4.7 Inverse Trig Functions Lesson

1

Precalculus
Chapter 4

Trigonometric Functions

Section 4.7

Inverse Trigonometric

Functions

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2

Multiple Choice

sec 3π/2

1

0

2

undefined

3

1

4

-1

3

Multiple Choice

tan 7π/4

1

-1

2

1

3

√2/2

4

undefined

4

Multiple Choice

tanθ is positive in

1

the 1st and 2nd quadrants

2

the 1st and 3rd quadrants

3

the 1st and 4th quadrants

4

the 2nd and 3rd quadrants

5

Multiple Choice

sin (π/2)

1

√3/2

2

√2/2

3

1/2

4

1

6

Multiple Choice

cos (π/4)

1

√3/2

2

√2/2

3

1/2

4

1

7

Multiple Choice

cos(30^o)

1

- ½

2

√3/2

3

2√3 /3

4

½

8

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4.7 A-2

Objectives

Evaluate the Inverse Sine Function

Evaluate the Inverse Cosine and Tangent
Functions

Approximate Inverse Trigonometric
Functions on a Calculator

Compose Trigonometric Functions and
Inverse Trigonometric Functions

Apply Inverse Trigonometric Functions

Evaluate the Inverse Secant, Cosecant, and
Cotangent Functions

9

Recall:

Previously this year, we discussed the idea that some functions have inverse. That is, some functions have another function that will "undo" it. We also learned a couple of ways to determine if a function is invertible. They are:

The function is _____________________.
The graph of the function __________________________.

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A function maps inputs (x-values) to outputs (y-values), but an inverse function maps the outputs back to the inputs.

The domain of f is the range of f^-1 and vice versa.

Also recall:

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4.7 A-3

Objective 1

Evaluate the Inverse Sine

Function

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4.7 A-4

The Inverse Sine Function

For y=sin x, any horizontal line taken between -1
≤y≤1 intersects the graph infinitely many times.
Therefore, y=sin x is not a one-to-one function.

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4.7 A-5

The Inverse Sine Function

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4.7 A-6

The Inverse Sine Function

Inverse sine function

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4.7 A-7

The Inverse Sine Function

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4.7 A-8

Example 1

Evaluate.

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4.7 A-9

Example 2

Evaluate.

18

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4.7 A-10

Example 3

Evaluate.

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4.7 A-12

Objective 2

Evaluate the Inverse Cosine

and Tangent Functions

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4.7 A-13

The Inverse Cosine Function

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4.7 A-14

The Inverse Cosine Function

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4.7 A-15

The Inverse Tangent Function

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4.7 A-16

The Inverse Tangent Function

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4.7 A-17

Examples 4 and 5

Evaluate.

26

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4.7 A-18

Examples 6 and 7

Evaluate.

27

Multiple Choice

1

0

2

π/2

3

2π

4

3π

28

Multiple Choice

1

-π/3

2

4π/3

3

-2π/3

4

2π/3

29

Multiple Choice

1

3π/4

2

π/4

3

5π/4

4

-3π/4

30

Multiple Choice

1

A

2

B

3

C

4

D

31

Fill in the Blank

Evaluate the given expression using only your unit circle. Give answers in degrees!

$\tan^{-1}\left(\frac{\sqrt{3}}{3}\right)$ =___ $\degree$

Type your answer in the boxes

32

Fill in the Blank

Evaluate the given expression using only your unit circle. Give answers in degrees!

$\cos^{-1}\left(\frac{1}{2}\right)$ =___ $\degree$

Type your answer in the boxes

33

Fill in the Blank

Evaluate the given expression using only your unit circle. Give answers in degrees!

$\sin^{-1}\left(\frac{1}{2}\right)$ =___ $\degree$

Type your answer in the boxes

34

Fill in the Blank

Evaluate the given expression using only your unit circle. Give answers in degrees!

$\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)$ =___ $\degree$

Type your answer in the boxes

35

Multiple Choice

$\cos^{-1}$ (-√3/2)

1

5π/6

2

2π/3

3

-π/6

4

7π/6

36

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4.7 A-20

Objective 3

Approximate Inverse

Trigonometric Functions on

a Calculator

37

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4.7 A-21

Examples 8-10

Evaluate.

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4.7 A-22

Skill Practice 3

Use a calculator to approximate the function
values in both radians and degrees.

39

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4.7 A-23

Example 11

Use a calculator to approximate the degree
measure (to 1 decimal place) of the angle θ
where

40

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4.7 A-24

Skill Practice 4

Use a calculator to approximate the degree
measure (to 1 decimal place) or radian measure
(to 4 decimal places) of the angle θ subject to
given conditions.

41

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4.7 A-25

Objective 4

Compose Trigonometric
Functions and Inverse
Trigonometric Functions

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4.7 A-26

Compose Trigonometric Functions and

Inverse Trigonometric Functions

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4.7 A-27

Example 12

Evaluate.

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4.7 A-28

Example 13

Evaluate.

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4.7 A-29

Skill Practice 5

Find the exact values.

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4.7 A-30

Example 14

Find the exact value.

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4.7 A-31

Skill Practice 6

Find the exact value of

48

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4.7 A-32

Skill Practice 7

Find the exact value of

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4.7 A-33

Example 15

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4.7 A-34

Skill Practice 8

51

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4.7 A-35

Objective 5

Apply Inverse Trigonometric

Functions

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4.7 A-36

Example 16

The Eiffel Tower is 1063 feet
tall. A measurement is
taken of the length of the
shadow of the tower and is
found to be 310 feet.
Approximate the angle of
elevation of the Sun to the
nearest tenth of a degree.

53

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4.7 A-37

Skill Practice 9

For the construction of a house, a 16-ft by
6-ft wooden frame is made. Find the angle
that the diagonal beam makes with the
base of the frame. Round to the nearest
tenth of a degree.