Slide 1-1

6Inverse Trigonometric Functions

Y. Ath

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6.1 Inverse Circular Functions

6.2 Trigonometric Equations I

6.3 Trigonometric Equations II

6.4 Equations Involving Inverse Trigonometric Functions

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Vertical Line Test

Horizontal Line Test

If a function f is one-to-one on its domain, then f has an inverse function

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Inverse Function

The inverse function of the one-to-one function f is defined as follows.

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Caution

The –1 in f –1 is not an exponent.

1 1( )

( )f x

f x

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xxyy

xxyy

xxyy

tantan ,tan tan(3)

coscos ,coscos (2)

sinsin ,sinsin (1)

:Example

11-

11-

11-

yyff

xxff

)(

)(1

1

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Inverse Sine Function

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.

11 :Range22

:Domain

y

x

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2,

2Range ,11Domain ,sin

1,1Range ,2

,2

Domain ,sin

1

,-xy

yx

12

71.02

2

4

00

71.02

2

4

12

RangeDomain

6.12

1

8.04

71.0

00

8.04

71.0

6.12

1

RangeDomain

-1.5 -1 -0.5 0 0.5 1 1.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

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Find y in each equation.

Example FINDING INVERSE SINE VALUES

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Example FINDING INVERSE SINE VALUES (cont.)

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Example FINDING INVERSE SINE VALUES (cont.)

–2 is not in the domain of the inverse sine function, [–1, 1], so does not exist.

,2sin y 11 y

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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

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,0Range ,11Domain ,cos

1,1Range ,,0Domain ,cos1

,-xy

yx

1

71.02

2

4

3

02

71.02

2

4

10

RangeDomain

14.31

4.24

371.0

6.12

0

8.04

71.0

01

RangeDomain

-1.5 -1 -0.5 0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

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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

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The inverse tangent function is defined by

y = arctan x if and only if tan y = x. The domain of y = arctan x is .( , )

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

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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.

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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.

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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

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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

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Inverse Function Values

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Trigonometric Equations I6.2Solving by Linear Methods ▪ Solving by Factoring ▪ Solving by Quadratic Methods ▪ Solving by Using Trigonometric Identities

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Example

12sin 1, sin

2

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Example 1(b)

SOLVING A TRIGONOMETRIC EQUATION BY LINEAR METHODS

Solve the equation 2 sinθ + 1 = 0 for all solutions.

210 360 , 330 360 ,

where is any integer

n n

n

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Example SOLVING A TRIGONOMETRIC EQUATION BY FACTORING

Subtract sin θ.

Factor out sin θ.

Zero-factor property

Solution set: {0°, 45°, 180°, 225°}

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Trigonometric Equations II6.3Equations with Half-Angles ▪ Equations with Multiple Angles

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Example (a) over the interval

and

(b) for all solutions.

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In-class exercises (pp 270-271)

Solution set: {30°, 60°, 210°, 240°}

Solution set, where 180º represents the period of sin2θ:

{30° + 180°n, 60° + 180°n, where n is any integer}

(1)

(2)

(3)

Solve tan 3x + sec 3x = 2 over the interval (4)Solution set: {0.2145, 2.3089, 4.4033}

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Equations Involving Inverse Trigonometric Functions 6.4Solving for x in Terms of y Using Inverse Functions ▪ Solving Inverse Trigonometric Equations

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Example

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Example:

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Example

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In class exercise

2

2 :Ans arcsinarccos Solve )3

3

32 :Ans 2cscsec Solve )2

3 :Ans arctan2 Solve 1)

11

xx

x

x