4.5 identities
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Transcript of 4.5 identities
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
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
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
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
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. 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
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
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