Chapter 4.4 Inverse Circular

Functions

1

Recall

For an inverse of a function to be in itself a

function, the function must be 1-1.

Are circular functions 1-1?

2

x

y

Restricting the Domain

Given sin we consider , as2 2

the restricted domain.

f x x

2

2

3

Inverse Sine Function

2 2

2 2

inverse sine functio

Let be the sine function with domain , .

Then the is defined as

Arcsin if and only i

n

f sin

where 1,1 and , .

f

y x x y

x y

4

Inverse Sine Function

2

2

1

1

11

2

2

5

Inverse Sine Function

2

2

1

1

6

Example 4.4.1

Evaluate the following.

1. Arcsin 0 0

2. Arcsin 12

13. Arcsin

2 6

34. Arcsin

2 3

25. Arcsin

2 4

7

Restricting the Domain

Given cos we consider 0, as

the restricted domain.

f x x

8

Inverse Cosine Function

inverse cosine funct

Let be the cosine function with domain 0, .

Then the is defined as

Arccos if and only if cos

where 1,1 an

ion

d 0, .

f

y x x y

x y

9

Inverse Cosine Function

0

1

111 0

10

Inverse Cosine Function

0

1

1

11

Example 4.4.2

Evaluate the following.

1. Arccos 02

2. Arccos 1 0

1 23. Arccos

2 3

34. Arccos

2 6

25. Arccos

2 4

12

Restricting the Domain

2 2Given tan we consider , as

the restricted domain.

f x x

2

2

13

Inverse Tangent Function

2 2

2 2

inverse tangent fu

Let be the tangent function with domain , .

Then the is defined as

Arctan if and only if tan

where and , .

nction

f

y x x y

x y

14

Inverse Tangent Function

2

2

2

2

15

Inverse Tangent Function

16

Example 4.4.3

Evaluate the following.

1. Arctan 0 0

2. Arctan 14

33. Arctan

3 6

4. Arctan 33

17

Restricting the Domain

Given cot we consider 0, as

the restricted domain.

f x x

0

18

Inverse Cotangent Function

inverse cotangent

Let be the cotangent function with domain 0, .

Then the is defined as

Arccot if and only if cot

where a

functio

n 0, .

n

d

f

y x x y

x y

19

Inverse Cotangent Function

0

0

20

Inverse Cotangent Function

21

Example 4.4.4

Evaluate the following.

31. Arccot 1

4

3 22. Arccot

3 3

3. Arccot 36

4. Arccot 0 =2

22

Restricting the Domain

2 2Given sec we consider 0, , as

the restricted domain.

f x x

0

2

1

1

23

Inverse Secant Function

2 2

2 2

inverse secant function

Let be the secant function with domain 0, , .

Then the is defined as

Arcsec if and only if sec

where , 1 1, and 0, , .

f

y x x y

x y

24

Inverse Secant Function

0

2

1

1 101

2

25

Inverse Secant Function

26

Example 4.4.5

Evaluate the following.

1. Arcsec 0 is undefined.

2. Arcsec 1 0

33. Arcsec 2

4

4. Arcsec 23

27

Restricting the Domain

2 2Given csc we consider ,0 0, as

the restricted domain.

f x x

0

2

2

1

1

28

Inverse Cosecant Function

2 2

2 2

inverse cosecant functi

Let be the cosecant function with domain ,0 0, .

Then the is defined as

Arccsc if and only if csc

where , 1 1, and

o

,0

n

0, .

f

y x x y

x y

29

Inverse Cosecant Function

2

2

1

1

10

01

2

2

30

Inverse Cosecant Function

31

Example 4.4.5

Evaluate the following.

2 31. Arccsc

3 3

2. Arccsc 12

3. Arccsc 24

4. Arccsc 26

32

Challenge!

Evaluate the following.

1 31. Arccos Arcsin

2 2

2. Arccos cos3

33. Arcsec sin

2

33

SUMMARY Function Domain Range

Arcsinx

Arccos x

Arctan x

Arccot x

Arccsc x

Arcsec x

,2 2

,2 2

0,

0,

, 02 2

0,2

1,1

1,1

, 1 1,

, 1 1,

34

End of Chapter 4.4

35