Copyright © 2009 Pearson Education, Inc.

CHAPTER 7: Trigonometric Identities, Inverse

Functions, and Equations7.1 Identities: Pythagorean and Sum and Difference

7.2 Identities: Cofunction, Double-Angle, and Half-Angle

7.3 Proving Trigonometric Identities

7.4 Inverses of the Trigonometric Functions

7.5 Solving Trigonometric Equations

Copyright © 2009 Pearson Education, Inc.

7.4Inverses of the Trigonometric Functions

Find values of the inverse trigonometric functions. Simplify expressions such as sin (sin -1 x) and

sin -1 (sin x). Simplify expressions involving composition such as

sin (cos –1 1/2) without using a calculator. Simplify expressions such as sin arctan (a/b) by

making a drawing and reading off appropriate ratios.

Slide 7.4 - 4Copyright © 2009 Pearson Education, Inc.

Inverse Sine Function

The graphs of an equation and its inverse are reflections of each other across the line y = x.

However, the inverse is not a function as it is drawn.

Slide 7.4 - 5Copyright © 2009 Pearson Education, Inc.

Inverse Sine Function

We must restrict the domain of the inverse sine function.It is fairly standard to restrict it as shown here.

The domain is [–1, 1].

The range is [–π/2, π/2].

Slide 7.4 - 6Copyright © 2009 Pearson Education, Inc.

Inverse Cosine Function

The graphs of an equation and its inverse are reflections of each other across the line y = x.

However, the inverse is not a function as it is drawn.

Slide 7.4 - 7Copyright © 2009 Pearson Education, Inc.

Inverse Cosine Function

We must restrict the domain of the inverse cosine function.It is fairly standard to restrict it as shown here.

The domain is [–1, 1].

The range is [0, π].

Slide 7.4 - 8Copyright © 2009 Pearson Education, Inc.

Inverse Tangent FunctionThe graphs of an equation and its inverse are reflections of each other across the line y = x.

However, the inverse is not a function as it is drawn.

Slide 7.4 - 9Copyright © 2009 Pearson Education, Inc.

Inverse Sine Function

We must restrict the domain of the inverse tangent function.It is fairly standard to restrict it as shown here.

The domain is (–∞, ∞).

The range is (–π/2, π/2).

Slide 7.4 - 10Copyright © 2009 Pearson Education, Inc.

Inverse Trigonometric Functions

y sin 1 x

arcsin x, where x sin y

Function Domain Range

1, 1 2, 2

y cos 1 x

arccos x, where x cos y

1, 1 0,

y tan 1 x

arctan x, where x tan y

( , ) ( 2, 2)

Slide 7.4 - 11Copyright © 2009 Pearson Education, Inc.

Graphs of the Inverse Trigonometric Functions

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Graphs of the Inverse Trigonometric Functions

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Example

Find each of the following function values.

a) sin 1 2

2b) cos 1

1

2

c) tan 1 3

2

Find such that sin = .2 2

In the restricted range [–π/2, π/2], the only number with sine of is π/4.2 2

Solution:

sin 1 2

2

4

, or 45º .

Slide 7.4 - 14Copyright © 2009 Pearson Education, Inc.

Example

Solution continued:

Find such that cos = –1/2.

In the restricted range [0, π], the only number with cosine of –1/2 is 2π/3.

cos 1 1

2

23

, or 120º .

Slide 7.4 - 15Copyright © 2009 Pearson Education, Inc.

Example

Solution continued:

3 2.Find such that tan =

tan 1 3

3

6

, or 30º .

In the restricted range (–π/2, π/2), the only number with tangent of is –π/6. 3 2

Slide 7.4 - 16Copyright © 2009 Pearson Education, Inc.

Example

Approximate the following function value in both radians and degrees. Round radian measure to four decimal places and degree measure to the nearest tenth of a degree.

Solution:Press the following keys (radian mode):

cos 1 0.2689

Readout: Rounded: 1.8430

Rounded: 105.5º

Change to degree mode and press the same keys:

Readout:

Slide 7.4 - 17Copyright © 2009 Pearson Education, Inc.

Composition of Trigonometric Functions

sin sin 1 x x, for all x in the domain of sin–1

cos cos 1 x x, for all x in the domain of cos–1

tan tan 1 x x, for all x in the domain of tan–1

Slide 7.4 - 18Copyright © 2009 Pearson Education, Inc.

Example

Simplify each of the following.

Solution:

a) cos cos 1 3

2

b) sin sin 11.8

cos cos 1 3

2

3

2

a) Since is in the domain, [–1, 1], it follows that3 2

b) Since 1.8 is not in the domain, [–1, 1], we cannot evaluate the expression. There is no number with sine of 1.8. So, sin (sin–1 1.8) does not exist.

Slide 7.4 - 19Copyright © 2009 Pearson Education, Inc.

Special Cases

sin 1 sin x x, for all x in the range of sin–1

cos 1 cos x x, for all x in the range of cos–1

tan 1 tan x x, for all x in the range of tan–1

Slide 7.4 - 20Copyright © 2009 Pearson Education, Inc.

ExampleSimplify each of the following.

Solution:

a) tan 1 tan6

b) sin 1 sin34

tan 1 tan6

6

a) Since π/6 is in the range, (–π/2, π/2), it follows that

b) Since 3π/4 is not in the range, [–π/2, π/2], we cannot apply sin–1(sin x) = x.

sin 1 sin34

sin 1 2

2

4

Slide 7.4 - 21Copyright © 2009 Pearson Education, Inc.

Example

Find

Solution:

sin cot 1 x

2

.

cot–1 is defined in (0, π), so consider quadrants I and II. Draw right triangles with legs x and 2, so cot = x/2.

sin cot 1 x

2

2

x2 4