Recall

,

1sin csc , 0

1cos sec , 0

tan , 0 cot , 0

P t x y

t y t yy

t x t xx

y xt x t y

x y

1

Fundamental Identities

sin costan cot

cos sin

1 1sec csc

cos sin

1cot

tan

t tt t

t t

t tt t

tt

2

Fundamental Identities

2 2

2 2

2 2

2 2

2 2 2

2 2

, 1

cos sin 1

cos sin 1

cos sin 1

cos cos cos

1 tan sec

P t x y x y

t t

t t

t t

t t t

t t

3

Fundamental Identities

2 2

2 2

2 2 2

2 2

cos sin 1

cos sin 1

sin sin sin

cot 1 csc

t t

t t

t t t

t t

4

Fundamental Identities

2 2

2 2

2 2

cos sin 1

1 tan sec

cot 1 csc

t t

t t

t t

5

Challenge!

35

32

Identify the quadrant where lies.

1. sin 0 and cos 0

2. csc 0 and tan 0

3. sec 0 and sin 0

4. cot 0 and

5. Arcsin

6. Arccot 5

7. Arcsec

P

P QII

P QI

P QIII

t P QIV P QII

P QI

P QII

P QII6

2 2 2

2

2 2

2

3Given that cos and , find

5

the circular function values of .

1 5sec

cos 3

9cos sin 1 sin 1

25

3 16sin 1 sin

5 25

9 4sin 1 sin

25 5

P QIV

Example 4.5.1

7

4 3 5sin cos sec

5 5 3

1 5csc

sin 44

sin 45tan3cos 3

51 3

cottan 4

8

sin ce P QIV

Example 4.5.2

2

2 2

2

2

Find the other circular function values of

2if tan and cos 0.

3

1 3 13cot sec

tan 2 9

13sec

3

131 tan sec sec

3

2 1 3 3 131 sec cos

3 sec 1313

P QIII

9

sin ce P QIII

2 3 13tan cot sec

3 2 3

3cos

13

sintan

cos

2 3 2sin tan cos

3 13 13

1 13csc

sin 2

10

Example 4.5.3

2 2 2

2

2

2

2Evaluate cos Arcsin .

3

2Let Arcsin Find cos .

3

2sin and

3

5cos sin 1 cos

9

2 5cos 1 cos , since

3 3

4 2 5cos 1 cos Arcsin

9 3 3

P QI

P QI

11

Sum and Difference Identities

sin sin cos cos sin

cos cos cos sin sin

tan tantan

1 tan tan

u v u v u v

u v u v u v

u vu v

u v

12

Example 4.5.4

512

5 8 3 212 12 12 3 4

2 2 23 4 3 4 3 4

Find the exact value of cos .

cos cos cos

cos cos cos sin sin

1 2 3 2

2 2 2 2

2 6 6 2

4 4 4

13

Example 4.5.4

3 4

3 4 3 4 3 4

Find the exact value of sin 105 .

sin 105 sin 60 45 sin

sin sin cos cos sin

3 2 1 2

2 2 2 2

6 2 6 2

4 4 4

14

Example 4.5.5

2 13 3

23

13

Let sin and sin where and

terminates in .

1. Find sin

sin sin cos cos sin

sin

sin

u v u v

QI

u v

u v u v u v

u P u QI

v P v QI

15

23

13

2 2 2 2

2 2

2 2

2 2

2 2

sin sin cos cos sin

sin

sin

cos sin 1 cos sin 1

2 1cos 1 cos 1

3 3

4 1cos 1 cos 1

9 9

5 8cos cos

9 9

5 2 2cos ,sin cos ,sin

3 3

u v u v u v

u P u QI

v P v QI

u u v v

u v

u v

u v

u ce P u QI v ce P v QI16

523 3

2 213 3

sin sin cos cos sin

sin cos

sin cos

2 2 2 5 1sin

3 3 3 3

4 2 5

9 9

4 2 5

9

u v u v u v

u u

v v

u v

17

523 3

2 213 3

2. cos cos cos sin sin

sin cos

sin cos

5 2 2 2 1cos

3 3 3 3

2 10 2

9 9

2 10 2

9

u v u v u v

u u

v v

u v

18

3. Quadrant where lies.

4 2 5sin 0

9

2 10 2cos 0

9

P u v

u v

u v

P u v QI

19

Example 4.5.6

51213 4

51213 4

Evaluate cot Arctan Arcsec .

Let Arctan Arcsec

1 1 1 tan tancot

tan tantan tan tan1 tan tan

12tan

13

5sec

4

u v

u vu v

u vu v u vu v

u

v P v QI

20

2 2 2

2

2

2

1 tan tancot

tan tan

12tan

13

5sec

49

1 tan sec tan16

5 31 tan tan

4 4

25 3tan 1 tan , since

16 4

u vu v

u v

u

v P v QI

v v v

v v

v v P v QI

21

1 tan tancot

tan tan

12 3tan tan

13 4

12 3 361 1611613 4 52 52cot

12 3 48 39 87 8752

13 4 52 52

12 5 16cot Arctan Arcsec

13 4 87

u vu v

u v

u v

u v

22

Double-Measure Identities

2 2

sin 2 sin

sin cos cos sin

sin cos sin cos

2sin cos

cos 2 cos

cos cos sin sin

cos sin

u u u

u u u u

u u u u

u u

u u u

u u u u

u u

23

Double-Measure Identities

2 2

2

2

2

sin 2 2sin cos

cos 2 cos sin

2cos 1

1 2sin

2tantan 2

1 tan

u u u

u u u

u

u

uu

u

24

Example 4.5.7

2 2 2

2

2

2

3Given that tan and ,

4

1. Find sin 2

sin 2 2sin cos

3tan

425

1 tan sec sec16

3 51 sec sec

4 4

9sec 1

16

u P u QII

u

u u u

u P u QII

u u u

u u

u

25

sin 2 2sin cos

3 5tan sec

4 41 4

cossec 5

sintan

cos

3 4 3sin tan cos

4 5 5

3 4 24sin 2 2

5 5 25

u u u

u u

uu

uu

u

u u u

u

26

2

2

2. Find cos 2 .

4cos 2 2cos 1 cos

5

42 1

5

162 1

25

321

25

7

25

u

u u u

27

2

2. Find tan 2 .

24sin 2 2425tan 27cos 2 7

25

2tanverify using tan 2 .

1 tan

3. Find the quadrant where 2 lies.

7 24cos 2 0 sin 2 0

25 25

2

u

uu

u

uu

u

P u

u u

P u QIV

28

Example 4.5.8

32

32

2

32

2 2

22 23 52 4

2 594 2

Evaluate tan 2Arcsec .

Let Arcsec Find tan 2 .

2tantan 2

1 tan

sec and

1 tan sec

1 tan tan

tan 1 tan

u u

uu

u

u P u QI

u u

u u

u u

29

2

52

52

25

2

54

14

32

2tantan 2

1 tan

tan

2tan 2

1

5

1

5

4 5

tan 2Arcsec 4 5

uu

u

u

u

30

Half-Measure Identities

1 1 cos 1 1 cossin cos

2 2 2 2

1 1 cos sin 1 costan

2 1 cos 1 cos sin

u uu u

u u uu

u u u

31

Example 4.5.9

5Given that sin and terminates at , find

13

11. sin

2

:2

4 2 2

2

t t QII

t

P t QII t

t

tP QI

32

12132 2

2 252 13

2

5sin

13 2

1 1 cossin

2 2

11cos sin 1 sin

2 2

5cos 1

13 2

144 25cos

169 26

12 5 26cos

13 26

tt P t QII P QI

tt

t t t

t

t

t33

1213

113

5sin

13 2

12 1 5 26cos sin

13 2 26

1 1 cos2. cos

2 2

1

2

2

1 26

26 26

tt P t QII P QI

t t

tt

34

5sin

13 2

12 1 5 26 1 26cos sin cos

13 2 26 2 26

1sin

1 23. tan

12cos

2

5 26

26

26

26

5

tt P t QII P QI

t t t

t

t

t

35

Example 4.8.10

8

4

4

2 24 2

22

Find the exact value of csc .

1csc

2 8sin

2

1 1

1 cos

2 2

1 1

1 2 2

42

P QI

36

4 1csc

2 2 2

4

1

2 2

22

2 2

37

Reading Assignment

Read Chapter 7.2 of Leithold:

Proving Trigonometric Identities

Page 386-391.

Then answer odd-numbered items

Page 391-392.

Check your answers on A-57.38

End of Chapter 4.5

39