Inverse Trigonometric Functions

Digital Lesson

2

Inverse Sine Function

y

2

1

1

x

y = sin x

Sin x has an inverse function on this interval.

Recall that for a function to have an inverse, it must be a one-to-one function and pass the Horizontal Line Test.

f(x) = sin x does not pass the Horizontal Line Test

and must be restricted to find its inverse.

3

The inverse sine function is defined byy = arcsin x if and only if sin y = x.

Angle whose sine is x

The domain of y = arcsin x is [–1, 1].

Example:

1a. arcsin2 6

1 is the angle whose sine is .6 2

This is another way to write arcsin x.

The range of y = arcsin x is [–/2 , /2].

1 3b. sin2 3

The sine of what angle is = to √3/2?

More examples

• Arcsin 0

4

• Sin-1 (-√2/2)

• Arcsin 2

Graph the arcsin x

• First, rewrite as x = sin y

5

Tablex y

-/2

-/4

0/4

/2

/4

/2

/4

6

Inverse Cosine Function

Cos x has an inverse function on this interval.

f(x) = cos x must be restricted to find its inverse.

y

2

1

1

x

y = cos x

7

The inverse cosine function is defined byy = arccos x if and only if cos y = x.

Angle whose cosine is x

The domain of y = arccos x is [–1, 1].

Example: 1a.) arccos2 3

1 is the angle whose cosine is .3 2

1 3 5b.) cos2 6

This is another way to write arccos x.

The range of y = arccos x is [0 , ].

More examples

• Arccos 0

8

• cos-1 (-√2/2)

• Arccos 2

Graph the arccos x

• First, rewrite as x = cos y

9

Tablex y

0

/4

/23/4

3/4

/2

/4

10

Inverse Tangent Functionf(x) = tan x must be restricted to find its inverse.

Tan x has an inverse function on this interval.

y

x

2

3

2

32

2

y = tan x

11

The inverse tangent function is defined byy = arctan x if and only if tan y = x.

Angle whose tangent is x

The domain of y = arctan x is .( , ) The range of y = arctan x is [–/2 , /2].

More examples

• Arctan 0

12

• tan-1 (-√3/3)

• Arctan 1

13

Graphing Utility: Graph the following inverse functions.

a. y = arcsin x

b. y = arccos x

c. y = arctan x

–1.5 1.5

–

–1.5 1.5

2

–

–3 3

–

Set calculator to radian mode.

14

Graphing Utility: Approximate the value of each expression.

a. cos–1 0.75 b. arcsin 0.19

c. arctan 1.32 d. arcsin 2.5

Set calculator to radian mode.

15

Composition of Functions:f(f –1(x)) = x and (f –1(f(x)) = x.

If –1 x 1 and – /2 y /2, thensin(arcsin x) = x and arcsin(sin y) = y.

If –1 x 1 and 0 y , thencos(arccos x) = x and arccos(cos y) = y.

If x is a real number and –/2 < y < /2, thentan(arctan x) = x and arctan(tan y) = y.

Example: tan(arctan 4) = 4

Inverse Properties:

16

Example:

a. sin–1(sin (–/2)) = –/2

1 5b. sin sin3

53 does not lie in the range of the arcsine function, –/2 y /2.

y

x

53

3

5 23 3 However, it is coterminal with

which does lie in the range of the

arcsine function.

1 15sin sin sin sin3 3 3

17

Example:

2Find the exact value of tan arccos .3

x

y

3

2

adj2 2Let = arccos , then cos .3 hyp 3

u u

2 23 2 5

opp 52tan arccos tan3 adj 2

u

u

18

Example:

2Find the exact value of tan arccos .3

x

y

3

2

adj2 2Let = arccos , then cos .3 hyp 3

u u

2 23 2 5

opp 52tan arccos tan3 adj 2

u

u

19

Example:

xy

-5

12u

Find the exact value of csc[arc tan(-5/12)]Let u = arc tan (-5/12), then tan u = opp = -5

adj 12

122 + (-5)2= c2

c = 13

Csc u = hyp = 13 opp -5

20

Example:

x

y

x

1

u

Find the exact value of sin[arc tan(x)]

Let u = arc tan (x), then tan u = opp = x adj 1

x2 + (1)2= c2

c = x2 + 1

sin u = opp = x hyp x2 + 1