range 

`sin^(-1)y= x`


2) Inverse of cos-
cos: R→[-1,1]

cos: [0, π]→[-1,1]
cos (x)=y is a bijective function 
`cos^(-1): [-1,1]→[0,π]`, here `[0,π]` is the principle range
`cos^-y=x`
3) Inverse of tan-
`tan: "R"- {x:x= (2n+1)π/2,  n∈ "Z"}→"R"`

`tan: [(-π/2),  (π/2)]→"R"`


tan x= y is a bijective function


`"tan"^(-1): R→(-pi/2, pi/2)`, where `(-pi/2, pi/2)` is the principle range.


`tan^(-1)y= x`


4) Inverse of cot-
`"cot": "R"- {x:x= n pi, n∈"Z"}→R}`

`cot: (0, pi)→"R"`
cot x= y is a bijective function
`cot ^(-1):" R"→(0, pi)` , where `(0, pi)` is the principle range.
`cot :"R"→ (0, pi)`
`cot^(-1)y= x`
5) Inverse of sec-
`sec: "R"-{x:x= (2n+1) pi/2}→"R"`

`sec: [0, pi]- {pi/2}→"R"- (-1,1)`


sec x= y is bijective function


`sec^-1: "R"- (-1,1) →[0, pi]- {pi/2}`, where `[0, pi]- {pi/2}` is the principle range

6) Inverse of cosec-
`cosec: "R"- {x=n pi, n∈"Z"}→"R"-(-1,1)`

`cosec: [-pi/2, pi/2] -{0}→"R"-(-1,1)`


cosec x= y is bijective function

`cosec^-1: "R"-(-1,1)→[-pi/2, pi/2]-{0}`, where `[-pi/2, pi/2] -{0}` is the principle range.

The following table gives the inverse trigonometric function (principal value branches) along with their domains and ranges.

If you would like to contribute notes or other learning material, please submit them using the button below.

Video Tutorials

We have provided more than 1 series of video tutorials for some topics to help you get a better understanding of the topic.

Series 1


Series 2


Series 3


Shaalaa.com | Inverse Trigonometry Functions part 2 (Natural domain Range)

Shaalaa.com


Next video


Shaalaa.com


Inverse Trigonometry Functions part 2 (Natural domain Range) [00:04:24]
S
Series: 1
0%


Advertisement
Share
Notifications

View all notifications
Login
Create free account


      Forgot password?
View in app×