5.1 Using Fundamental Identities. Fundamental Trigonometric Identities.

5.1 Using Fundamental Identities

Fundamental Trigonometric Identities

uu

uu

uu

cot

1tan

sec

1cos

csc

1sin

uu

uu

uu

tan

1cot

cos

1sec

sin

1csc

Fundamental Trigonometric Identities

u

uu

u

uu

sin

coscot

cos

sintan

uu

uu

uu

22

22

22

csccot1

sectan1

1cossin

Co functions

uuuu

uuuu

uuuu

sec2

csc;csc2

sec

tan2

cot;cot2

tan

sin2

cos;cos2

sin

Even or Odd functions

uu

uu

Odd

csc)csc(

sinsin

uu

uu

Even

sec)sec(

cos)cos(

uu

uu

Odd

cot)cot(

tan)tan(

Simplify the Trigonometric Identity

Given: y

y

sin1

cos2

Simplify the Trigonometric Identity

Given:

y

y

y

y

y

y

sin1

sin1

sin1

cos

sin1

cos

2

2

Simplify the Trigonometric Identity

Given:

y

yy

y

y

y

y

y

y

2

2

2

2

sin1

sin1cos

sin1

sin1

sin1

cos

sin1

cos

Simplify the Trigonometric Identity

Given:

y

yy

y

y

y

y

y

y

2

2

2

2

sin1

sin1cos

sin1

sin1

sin1

cos

sin1

cos

yy

yy

22

22

sin1cos

1sincos

Simplify the Trigonometric Identity

Given:

y

y

yy

y

yy

y

y

y

y

y

y

sin1cos

sin1cos

sin1

sin1cos

sin1

sin1

sin1

cos

sin1

cos

2

2

2

2

2

2

Simplify the Trigonometric Identity

Given: 1sec

1sec2

x

x

Simplify the Trigonometric Identity

Given:

1sec

1sec1sec

1sec

1sec2

x

xx

x

x

Simplify the Trigonometric Identity

Given:

1sec

1sec

1sec1sec

1sec

1sec2

x

x

xx

x

x

Evaluate the function

Find the six function values

Given:

Cot

Tan

Cos

Sin

Find

5

53csc;

2

3sec

Evaluate the function

Find the six function values

Given:

2

5

5

52

353

2

3

5

3

2

5

53csc;

2

3sec

Cot

Tan

Cos

Sin

Find

Simplify a Trigonometric expression

Given: SinCscSin

Simplify a Trigonometric expression

Given:

Sin

Sin

SinSin

SinCscSin

21

Simplify a Trigonometric expression

Given:

22

2

2

1

1

CosSin

CosSin

Sin

SinSin

Sin

Sin

SinSin

SinCscSin

Using Substitution

111

1111

111

1

1

22

22

2

23

23

CscxxCotorCscxCscx

CscxxCscuu

uuu

uuu

CscxuSubstitute

CscxxCscxCsc

Using Substitution

Sin

Sin

SinSin

xx

2

2

4

22

422

44

cos116cos1616

)cos4(416

cos2416

cos2;41622

2

22

2

2

2

Using Substitution

Sin

Sin

SinSin

xx

2

2

4

22

422

44

cos116cos1616

)cos4(416

cos2416

cos2;41622

2

22

2

2

2

2

2

2

1

12

1

12

2

2

2

2

2

Cos

Cos

Cos

Cos

Homework

Page 359-361

# 1, 17, 27, 37, 47,

53, 57, 65, 77,

87, 93, 107

Homework

Page 359-361# 10, 26, 36, 48,

54, 58, 68, 78, 88, 94, 120