1. 2
( 12 )
0 90
0 90

2. 95
1.
2.
0 89
sin cos tan
30
2
1
2
3
3
3
45
2
2
2
2 1
60
2
3
2
1
3
3.
x
1) 2) 3)
4) 5) 6)
x
18
39
32
51
x
942
21
x
75
12
x
44x
140
25
16 x

3. 96
1.
1) ABC C 90 A 30
BC
AB
BC
AB
2)
A
A
C
30
B
A C
3
30
B
A C
5
B C
A
30
2

4. 97
A C
B
3 5 10
30
12 15

5. 98
1.
1) 1) sin A
2) cos B
3) tan A
4) tan B
2)
1) sin A sin B
2) cos A cos B
3) tan A tan B
3)
1) sin A sin B
2) cos A cos B
3) tan A tan B
A C
B
c
a
b
A C
B
25
20
15
8 15
17B
C
A

6. 99
2. x
1) 2)
3) 4)
5) 6)
3. ABC
1) sin 2) cos
3) tan 4) AC
18
x
30
12
60
x
x
60
A C
B
50 21
42x
50
32
42
25
10
x
x 73 x

7. 100
4. a, b c
1) 2)
3) 4)
5. ABC AB 18 AD A
AD BC AD
6. 1)
C
A B
C
D
18
100
30
30
C
A B8
b a
60
c
b6
B A45 45
b
B
a
C A
45
30
A
BaC
c7
60

8. 101
2)
3)
60
x
12
30
x
30

9. 102
1.
1) 1) sin A =
c
a
2) cos B =
c
a
3) tan A =
b
a
4) tan B =
a
b
2)
sin A =
25
20
=
5
4
, sin B =
25
15
=
5
3
cos A =
25
15
=
5
3
, cos B =
25
20
=
5
4
tan A =
15
20
=
3
4
, tan B =
20
15
=
4
3
3)
sin A =
17
8
, sin B =
17
15
cos A =
17
15
, cos B =
17
8
tan A =
15
8
, tan B =
8
15
A C
B
c
a
b
A C
B
25
20
15
8 15
17B
C
A

10. 103
2. x
1) sin 30 =
18
x
x = 18 sin 30 = 18 0.50 = 9
2) cos 60 =
12
x
x = 12 cos 60 = 12 0.5 = 6
3) tan 60 =
42
x
x = 42 tan 60 = 42 1.732 = 72.744
4) tan x =
50
32
= 0.64
tan 33 = 0.649
x 33
5) cos x =
7
3
= 0.42
cos 65 = 0.423
x 65
6) sin x =
25
10
= 0.40
sin 24 = 0.407
x 24
18
x
30
12
60
x
x
60
42
X
50
32
73
X
25
X
10

11. 104
3. ABC
1) sin
sin =
50
21
0.42
2) cos
sin 25 = 0.423
25
cos 25 = 0.906
3) tan
tan = tan 25
tan 25 = 0.466
4) AC
cos =
50
AC
AC = 50 cos
= 50 cos 25
= 50 0.906
= 45.3
4. a, b c
1) sin 30 = 8
a
, a = 8 sin 30
= 2
1
8 = 4
sin 60 = 8
b
, b = 8 sin 60
= 8 0.866 = 6.92830
C
A B8
b a
60
A C
B
50 21

12. 105
2) tan 60 =
7
a
a = 7 1.732
= 12.124
B = 90 – 60 = 30
sin B = sin 30 = c
7
c = 7 sin 30
= 7 0.5 = 3.5
3) sin A = c
6
c =
Asin
6
=
45
6
sin
=
0.707
6 = 8.487
ABC
(A = B = 45 )
b = 6
4) sin 45 =
30
b
b = 30 sin 45 = 30 0.707
= 21.21
A = 90 – 45 = 45
A = B = 45
ABC
a = b = 21.21
5. ABC AB 18 AD A
AD BC AD
A B
C
D
18
60
a
bC
B
A
45
30
cB A
b6
C
45 45

13. 106
ABC
AB = AC = BC = 18
BAC = CBA = ACB = 60
AB = AC
ABC
BD = CD = 2
18
9
ABD D
sin B =
18
AD
AD = 18 sin B
= 18 sin 60
sin 60 = 0.866
AD = 18 0.866
= 15.588
6. 1) x
tan 30 =
100
x
x = 100 tan 30
= 100 0.577 57.77
57.77
2) x
tan 60 =
12
x
x = 12 tan 60
= 12 1.732
= 20.784
20.8
18
9
A
D
B
100
30
x
12
60

14. 107
30
x
30
3) x
tan 30 =
30
x
x = 30 tan 30
= 30 0.577
= 17.31
17.3

15. 108
2.1
1. 1) sin A =
13
5
cos A =
13
12
sin B =
13
12
cos B =
13
5
tan A =
12
5
tan B =
5
12
2) sin A =
5
3
cos A =
5
4
sin B =
5
4
cos B =
5
3
tan A =
4
3
tan B =
3
4
3) sin A =
41
4
cos A =
41
5
sin B =
41
5
cos B =
41
4
tan A =
5
4
tan B =
4
5
2. 1) sin A =
AB
BC
2) sin B =
AB
AC
3) cos A =
AB
AC
4) cos B =
AB
BC
5) tan B =
BC
AC
6) tan A =
AC
BC
A
B
C
13
12
5
C
3
5
4
C
4 5
41
B A
B C
A
A
B

16. 109
3.
1) sin A = c
a
2) cos B = c
a
3) cos A = c
b
4) sin B = c
b
4. 1) sin 32 = 0.530 2) sin 4 = 0.070
3) sin 81 = 0.988 4) sin 17 = 0.292
5) cos 29 = 0.875 6) tan 18 = 0.325
7) tan 81 = 6.314 8) sin 51.5 0.7825
9) cos 67.5 0.383 10) tan 42.5 0.9165
5. 1) sin = 0.53
sin 32 = 0.53
= 32
2) sin = 0.899
sin 64 = 0.899
= 64
3) sin =
5
2
0.4
sin 24 = 0.407
= 24 ( )
4) sin =
8
5
0.625
sin 39 = 0.629
= 39 ( )
5) sin =
23
15
0.652
sin 41 = 0.656
= 41 ( )
C
b a
A Bc

17. 110
6. 1) cos = 0.5
cos 60 = 0.500
= 60
2) cos = 0.99
cos 8 = 0.990
= 8
3) cos = 0.75
cos 41 = 0.755
= 41 ( )
4) cos =
5
3
0.6
cos 53 = 0.602
= 53 ( )
5) cos =
13
9
0.692
cos 46 = 0.695
= 46 ( )
7. 1) tan = 0.404
tan 22 = 0.404
= 22
2) tan = 4.011
tan 76 = 4.011
= 76
3) tan =
4
9
2.250
tan 66 = 2.246
= 66 ( )
4) tan =
7
4
0.571
tan 30 = 0.577
= 30 ( )
5) tan =
13
28
2.154
tan 65 = 2.145
= 65 ( )

18. 111
8.
1) cos = 0.616 cos 52 = 0.616 = 52
2) tan = 0.488 tan 26 = 0.488 = 26
3) sin = 0.982 sin 79 = 0.982 = 79
4) tan = 2.356 tan 67 = 2.356 = 67
5) cos = 0.707 cos 45 = 0.707 = 45
9. 1
AB2
+ BC2
= AC2
BC2
= AC2
– AB2
= 152
– 92
= 144
BC = 12
BC BC = 12
sin A =
AC
BC
=
15
12
0.8
tan A =
AB
BC
=
9
12
1.33
cos C =
AC
BC
=
15
12
0.8
tan C =
BC
AB
=
12
9
0.75
2
cos A =
15
9
0.6
cos 53 = 0.602
A = 53 ( )
C = 180 – 90 – 53 = 37 ( )
sin A = sin 53 = 0.799 0.8
tan A = tan 53 = 1.327 1.33
A B
C
15
9

19. 112
cos C = cos 37 = 0.799 0.8
tan C = tan 37 = 0.754 0.75
10. 1) sin 45 =
14
x
x = 14 sin 45
sin 45 = 0.707
x = 14 0.707
= 9.898
2) tan 30 =
35
x
x = 35 tan 30
=
3
3
35
= 5
cos 30 =
y
35
y =
30cos
35
=
2
3
35
= 10
14 x
45
30 x
y
35

20. 113
3) cos 60 =
8
x
x = 8 cos 60
cos 60 = 0.500
x = 8 0.500
= 4
sin 60 =
8
y
y = 8 sin 60
sin 60 = 0.866
y = 8 0.866
= 6.928
tan 45 =
z
y
z =
45tan
y
45tan
928.6
tan 45 = 1.0
z = 6.928
4) sin 60 =
8
x
x = 8 sin 60
sin 60 = 0.866
x = 8 0.866
= 6.928
sin 30 =
8
y
y = 8 sin 30
sin 30 = 0.50
y = 8 0.5
= 4
11. ABC C cos A =
2
1
cos A =
2
1
sin B = cos A
45
y
z
A
B C
2
1
sin B =
2
1
0.5
y 8
60
x
x
30 60
y
8

21. 114
sin 30 = 0.5 B = 30
cos 60 = 0.5 A = 60
1) sin A = sin 60
sin 60 = 0.866
2) tan A = tan 60
tan 60 = 1.732
3) sin B = cos A =
2
1
0.5
4) cos B = sin A = 0.866
5) tan B = tan 30
tan 30 = 0.577
12. 1)
ABD
BD2
= AB2
+ AD2
BD2
= 82
+ 62
BD2
= 64 + 36
BD2
= 100
BD = 10
BD
BD = 10
2) 1ˆsin =
BD
AB
=
10
8
5
4
3) 1ˆcos =
BD
AD
=
10
6
5
3
4) 2ˆsin =
BD
AD
=
10
6
5
3
5) 2ˆcos =
BD
AB
=
10
8
5
4
A B
D C
8
6
2 4
31

22. 115
ABCD
BC = 6 DC = 8
6) 3ˆsin =
BD
BC
=
10
6
5
3
7) 3ˆcos =
BD
CD
=
10
8
5
4
8) 4ˆsin =
BD
CD
=
10
8
5
4
9) 1ˆsin =
5
4
0.8
sin 53 = 0.799
1ˆ 53
10) 2ˆsin =
5
3
0.6
sin 37 = 0.602
2ˆ 37
1ˆ 53
1ˆ + 2ˆ = 90
53 + 2ˆ 90
2ˆ 37
11) 3ˆsin =
5
3
0.6
sin 37 = 0.602
3ˆ 37
12) 4ˆsin =
5
4
0.8
sin 53 = 0.799
4ˆ 53

23. 116
13.
1) tan A = 2 a = 10
tan A =
b
a
2 =
b
10
b =
2
10
=
2
52
= 25
2) tan A =
4
1
b = 3
tan A =
b
a
4
1
=
3
a
a =
4
3
3) a = 5 b = 12
sin A =
c
a
c c2
= a2
+ b2
c2
= 52
+ 122
c2
= 25 + 144
c2
= 169
c = 13
c
b
a
B
C A

24. 117
c
c = 13
sin A =
13
5
4) cos B =
3
2
a = 5
cos B =
c
a
3
2
=
c
5
c =
2
35
7.5
5) a = 4 c = 6
tan B =
a
b
b c2
= a2
+ b2
62
= 42
+ b2
b2
= 62
– 42
b2
= 36 – 16
b2
= 20
b = 52
b b = 52
tan B =
4
52
2
5
14. 1)
BG = 12 GH = 8
B =
A C
D
G
B
H
E

25. 118
BGH
sin =
BG
GH
=
12
8
= 0.667
sin 42 = 0.669
42
BDE
BE x
cos =
BD
BE
cos 42 =
15
x
cos 42 = 0.743
0.743 =
15
x
x = 0.743 15 11.145
BE 11.145
2) AC = 9 DE = 6 AB = 15
B
G H8
12
x
B
D E
15
C
E
B
D
A
6
9
15

26. 119
BC
AB2
= AC2
+ BC2
152
= 92
+ BC2
BC2
= 152
– 92
BC2
= 225 – 81
BC2
= 144
BC = 12
BC BC = 12
EC = BC – BE = 12 – 11.145 = 0.855
3) A = 50 BH = 4
ABC BGH
Aˆ = Gˆ = 50
tan 50 =
GH
4
tan 50 = 1.192
1.192 =
GH
4
GH =
192.1
4
GH = 3.356
B
G H
4
50

27. 120
4) AB = 210
AC = 56
ABC A =
cos =
210
56
= 0.949
cos 18 = 0.951
18
ABC = 180
CBˆA 180 – 90 – 18 72
15. DEF DG DE 18
DEF
60
DGE E = 60
sin E =
18
DG
sin 60 =
18
DG
DG = 18 sin 60
= 18 0.866
= 15.588
18
E
G
F
D
18
A C
B
210
56

28. 121
16. JKL LM JL =16
JKL J = K
J = K =
180 = 90 + +
= 45
JLM
sin J =
16
LM
LM = 16 sin 45
sin 45 = 0.707
LM = 16 0.707
= 11.312
17. ABC ACBC Aˆ = 27 BC = 10
ABC AB
sin A =
AB
BC
sin 27 =
AB
10
AB =
27sin
10
sin 27 = 0.454
L
M K
16
J
27
A
BC
10

29. 122
AB =
454.0
10
22.03
AC
tan A =
AC
BC
tan 27 =
AC
10
AC =
27tan
10
tan 27 = 0.510
AC =
510.0
10
AC 19.61
ABC = AB + BC + AC
22.03 + 10 + 19.61
51.64
18. 14, 14 18
AD CAˆB ABC
AD CAˆB ABC AD
BC
A
CB D
18
CB
14 14
A

30. 123
BD = DC = 9
B = C =
cos B =
AB
BD
cos =
14
9
cos = 0.643
cos 50 = 0.643
= 50
180 = BAˆCACˆBCBˆA
180 = 50 + 50 + BAˆC
BAˆC = 180 – 100
BAˆC = 80
50CBˆA , 50ACˆB 80BAˆC

31. 124
2.2
1. 40
20
A
B
C
tan B =
BC
AC
AC = 40 tan 20
tan 20 = 0.364
AC = 40 0.364
AC = 14.56
14.56
40 .
20
A
CB 20
40

32. 125
2. 6.5
4
AB
AC 6.5
BC 4
BCˆA BCˆA =
cos =
AC
BC
=
5.6
4
= 0.615
cos 52 = 0.616
52
sin =
AC
AB
AB = 6.5 sin 52
sin 52 = 0.788
AB = 6.5 0.788
= 5.122
5.122 5
C
4
A
B
6.5

33. 126
3. ABCD 6 55
ABC
sin A =
AC
BC
BC = AC sin 55
sin 55 = 0.819
BC = 6 0.819
= 4.914
cos A =
AC
AB
AB = AC cos 55
sin 55 = 0.574
AB = 6 0.574
= 3.444
4.914
3.444
55
A B
D C
6

34. 127
4.
AB
AC 16
cos A =
AB
AC
AB =
39cos
16
cos 39 = 0.777
AB =
777.0
16
= 20.592
20.6 ( )
5.
AB 2
BC
sin A =
AB
BC
BC = AB sin A
= 2 sin 7
sin 7 = 0.122
BC = 2 0.122
= 0.244
= 0.244 1000
= 244
244
C
39
16A
B
A
B
C
7
2 .

35. 128
6.
AB 9.5
BC
sin A =
AB
BC
BC = 9.5 sin 58
sin 58 = 0.848
BC = 9.5 0.848
= 8.056
8.056 + 1.2 9.2 ( )
7. 45 12
1.5
AB
BC ( )
45
12
A
CB
A C
B
58
9.5
1.5 .
12 .
45

36. 129
ABC
tan C =
BC
AB
AB = BC tan 45
AB = 12 1 12
12 + 1.5 13.5
8.
AC 38
BC + CD
tan 13 =
AC
BC
BC = AC tan 13
tan 13 = 0.231
BC = 38 0.231
= 8.778
tan 16 =
AC
CD
CD = AC tan 16
tan 16 = 0.287
CD = 38 0.287
= 10.906
8.778 + 10.906 = 19.684 19.7
9.
AA // CD
CAA = DCA = 40 ( )
CD // BB
BCD = BBC = 55 ( )
D
A
B
C 13
16
40
55
C
A
B
.
.
x
5
D40
55
A
B

37. 130
AB
BD 5
BCD BDC
tan 55 =
CD
BD
CD =
55tan
5
tan 55 = 1.428
CD =
428.1
5
= 3.5
ACD CDA
tan 40 =
CD
AD
AD = 3.5 tan 40
tan 40 = 0.839
AD = 3.5 0.839
= 2.9365 2.9
5 + 2.9 7.9 ( )
10.
OB
OA
15.07
1
2
3
4
567
8
9
10
11 12
AO
9 .
B

38. 131
12 3 15 1
15.07 . 7
7
15
90
42
cos =
OB
OA
OA = OB cos 42
cos 42 = 0.743
OA = 9 0.743
= 6.687
6.7
12
3