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Pre-CalculusChapter 4

Trigonometric Functions

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4.7 Inverse Trigonometric Functions

Objectives: Evaluate inverse sine functions. Evaluate other inverse trigonometric functions.

Evaluate compositions of trigonometric functions.

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Inverse Functions Recall that a function and its

inverse reflect over the line y = x. What must be true for a function

to have an inverse? It must be one-to-one, that is, it

must pass the horizontal line test.

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More Inverse Functions Are sine, cosine, and tangent one-

to-one? If not, what must we do so that

these functions will have inverse functions?

Hint: Consider y = x2. We must restrict the domain of the original function.

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Sine and Its Inverse f(x) = sin x does not pass the

Horizontal Line Test It must be restricted to find its

inverse. y

2

1

1

x

y = sin x

Sin x has an inverse function on this interval. 22

x

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Inverse Sine Function The inverse sine function is defined by

y = arcsin x if and only if sin y = x.

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

_____________. Why are the domain and range

defined this way?

Angle whose sine is x

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What Does “arcsin” Mean? In an inverse function, the x-values

and the y-values are switched. So, arcsin x means the angle (or arc)

whose sin is x. Notation for inverse sine

arcsin x sin -1 x

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Examples If possible, find the exact value.

2sin.3

2

3sin.2

2

1arcsin.1

1

1

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Graphing Arcsine Create a table for sin y = x for –π/2 ≤ y ≤

π/2.

Graph x on horizontal axis and y on vertical axis.

y –π/2 –π/4 –π/6 0 π/6 π/4 π/2x

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Graph of Arcsine

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Inverse Cosine Function f(x) = cos x must be restricted to

find its inverse.

Cos x has an inverse function on this interval.

y

2

1

1

x

y = cos x

x0

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Inverse Cosine Function The inverse cosine function is

defined byy = arccos x if and only if cos y

= x.

The domain of y = arccos x is [–1, 1]. The range of y = arccos x is [0, π]. Notation for inverse cosine:

arccos x or cos -1 x

Angle whose cosine is x

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Examples If possible, find the exact value

2

3cos.2

2

1arccos.1

1

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Graphing Arccos Create a table for cos y = x for 0 ≤ y ≤ π.

Graph x on horizontal axis and y on vertical axis.

y 0 π/6 π/3 π/2 2π/3 5π/6 πx

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Graph of Arccos

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Inverse Tangent Function f(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

22

x

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Inverse Tangent Function The inverse tangent function is

defined byy = arctan x if and only if tan y

= x.

The domain of y = arctan x is (–∞, ∞). The range of y = arctan x is (–π/2, π/2). Notation for inverse tangent:

arctan x or tan -1 x

Angle whose tangent is x

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Examples If possible, find the exact value

3tan.2

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3arctan.1

1

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Graph of Arctan

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Examples Evaluate using your calculator.

(What mode should the calculator be in?)

5.2arcsin.4

32.1arctan.3

19.0arcsin.2

75.0cos.1 1

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Summary

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Composition of Functions Given the restrictions specified in the

previous slide, we have the following properties of inverse trig functions.

sin(arcsin ) and arcsin(sin )

cos(arccos ) and arccos(cos )

tan(arctan ) and arctan(tan )

x x y y

x x y y

x x y y

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Examples If possible, find the exact value.

1coscos.3

3

5sinarcsin.2

5arctantan.1

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Example 2Find the exact value of tan arccos .

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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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Example

Find the exact value of .

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3arcsincos

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Homework 4.7 Worksheet 4.7