4.7 Inverse Trig Functions Lesson Slides: 50 Slides

1

Precalculus
Chapter 4

Trigonometric Functions

Section 4.7

Inverse Trigonometric

Functions

without the prior written consent of McGraw-Hill Education.

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

without the prior written consent of McGraw-Hill Education.

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 __________________________.

10

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:

11

without the prior written consent of McGraw-Hill Education.

4.7 A-3

Objective 1

Evaluate the Inverse Sine

Function

12

without the prior written consent of McGraw-Hill Education.

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.

13

without the prior written consent of McGraw-Hill Education.

4.7 A-5

The Inverse Sine Function

14

without the prior written consent of McGraw-Hill Education.

4.7 A-6

The Inverse Sine Function

Inverse sine function

15

without the prior written consent of McGraw-Hill Education.

4.7 A-7

The Inverse Sine Function

16

without the prior written consent of McGraw-Hill Education.

4.7 A-8

Example 1

Evaluate.

17

without the prior written consent of McGraw-Hill Education.

4.7 A-9

Example 2

Evaluate.

18

19

without the prior written consent of McGraw-Hill Education.

4.7 A-10

Example 3

Evaluate.

20

without the prior written consent of McGraw-Hill Education.

4.7 A-12

Objective 2

Evaluate the Inverse Cosine

and Tangent Functions

21

without the prior written consent of McGraw-Hill Education.

4.7 A-13

The Inverse Cosine Function

22

without the prior written consent of McGraw-Hill Education.

4.7 A-14

The Inverse Cosine Function

23

without the prior written consent of McGraw-Hill Education.

4.7 A-15

The Inverse Tangent Function

24

without the prior written consent of McGraw-Hill Education.

4.7 A-16

The Inverse Tangent Function

25

without the prior written consent of McGraw-Hill Education.

4.7 A-17

Examples 4 and 5

Evaluate.

26

without the prior written consent of McGraw-Hill Education.

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 Blanks

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

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

\degree

32

Fill in the Blanks

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

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

\degree

33

Fill in the Blanks

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

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

\degree

34

Fill in the Blanks

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

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

\degree

35

Multiple Choice

$\cos^{-1}$ (-√3/2)

1

5π/6

2

2π/3

3

-π/6

4

7π/6

36

without the prior written consent of McGraw-Hill Education.

4.7 A-20

Objective 3

Approximate Inverse

Trigonometric Functions on

a Calculator

37

without the prior written consent of McGraw-Hill Education.

4.7 A-21

Examples 8-10

Evaluate.

38

without the prior written consent of McGraw-Hill Education.

4.7 A-22

Skill Practice 3

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

39

without the prior written consent of McGraw-Hill Education.

4.7 A-23

Example 11

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

40

without the prior written consent of McGraw-Hill Education.

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

without the prior written consent of McGraw-Hill Education.

4.7 A-25

Objective 4

Compose Trigonometric
Functions and Inverse
Trigonometric Functions

42

without the prior written consent of McGraw-Hill Education.

4.7 A-26

Compose Trigonometric Functions and

Inverse Trigonometric Functions

43

without the prior written consent of McGraw-Hill Education.

4.7 A-27

Example 12

Evaluate.

44

without the prior written consent of McGraw-Hill Education.

4.7 A-28

Example 13

Evaluate.

45

without the prior written consent of McGraw-Hill Education.

4.7 A-29

Skill Practice 5

Find the exact values.

46

without the prior written consent of McGraw-Hill Education.

4.7 A-30

Example 14

Find the exact value.

47

without the prior written consent of McGraw-Hill Education.

4.7 A-31

Skill Practice 6

Find the exact value of

48

without the prior written consent of McGraw-Hill Education.

4.7 A-32

Skill Practice 7

Find the exact value of

49

without the prior written consent of McGraw-Hill Education.

4.7 A-33

Example 15

50

without the prior written consent of McGraw-Hill Education.

4.7 A-34

Skill Practice 8

51

without the prior written consent of McGraw-Hill Education.

4.7 A-35

Objective 5

Apply Inverse Trigonometric

Functions

52

without the prior written consent of McGraw-Hill Education.

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

without the prior written consent of McGraw-Hill Education.

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.