Section 5.5 Inverse Trigonometric Functions and Their Graphs

Section 5.5 Inverse Trigonometric Functions and Their Graphs

DEFINITION: The inverse sine function, denoted by sin-1 x (or arcsin x), is defined to be

the inverse of the restricted sine function

sin x, - x

2

2

DEFINITION: The inverse cosine function, denoted by cos-1 x (or arccos x), is defined to be the inverse of the restricted cosine function

cos x, 0 x

DEFINITION: The inverse tangent function, denoted by tan-1 x (or arctan x), is defined

to be the inverse of the restricted tangent function

tan x, - < x <

2

2

DEFINITION: The inverse cotangent function, denoted by cot-1 x (or arccot x), is defined to be the inverse of the restricted cotangent function

cot x, 0 < x <

1

DEFINITION: The inverse secant function, denoted by sec-1 x (or arcsec x), is defined to

be the inverse of the restricted secant function

[

]

sec x, x [0, /2) [, 3/2) or x [0, /2) (/2, ] in some other textbooks

DEFINITION: The inverse cosecant function, denoted by csc-1 x (or arccsc x), is defined

to be the inverse of the restricted cosecant function

[

]

csc x, x (0, /2] (, 3/2] or x [-/2, 0) (0, /2] in some other textbooks

IMPORTANT: Do not confuse sin-1 x, cos-1 x, tan-1 x, cot-1 x, sec-1 x, csc-1 x

with

1

1

1

1

1

1

,

,

,

,

,

sin x cos x tan x cot x sec x csc x

FUNCTION sin-1 x cos-1 x tan-1 x cot-1 x sec-1 x csc-1 x

DOMAIN [-1, 1] [-1, 1]

(-, +) (-, +) (-, -1] [1, +) (-, -1] [1, +)

RANGE [-/2, /2]

[0, ] (-/2, /2)

(0, ) [0, /2) [, 3/2) (0, /2] (, 3/2]

2

FUNCTION sin-1 x cos-1 x tan-1 x cot-1 x sec-1 x csc-1 x

DOMAIN [-1, 1] [-1, 1]

(-, +) (-, +) (-, -1] [1, +) (-, -1] [1, +)

RANGE [-/2, /2]

[0, ] (-/2, /2)

(0, ) [0, /2) [, 3/2) (0, /2] (, 3/2]

EXAMPLES:

(a)

sin-1 1

=

,

since

sin

=

1

and

[ -,

] .

2

2

2

22

(b)

sin-1(-1)

=

-,

since

( ) sin -

=

-1

and

-

[ -,

] .

2

2

2

22

(c)

sin-1

0

=

0,

since

sin

0

=

0

and

0

[ -,

] .

22

(d)

sin-1

1

=

,

since

sin

=

1

and

[ -,

] .

26

62 6

22

(e)

sin-1

3

=

,

since

sin

=

3

and

[ -,

] .

23

32

3

22

(f)

sin-1

2

=

,

since

sin

=

2

and

[ -,

] .

24

42

4

22

EXAMPLES:

cos-1

0

=

,

2

tan-1

1

=

,

4

cos-1 1 = 0,

cos-1(-1) = ,

cos-1

1

=

,

cos-1

3 =,

cos-1

2 =

23

26

24

tan-1(-1)

=

-

,

tan-1 3

=

,

tan-1 1

=,

()

tan-1 - 1

=-

4

3

36

3

6

EXAMPLES: Find sec-1 1, sec-1(-1), and sec-1(-2).

3

FUNCTION sin-1 x cos-1 x tan-1 x cot-1 x sec-1 x csc-1 x

DOMAIN [-1, 1] [-1, 1]

(-, +) (-, +) (-, -1] [1, +) (-, -1] [1, +)

RANGE [-/2, /2]

[0, ] (-/2, /2)

(0, ) [0, /2) [, 3/2) (0, /2] (, 3/2]

EXAMPLES: Find sec-1 1, sec-1(-1), and sec-1(-2).

Solution: We have sec-1 1 = 0,

sec-1(-1) = ,

sec-1(-2) = 4 3

since

sec 0 = 1,

sec = -1,

4 sec = -2

3

and

4 [ ) [ 3 )

0, , 0, ,

3

2

2

2 Note that sec 3 is also -2, but

sec-1(-2) = 2 3

since

2 [ ) [ 3 )

0, ,

3

2

2

EXAMPLES: Find

tan-1 0

cot-1 0

cot-1 1

sec-1 2

csc-1 2

csc-1 2 3

4

FUNCTION sin-1 x cos-1 x tan-1 x cot-1 x sec-1 x csc-1 x

DOMAIN [-1, 1] [-1, 1]

(-, +) (-, +) (-, -1] [1, +) (-, -1] [1, +)

RANGE [-/2, /2]

[0, ] (-/2, /2)

(0, ) [0, /2) [, 3/2) (0, /2] (, 3/2]

EXAMPLES: We have

tan-1 0 = 0,

cot-1

0

=

,

2

cot-1

1

=

,

4

sec-1

2

=

,

4

csc-1

2

=

,

6

csc-1 2

=

33

EXAMP(LES: Eva)luate ( )

(

)

7

(a) sin arcsin , arcsin sin , and arcsin sin .

6

6

6

(

)

( )

(

)

8

(b) sin arcsin , arcsin sin , and arcsin sin .

7

7

7

(

( ))

(

)

(

)

2

2

9

(c) cos arccos - , arccos cos , and arccos cos .

5

5

5

Solution: Since arcsin x is the inverse of the restricted sine function, we have

sin(arcsin x) = x if x [-1, 1] and arcsin(sin x) = x if x [-/2, /2]

Therefo(re

)

( )

(a) sin arcsin = arcsin sin = , but

6

66

(

)

()

7

1

arcsin sin = arcsin - = -

6

2

6

or

(

)

7

( ( ))

(

)

( )

arcsin sin = arcsin sin + = arcsin - sin = - arcsin sin = -

6

6

6

6

6

(

)

( )

(b) sin arcsin = arcsin sin = , but

7

77

(

)

8

( ( ))

(

)

( )

arcsin sin 7

= arcsin sin + 7

= arcsin

- sin 7

= - arcsin

sin 7

= -7

(c) Similarly, since arccos x is the inverse of the restricted cosine function, we have

cos(arccos x) = x if x [-1, 1] and arccos(cos x) = x if x [0, ]

(

( ))

(

)

2

2

2 2

Therefore cos arccos - 5

= - 5 and arccos

cos 5

= , but 5

(

)

9

((

))

( )

arccos cos = arccos cos 2 - = arccos cos =

5

5

55

5

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