Mathematics Introduction, Basic concept and Graph of inverse trigonometry function For CBSE-NCERT
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# Graphs of Trigonometric Functions

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# Graphs of Inverse Trigonometric Functions

### Topic Covered

star Introduction
star Basic Concept and graph
star Graph of Inverse trigonometry function

### Introduction

\color{green} ✍️ As we've studied, the inverse of a function f, denoted by f^(–1), exists if f is one-one and onto.

\color{green} ✍️ As we know that trigonometric functions are not one-one and onto over their natural domains and ranges so their inverses do not exist.

\color{green} ✍️ Here we will study, how to restrict domains and ranges of trigonometric functions which ensure the existence of their inverses.

### Basic Graphs of Trigonometry

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

1. sine function, i.e.,
sin : R → [– 1, 1]

2. cosine function, i.e.,
 cos : R → [– 1, 1]

3. tangent function, i.e., tan : R – { x : x = (2n + 1) π/2 , n ∈ Z} → R

4. cotangent function, i.e.,
cot : R – { x : x = nπ, n ∈ Z} → R

5.secant function, i.e., sec : R – { x : x = (2n + 1)π/2 , n ∈ Z} → R – (– 1, 1)

6. cosecant function, i.e., cosec : R – { x : x = nπ, n ∈ Z} → R – (– 1, 1)

### Plotting of sin^(-1) graph

=> Let us first take Sine function.

y= sin x, quad quad R-> [-1,1]

=> This function is not one-one and hence cannot be inverted.

But we can make a function one-one by restricting the domain.

=> We can make the function one-one by restricting the domain to [-(3pi)/2 , -pi/2] , [ - pi/2 , pi/2] [ ( pi/2 , (3pi)/2]

\color{blue} ✍️ \color{green}(KEY \ CONCEPT) :  The branch with range [- \ pi/2 , pi/2] is called principal value of branch, whereas other intervals as range give different branches of sin^(–1).

\color { maroon} ® \color{maroon} ul (" REMEMBER") : Whenever we refer to the function sin^(–1), we take it as the function whose domain is [–1, 1] and range is [-pi/2,pi/2]

sin^(-1) : [-1 , 1]-> [ - \ pi/2 , pi/2]

Remarks :

(i) If y = f (x) is an invertible function, then x = f^(–1) (y). Thus, the graph of sin^(–1) function can be obtained from the graph of original function by interchanging x and y axes, i.e., if (a, b) is a point on the graph of sine function, then (b, a) becomes the corresponding point on the graph of inverse the graph of y = sin x by interchanging x and y axes. The graphs of y = sin x and y = sin^(–1) x are as given in . The dark portion of the graph of y = sin^(–1) x represent the principal value branch.

(ii) It can be shown that the graph of an inverse function can be obtained from the corresponding graph of original function as a mirror image (i.e., reflection) along the line y = x. This can be visualised by looking the graphs of y = sin x and y = sin^(–1) x as given in the same axes

### Plotting of cos^(-1) graph

=> Let us first take cos function, same as sin function

y= cos x, quad quad R-> [-1,1]

=> This function is not one-one and hence cannot be inverted.

But we can make a function one-one by restricting the domain.

=> We can make the function one-one by restricting the domain to [- pi , 0] , [ 0 , pi] [ pi , 2pi]

\color{green} ✍️ \color{green} (KEY \ CONCEPT) :  The branch with range [0 , pi] is called principal value of branch, whereas other intervals as range give different branches of cos^(–1).

\color { maroon} ® \color{maroon} ul (" REMEMBER") : Whenever we refer to the function cos^(–1), we take it as the function whose domain is [–1, 1] and range is [0,pi]

cos^(-1) : [-1 , 1]-> [ 0 , pi]

### Plotting of tan^(-1) graph

=> Let us first take tan function,

tan : R – { x : x = (2n + 1) π/2 , n ∈ Z} → R

=> This function is not one-one and hence cannot be inverted.

But we can make a function one-one by restricting the domain.

=> We can make the function one-one by restricting the domain to (- (3pi)/2 , -pi/2) , ( -pi/2 , pi/2), ( ( pi/2 , (3pi)/2))

\color{green} ✍️ \color{green} (KEY \ CONCEPT) :  The branch with range (-pi/2 , pi/2) is called principal value of branch, whereas other intervals as range give different branches of tan^(-1).

\color { maroon} ® \color{maroon} ul (" REMEMBER") : Whenever we refer to the function tan^(–1), we take it as the function whose domain is R and range is (-pi/2,pi/2)

tan^(-1) : R-> ( -pi/2 , pi/2)

### Plotting of cosec^(-1) graph

=> Let us first take Sine function.

cosec : R – { x : x = nπ, n ∈ Z} → R – (– 1, 1)

=> This function is not one-one and hence cannot be inverted.

But we can make a function one-one by restricting the domain.

=> We can make the function one-one by restricting the domain to [-(3pi)/2 , -pi/2] - {pi}, [ -pi/2 , pi/2] - {0}, [ pi/2 , (3pi)/2]-{pi}

\color{green} ✍️ \color{green} (KEY \ CONCEPT) :  The branch with range [- pi/2 , pi/2] - {0} is called principal value of branch, whereas other intervals as range give different branches of cosec^(–1).

\color { maroon} ® \color{maroon} ul (" REMEMBER") : Whenever we refer to the function cosec^(–1), we take it as the function whose domain is R-[–1, 1] and range is [-pi/2,pi/2] - {0}

cosec^(-1) : R - [-1 , 1]-> [ - \ pi/2 , pi/2] - {0}

### Plotting of sec^(-1) graph

=> Let us first take sec function,

sec : R – { x : x = (2n + 1)π/2 , n ∈ Z} → R – (– 1, 1)

=> This function is not one-one and hence cannot be inverted.

But we can make a function one-one by restricting the domain.

=>We can make the function one-one by restricting the domain to [- pi , 0]-{-pi/2} , [ 0 , pi]-{pi/2}, [ pi , 2pi] - {(3pi)/2}

\color{green} ✍️ \color{green} (KEY \ CONCEPT) :  The branch with range [0 , pi] - {pi/2} is called principal value of branch, whereas other intervals as range give different branches of sec^(–1).

\color { maroon} ® \color{maroon} ul (" REMEMBER") : Whenever we refer to the function sec^(–1), we take it as the function whose domain is R - [–1, 1] and range is [0,pi]-{pi/2}

sec^(-1) :R- [-1 , 1]-> [ 0 , pi]-{pi/2}

### Plotting of cot^(-1) graph

=> Let us first take cot function, same as sin function

cot : R - { x : x = nπ, n ∈ Z} → R

=> This function is not one-one and hence cannot be inverted.

But we can make a function one-one by restricting the domain.

=> We can make the function one-one by restricting the domain to (- pi , 0) , ( 0 , pi), ( pi , 2pi)

\color{green} ✍️ \color{green} (KEY \ CONCEPT) :  The branch with range (0 , pi) is called principal value of branch, whereas other intervals as range give different branches of cot^(–1).

\color { maroon} ® \color{maroon} ul (" REMEMBER") : Whenever we refer to the function cot^(–1), we take it as the function whose domain is R and range is (0,pi)

cot^(-1) : R -> (0 , pi)