Warm-Up: 9/14/12 Find the amplitude, period, vertical asymptotes, domain, and range. Sketch the graph.

Warm-Up: 9/14/12Find the amplitude, period, vertical asymptotes, domain, and range. Sketch the graph.

3sec 2 24

y x

3sec 2 24

y x

3sec 2 24

y x

4.7 – Inverse Trigonometric Functions

In this section, you will learn to:

Evaluate the inverse trigonometric functions

Evaluate the composition of trigonometric functions

Functions: In order for a relation to be a function, it must

pass the vertical line test.

For a function to have an inverse, it must pass the horizontal line test.

Different values of x cannot yield the same values of y.

Function or not a function?

1)

2) 3)

1,2 1, 2 2,3 2, 5

1 2 3 4 5-1-2

1

2

3

-1

-2

-3

x

y

1 2 3-1-2-3-4-5

1

2

3

-1

-2

-3

x

y

Which function has an inverse?

1)

2) 3)

1,2 1, 2 2,3 2, 5

1 2 3 4 5-1-2

1

2

3

-1

-2

-3

x

y

1 2 3-1-2-3-4-5

1

2

3

-1

-2

-3

x

y

Inverse Sine Function: The sine function does not pass the horizontal

test, therefore it does not have an inverse.

However, if we restrict the domain, then it will pass the horizontal line test.

Inverse Sine Function:

Inverse Sine Function:

Inverse Sine Function: If we restrict the domain to the interval 

 , then it will pass the  horizontal 2 2

line test. On this restricted interval, sin

has a unique inverse called the inverse sine

function denoted as ar

x

y x

y

1csin or sin .x y x

Definition of an Inverse Sine Function:

1

The inverse sine function is defined by

arcsin   or  sin  if and only if

sin   where   1 1  and .2 2

y x y x

y x x y

The domain of   arcsin is 1,1 and  the

range is , .2 2

y x

Graphing an Inverse Sine Function: To sketch the graph of an inverse sine

function, interchange the domain and the range of the original sine function.

y

sinx y 1

2

6

1

2

2

11

2

6

Graphing an Inverse Sine Function: Inverse functions are reflected about the

line y = x.

Inverse Cosine Function: The cosine function does not pass the horizontal

test, therefore it does not have an inverse.

However, if we restrict the domain, then it will pass the horizontal line test.

Inverse Cosine Function:

Inverse Cosine Function:

Inverse Cosine Function: If we restrict the domain to the interval

0 , then it will pass the  horizontal

line test. On this restricted interval,  cos

has a unique inverse called the inverse cosine

function denoted as  arc

x

y x

y

1cos   or   cos .   x y x

Definition of an Inverse Cosine Function:

1

The inverse cosine function is defined by

arccos   or  cos  if and only if

cos   where   1 1  and 0 .

y x y x

y x x y

The domain of   arccos is 1,1 and  the

range is 0, .

y x

Graphing an Inverse Cosine Function:

To sketch the graph of an inverse cosine function, interchange the domain and the range of the original cosine function.

y

cosx y 1

0 3

4

2

2

12

2

4

Graphing an Inverse Cosine Function: Inverse functions are reflected about the

line .y x

Inverse Tangent Function: The tangent function does

not pass the horizontal test, therefore it does not have an inverse.

However, if we restrict the domain, then it will pass the horizontal line test.

Inverse Tangent Function:

Inverse Tangent Function:

Inverse Tangent Function: If we restrict the domain to the interval

, then it will pass the  horizontal 2 2

line test. On this restricted interval,  tan

has a unique inverse called the inverse tangent

function denoted as 

x

y x

y

1arctan   or   tan .   x y x

Definition of an Inverse Tangent Function:

1

The inverse tangent function is defined by

arctan   or  tan  if and only if

tan   where     and .2 2

y x y x

y x x y

The domain of   arctan is , and  the

range is , .2 2

y x

Graphing an Inverse Tangent Function:

To sketch the graph of an inverse tangent function, interchange the domain and the range of the original sine function.

y

tanx y .undef

2

4

1 .undef1

4

2

Graphing an Inverse Tangent Function: Inverse functions are reflected about the

line .y x

Find the exact value of the inverse functions:

21) arccos :

2

32) arctan :

3

3) arccos 1 :

Solutions:

2 31) arccos ; 0

2 4y

3) arcsin 1 ;2 2 2

y

32) arctan ;

3 6 2 2y

Composition of Functions:

1) If   1 1 and ,  then2 2

sin arcsin and arcsin sin

x y

x x x x

2) If    1 1 and 0 ,   then

cos arccos and arccos cos

x y

x x x x

3) If    1 1 and , then2 2

tan arctan and arctan tan

x y

x x x x

Composition of Function Examples:

21) sin arcsin :

2

12) cos arcsin :

2

3) tan arcsin 1 :

2

2

3cos

6 2

tan undefined2