Sullivan Algebra and Trigonometry: Section 7.1 The Inverse Sine, Cosine, and Tangent Functions Objectives of this Section Find the Exact Value of the Inverse.

Sullivan Algebra and Trigonometry: Section 7.1

The Inverse Sine, Cosine, and Tangent Functions

Objectives of this Section

• Find the Exact Value of the Inverse Sine, Cosine, and Tangent Functions

• Find the Approximate Value of the Inverse Sine, Cosine, and Tangent Functions

Let f denote a one-to-one function y = f (x). The inverse of f, denoted f -1, is a function such that f -1(f (x)) = x for every x in the domain f and f (f -1(x)) = x for every x in the domain of f -1.

In other words, the function f maps each x in its domain to a unique y in its range. The inverse function f -1 maps each y in the range back to the x in the domain.

Recall the Definition of the Inverse Function

To find the inverse of the sine function, first examine the graph to see if the function is one - to - one, using the horizontal line test.

6.28 3.14 0 3.14 6.28

1.5

0.75

0.75

1.5 y = b

-1< b < 1

Since the sine function is not one - to - one for all real numbers, we must restrict the domain to an interval where the function is one - to - one.

x

y

2

2

-1

1

The inverse sine of x means

where and

y x x y

y x

sin sin1

2 21 1

sin sin 1

2u u u where

2

sin sin 1 1 1v v v where

Characteristics of y x sin 1

Domain of is the Range of

y x y x

x

sin sin :1

1 1

Range of is the Domain of

y x y x

y

sin sin :1

2 2

1 0 1

2,1

1,

2

2,1

1,2

y xsin

y x sin 1

2

3sin of eexact valu theFind 1y

22

2

3sin 1

22

2

3sin

3

y

.2

2sin of eexact valu theFind 1

22

2

2sin 1

22

2

2sin

4

y

To find the inverse of the cosine function, first examine the graph to see if the function is one - to - one, using the horizontal line test.

6.28 0 6.28

1.5

1.5

2 2

y = b -1 < y < 1

Since the cosine function is not one - to - one for all real numbers, we must restrict the domain to an interval where the function is one - to - one.

0 1.57 3.14

1.5

1.5

0 1,

, 1

The inverse cosine of x

means

where and

y x x y

y x

cos cos1

0 1 1

cos cos 1 0u u u where

cos cos 1 1 1v v v where

Domain of is the Range of

y x y x

x

cos cos :1

1 1

Characteristics of y x cos 1

Range of is the Domain of

y x y x

y

cos cos :1

0

1 0.38 1.76 3.14

1

0.38

1.76

3.14

, 1

1 0,

1,

0 1,

y xcos

y x cos 1

.2

3cos of eexact valu theFind 1

0 where

2

3cos 1

0 where2

3cos

6

.2

2cos of eexact valu theFind 1

0 where

2

2cos 1

0 where2

2cos

34

To find the inverse of the tangent function, first examine the graph to see if the function is one - to - one, using the horizontal line test.

7.85 0 7.85

5

5

Since the tangent function is not one - to - one for all real numbers, we must restrict the domain to an interval where the function is one - to - one.

1.57 0 1.57

5

5

2

2

The inverse tangent of x

means

where and

y x x y

y x

tan tan1

2 2

tan tan 1

2 2u u u where

vvv wheretantan 1

Characteristics of y x tan 1

Range of is the Domain of

2

y x y x

y

tan tan :1

2

x

xyxy

:tan of Range theis tan ofDomain 1

2 0 2

y xtan

y x tan 1

x 2

x 2

y 2

y 2

Find the exact value of tan . 1 3

tan 1 32 2

where

tan 32 2

where

3

Find the exact value of tan . 1 3

tan 1 32 2

where

tan 32 2

where

3