5.3 solving trigonometric equations. what are two values of x between 0 and when cos x = ½ x =...

5.3 Solving 5.3 Solving Trigonometric Trigonometric

EquationsEquations

What are two values of x between 0 and

When Cos x = ½

x = arccos ½

2

3

5

3

xandx

How do you find

What is the period of the Cosine function?

3

5x

How to you find

What is the period of the Cosine function?

minus

3

5x

2

23

Why does this work?

Look at the Cosine function, there are many times it equals ½

We limited it to 0 to

2

1,

3

2

1,

3

5

2

If we wanted all the answers to cos x = ½

The answers would be

Where n is an integer

23

5

23

n

n

Solve for x

xx tantan3 3

Solve for x

0tantan3

tantan3

3

3

xx

xx

Solve for x for

)2,0[

01tantan

0tantan3

tantan3

2

3

3

xx

xx

xx

Set each factor equal to 0

Solve ( remember the period for tan is )

nx

x

0

0tan

3

3tan

3

1tan

1tan3

01tan3

2

2

2

x

x

x

x

Set each factor equal to 0

Solve ( remember the period for tan is )

nx

x

x

0

0arctan

0tan

3

3tan

3

1tan

1tan3

01tan3

2

2

2

x

x

x

x

n

n

6

5

3

3arctan

63

3arctan

When solving a trig equation, you must have only one function

Solve for x where the domain is )2,0[

2tansincos xxx



2cos

sinsincos

2tansincos

x

xxx

xxx



2cos

sin

cos

cos

2cos

sinsincos

2tansincos

22

x

x

x

x

x

xxx

xxx



2cos

sincos

2cos

sin

cos

cos

2cos

sinsincos

2tansincos

22

22

x

xx

x

x

x

x

x

xxx

xxx



2cos

sincos

2cos

sin

cos

cos

2cos

sinsincos

2tansincos

22

22

x

xx

x

x

x

x

x

xxx

xxx

xx

cos2

1;2

cos

1



2cos

sincos

2cos

sin

cos

cos

2cos

sinsincos

2tansincos

22

22

x

xx

x

x

x

x

x

xxx

xxx

3

5,

3;

2

1arccos

cos2

1;2

cos

1

xx

xx

Solve for x

Let 2x = u

3

5,

3

2

1cos

2

12cos

u

u

x

Solve for x

Let 2x = u

3

5,

3

2

1cos

2

12cos

u

u

x

6

5

2

1

3

5;

3

52

62

1

3;

32

xx

xx

Solve for x

Let 2x = u

3

5,

3

2

1cos

2

12cos

u

u

x

26

5;2

6

6

5

2

1

3

5;

3

52

62

1

3;

32

nxnxSo

xx

xx

Solve for y: Domain

Let sin y = u

01334

0341343

034912

031312

03sin13sin12

2

2

2

uu

uuu

uuu

uu

yy

)2,0[

Solve for y Range

Let sin y = u

01334

0341343

034912

031312

03sin13sin12

2

2

2

uu

uuu

uuu

uu

yy

01sin33sin4 yy

y

y

y

y

4

3arcsin

4

3sin

3sin4

03sin4

Solve for y

Let sin y = u

01334

0341343

034912

031312

03sin13sin12

2

2

2

uu

uuu

uuu

uu

yy

01sin33sin4 yy

y

y

y

y

4

3arcsin

4

3sin

3sin4

03sin4

y

y

y

y

3

1arcsin

3

1sin

1sin3

01sin3

3398.0y 8481.0y

Solve for y: domain

We have to give all the answers

3398.0

8481.0

)2,0[

3

1

4

3 ?

?

Solve for y: domain

We have to give all the answers

3398.0

8481.0

)2,0[

3

1

4

3 294.28481.0

803.2339.0

Homework Homework

Page 376 – 379 Page 376 – 379

##5, 9, 17, 22,5, 9, 17, 22,

28, 35, 57,28, 35, 57,

61, 7361, 73

Homework

Page 376 – 379

# 3, 7, 12, 20,

25, 31, 56,

60, 71, 87

5.3 solving trigonometric equations. what are two values of x between 0 and when cos x = ½ x =...

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