P-BLTZMC04 459-584-hr 21-11-2008 13:06 Page 550 Section ...

550 Chapter 4 Trigonometric Functions

S e c t i o n 4.7

Objectives

Understand and use the

inverse sine function.

Understand and use the

inverse cosine function.

Understand and use the

inverse tangent function.

Use a calculator to evaluate

inverse trigonometric functions.

Find exact values of

composite functions with inverse trigonometric functions.

Inverse Trigonometric Functions

In 2008, director Christopher Nolan pulled out all the stops with The Dark Knight, the sequel to Batman Begins, that made comic-book films before it look like kid stuff. The movie is being shown at a local theater, where you can experience the stunning force of its action scenes that teeter madly out of control. Where in the theater should you sit to maximize the visual impact of the director's vision of good and evil? In this section's exercise set, you will see how an inverse trigonometric function can enhance your moviegoing experiences.

Study Tip

Here are some helpful things to remember from our earlier discussion of inverse functions.

? If no horizontal line intersects the graph of a function more than once, the function is one-to-one and has an inverse function.

? If the point 1a, b2 is on the graph of f, then the point 1b, a2 is on the graph of the inverse function, denoted f-1. The graph of f-1 is a reflection of the graph of f about the line y = x.

Understand and use the

inverse sine function.

The Inverse Sine Function

Figure 4.86 shows the graph of y = sin x. Can you see that every horizontal line that can be drawn between -1 and 1 intersects the graph infinitely many times? Thus, the sine function is not one-to-one and has no inverse function.

y

1

y = sin x

y

y

=

sin

x,

-

p 2

x

p 2

1

x

-q

q

-1

Figure 4.87 The restricted sine function passes the horizontal line test. It is one-to-one and has an inverse function.

-w -p

-q

x

q

p

w

2p r

-1

Figure 4.86 The horizontal line test shows that the sine function is not one-to-one and has no inverse function.

In Figure 4.87, we have taken a portion of the sine curve, restricting the

p

p

domain of the sine function to - ... x ... . With this restricted domain, every

2

2

horizontal line that can be drawn between -1 and 1 intersects the graph

exactly once. Thus, the restricted function passes the horizontal line test and is

one-to-one.

y = sin x

-

p 2

x

p 2

y

(q, 1)

1

-q

(0, 0)

x q

-1

(-q, -1)

Domain: [-q, q]

Range: [-1, 1]

Figure 4.88 The restricted sine function

Section 4.7 Inverse Trigonometric Functions 551

On the restricted domain - p ... x ... p , y = sin x has an inverse function.

2

2

The inverse of the restricted sine function is called the inverse sine function. Two

notations are commonly used to denote the inverse sine function:

y = sin-1 x or y = arcsin x.

In this book, we will use y = sin-1 x. This notation has the same symbol as the inverse function notation f-11x2.

The Inverse Sine Function

The inverse sine function, denoted by sin-1, is the inverse of the restricted sine

p

p

function y = sin x, - ... x ... . Thus,

2

2

y = sin-1 x means sin y = x,

where

p -

...

y

...

p

and

-1

...

x

...

1.

We

read

y

=

sin-1 x

as

"y

equals

the

2

2

inverse sine at x."

Study Tip

The notation y = sin-1 x does not mean y =

1 . The notation y =

1 , or the reciprocal

sin x

sin x

of the sine function, is written y = 1sin x2-1 and means y = csc x.

Inverse sine function

y=sin?1 x

Reciprocal of sine function

y=(sin

x)?1=

1 sin

x=csc

x

One way to graph y = sin-1 x is to take points on the graph of the restricted sine function and reverse the order of the coordinates. For example, Figure 4.88

shows

that

a-

p ,

-1b, 10, 02,

and

ap, 1b

are

on

the

graph

of

the

restricted

sine

2

2

function. Reversing the order of the coordinates gives a -1, - p b, 10, 02, and 2

a 1, p b . We now use these three points to sketch the inverse sine function. The graph 2

of y = sin-1 x is shown in Figure 4.89. Another way to obtain the graph of y = sin-1 x is to reflect the graph of the

restricted sine function about the line y = x, shown in Figure 4.90. The red graph is the restricted sine function and the blue graph is the graph of y = sin-1 x.

y

q

(1, q)

(0, 0) -1

y = sin-1 x

x 1

(-1, -q)

Domain: [-1, 1]

-q Range: [-q, q]

Figure 4.89 The graph of the inverse sine function

y

q 1 y = sin-1 x

-q -1

(-q, -1)

-1

-q

(-1, -q)

y = x

(1, q)

(q, 1)

y = sin x x

1q

Figure 4.90 Using a reflection to obtain the graph of the inverse sine function

552 Chapter 4 Trigonometric Functions

y

q 1 y = sin-1 x

-q -1

(-q, -1)

-1

-q

(-1, -q)

y = x

(1, q)

(q, 1)

y = sin x x

1q

Figure 4.90 (repeated)

Exact values of sin-1 x can be found by thinking of sin1 x as the angle in the

interval c P , P d whose sine is x. For example, we can use the two points on the 22

blue graph of the inverse sine function in Figure 4.90 to write

sin?1(?1)=?

p 2

and

sin?1

1=

p 2

.

The angle whose sine is -1 is - p2 .

The angle whose sine is 1 is p2 .

Because we are thinking of sin-1 x in terms of an angle, we will represent such an angle by u.

Finding Exact Values of sin1 x

1. Let u = sin-1 x.

2.

Rewrite u

=

sin-1 x as sin u

=

x, where

p -

...

u

...

p.

2

2

3.

Use

the

exact

values

in

Table

4.7

to

find

the

value

of

u

in

c

-

p 2

,

p 2

d

that

satisfies sin u = x.

Table 4.7 Exact Values for sin U, P U P

2

2

p

p

p

p

p

U

-

-

2

3

4

6

0

6

sin U

-1

23 -

2

22 -

2

1 -

2

0

1 2

p

pp

4

3

2

22 23 1

2

2

EXAMPLE 1 Finding the Exact Value of an Inverse Sine Function

Find

the

exact

value

of

sin-1

22 .

2

Solution

Step 1 Let U sin1 x. Thus,

u

=

sin-1

22 .

2

We must find the angle u,

p -

2

...

u

...

p, 2

whose sine equals

22 .

2

Step 2 Rewrite U sin1 x as sin U x, where P U P . Using the

2

2

definition of the inverse sine function, we rewrite u = sin-1 22 as 2

sin u

=

22 , where - p

2

2

...

u

...

p. 2

Step 3

Use

the

exact

values

in

Table

4.7

to

find

the

value

of

U

in

c

P, 2

P 2

d

that

satisfies sin U

x.

Table 4.7 shows that the only angle in the interval

c-

p, 2

pd 2

that

satisfies sin u

=

22 is p . Thus, u

=

p . Because u, in step 1, represents sin-1

22 ,

24

4

2

we conclude that

sin-1 22 = p . 24

The

angle

in

c

-

p, 2

p 2

d

whose

sine

is

22 2

is

p. 4

Section 4.7 Inverse Trigonometric Functions 553

Check

Point

1

Find the exact value of sin-1

23 .

2

Understand and use the inverse

cosine function.

EXAMPLE 2 Finding the Exact Value of an Inverse Sine Function

Find the exact value of sin-1 a - 1 b . 2

Solution Step 1 Let U sin1 x. Thus,

u

=

sin-1 a -

1b. 2

We must find the angle u,

p -

2

...

u

...

p, 2

whose sine equals

-

1 .

2

Step 2 Rewrite U sin1 x as sin U x, where P U P . We rewrite

u

=

sin-1 a -

1b 2

and obtain

2

2

sin u

=

1 - , where

2

p -

2

...

u

...

p. 2

Step 3

Use

the

exact

values

in

Table

4.7

to

find

the

value

of

U

in

c P, 2

Pd 2

that satisfies sin U x. Table 4.7 shows that the only angle in the interval

c- p, pd 22

that satisfies sin u

=

1 - is

2

- p . Thus, 6

sin-1 a - 1 b

=

p - .

2

6

Check Point 2 Find the exact value of sin-1a - 22 b. 2

Some inverse sine expressions cannot be evaluated. Because the domain of the inverse sine function is 3 - 1, 14, it is only possible to evaluate sin-1x for values

of x in this domain. Thus, sin-1 3 cannot be evaluated. There is no angle whose sine is 3.

The Inverse Cosine Function

Figure 4.91 shows how we restrict the domain of the cosine function so that it becomes one-to-one and has an inverse function. Restrict the domain to the interval 30, p4, shown by the dark blue graph. Over this interval, the restricted cosine function passes the horizontal line test and has an inverse function.

y

y = cos x

1

0 x p

Figure 4.91 y = cos x is one-to-one on the interval 30, p4.

-2p -w

-p

-q -1

x

q

p

w 2p

554 Chapter 4 Trigonometric Functions

The Inverse Cosine Function The inverse cosine function, denoted by cos-1, is the inverse of the restricted cosine function y = cos x, 0 ... x ... p. Thus,

y = cos-1 x means cos y = x,

where 0 ... y ... p and -1 ... x ... 1.

One way to graph y = cos-1 x is to take points on the graph of the restricted

cosine function and reverse the order of the coordinates. For example, Figure 4.92

shows

that

(0,

1),

ap, 2

0b,

and

1p,

- 12

are

on

the

graph

of

the

restricted

cosine

function. Reversing the order of the coordinates gives (1, 0),

a

0,

p 2

b

,

and

1

-

1,

p2.

We now use these three points to sketch the inverse cosine function. The graph

of y = cos-1 x is shown in Figure 4.93. You can also obtain this graph by reflecting

the graph of the restricted cosine function about the line y = x.

y

y (0, 1)

1

y = cos x 0 x p

(q, 0)

q

p x

-1 (p, -1)

Domain: [0, p] Range: [-1, 1]

Figure 4.92 The restricted cosine function

(-1, p)

p

( ) q 0, q y = cos-1 x

(1, 0)

x

-1

1

Domain: [-1, 1] Range: [0, p]

Figure 4.93 The graph of the inverse cosine function

Exact values of cos-1 x can be found by thinking of cos1 x as the angle in the interval [0, P] whose cosine is x.

Finding Exact Values of cos1 x

1. Let u = cos-1 x. 2. Rewrite u = cos-1 x as cos u = x, where 0 ... u ... p. 3. Use the exact values in Table 4.8 to find the value of u in 30, p4 that satisfies

cos u = x.

Table 4.8 Exact Values for cos U, 0 U P

U

0

p

6

p

p p 2p

3p

4

32

3

4

5p p

6

cos U 1

23 2

22 2

1 2

0

1 -

2

22 -

2

23 -

2

-1

EXAMPLE 3 Finding the Exact Value of an Inverse Cosine Function

Find the exact value of cos-1 a - 23 b . 2

 Understand and use the

inverse tangent function.

Section 4.7 Inverse Trigonometric Functions 555

Solution Step 1 Let U cos1 x. Thus,

u = cos-1 a - 23 b . 2 23

We must find the angle u, 0 ... u ... p, whose cosine equals - 2 . Step 2 Rewrite U cos1 x as cos U x, where 0 U P. We obtain

23 cos u = - 2 , where 0 ... u ... p. Step 3 Use the exact values in Table 4.8 to find the value of U in 30, P4 that

satisfies cos U x. The table on the previous page shows that the only angle in the

interval 30, p4 that satisfies cos u

=

-

23 2

is

5p 6

.

Thus,

u

=

5p and 6

cos-1 a - 23 b = 5p .

2

6

The angle in [0, p] whose cosine is - 23 is 5p .

2 6

Check Point 3 Find the exact value of cos-1a - 1 b. 2

The Inverse Tangent Function

Figure 4.94 shows how we restrict the domain of the tangent function so that it becomes one-to-one and has an inverse function. Restrict the domain to the interval a - p , p b, shown by the solid blue graph. Over this interval, the restricted tangent

22 function passes the horizontal line test and has an inverse function.

y

y

=

tan

x,

-

p 2

<

x

<

p 2

3

2p

-p

-w

-q

-3

p

2p x

q

w

Figure 4.94 y = tan x is

one-to-one on the interval a - p, pb.

22

The Inverse Tangent Function

The inverse tangent function, denoted by tan-1, is the inverse of the restricted

tangent function y

=

tan x,

p -

6

x

6

p . Thus,

2

2

y = tan-1 x means tan y = x,

where

p -

6

y

6

p and

-q

6

x

6

q.

2

2

556 Chapter 4 Trigonometric Functions

We graph y = tan-1 x by taking points on the graph of the restricted function

and

reversing

the

order

of

the

coordinates.

Figure

4.95

shows

that

a

-

p 4

,

-1b,

(0,

0),

and a p , 1b are on the graph of the restricted tangent function. Reversing the 4

order gives a - 1, - p b, (0, 0), and a1, p b. We now use these three points to graph

4

4

the inverse tangent function.The graph of y = tan-1 x is shown in Figure 4.96. Notice

that the vertical asymptotes become horizontal asymptotes for the graph of the

inverse function.

y

y

=

tan

x,

-

p 2

<

x

<

p 2

1

-q

(-p4 , -1)

(

p 4

,

1)

x q

y y = tan-1 x

q

(-1, )-p4

x

-1

1

(1,

)p

4

-q

( ) Domain: -q, q

Range: (-, )

Domain: (-, )

( ) Range: -q, q

Figure 4.95 The restricted tangent function

Figure 4.96 The graph of the inverse tangent function

Exact values of tan-1 x can be found by thinking of tan1 x as the angle in the interval a P , P b whose tangent is x.

22

Finding Exact Values of tan1 x

1. Let u = tan-1 x.

2.

Rewrite u

=

tan-1 x as tan u

=

x, where

p -

6

u

6

p.

2

2

3.

Use the exact values satisfies tan u = x.

in

Table

4.9

to

find

the

value

of

u

in

a-

p, 2

pb 2

that

Table 4.9

Exact

Values

for

tan

U,

P 2

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