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( 12 )
0 90
0 90
95
1.
2.
0 89
sin cos tan
302
1
2
3
3
3
452
2
2
2 1
602
32
13
3.
x
1) 2) 3)
4) 5) 6)
x18
39
32
51x
94221
x 75
12
x
44x
14025
16x
96
1.
1) ABC C 90 A 30
BC
AB
BC
AB
2)
A
A
C
30
B
AC
3
30
B
A C
5
B C
A
30
2
97
A C
B
3 5 10
3012 15
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
ca
b
A C
B
2520
15
8 15
17B
C
A
99
2. x
1) 2)
3) 4)
5) 6)
3. ABC
1) sin 2) cos
3) tan 4) AC
18x
30
12
60x
x
60
AC
B
5021
42x
50
32
42
25
10
xx 7
3x
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
101
2)
3)
60
x
12
30
x
30
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
ca
b
A C
B
2520
15
8 15
17B
C
A
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
18x
30
12
60
x
x
60
42
X
50
32
73
X
25X
10
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
18 = 4
sin 60 = 8
b, b = 8 sin 60
= 8 0.866 = 6.92830
C
A B8
b a
60
AC
B
5021
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 =30b
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
106
ABC
AB = AC = BC = 18
BAC = CBA = ACB = 60
AB = AC
ABC
BD = CD = 2
18 9
ABD D
sin B = 18AD
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 = 12x
x = 12 tan 60
= 12 1.732
= 20.784
20.8
18
9
A
D
B
100 30
x
12
60
107
30
x
30
3) x
tan 30 = 30x
x = 30 tan 30
= 30 0.577
= 17.31
17.3
108
2.1
1. 1) sin A = 13
5cos A =
13
12
sin B = 13
12cos B =
13
5
tan A = 12
5tan B =
5
12
2) sin A = 5
3cos A =
5
4
sin B = 5
4cos B =
5
3
tan A = 4
3tan B =
3
4
3) sin A = 41
4cos A =
41
5
sin B = 41
5cos B =
41
4
tan A = 5
4tan 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
35
4
C
4 5
41
B A
B C
A
A
B
109
3.
1) sin A =c
a2) cos B =
c
a
3) cos A =c
b4) 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
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 ( )
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 + BC
2= AC
2
BC2
= AC2 – AB
2
= 152 – 9
2
= 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
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
335
= 5
cos 30 =y
35
y =30cos
35
=
2
3
35
= 10
14x
45
30 x
y
35
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
21
sin B = 2
1 0.5
y 8
60x
x
30 60
y
8
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 = AB
2 + AD
2
BD2 = 8
2 + 6
2
BD2 = 64 + 36
BD2 = 100
BD = 10
BD
BD = 10
2) 1sin =BD
AB=
10
8
5
4
3) 1cos =BD
AD=
10
6
5
3
4) 2sin =BD
AD=
10
6
5
3
5) 2cos =BD
AB=
10
8
5
4
A B
D C
8
6
24
31
115
ABCD
BC = 6 DC = 8
6) 3sin =BD
BC=
10
6
5
3
7) 3cos =BD
CD=
10
8
5
4
8) 4sin =BD
CD=
10
8
5
4
9) 1sin = 5
4 0.8
sin 53 = 0.799
1 53
10) 2sin =5
30.6
sin 37 = 0.602
2 37
1 53
1 + 2 = 90
53 + 2 90
2 37
11) 3sin =5
30.6
sin 37 = 0.602
3 37
12) 4sin =5
40.8
sin 53 = 0.799
4 53
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 = a
2 + b
2
c2
= 52 + 12
2
c2
= 25 + 144
c2
= 169
c = 13
c
b
a
B
C A
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 = a
2 + b
2
62
= 42 + b
2
b2
= 62 – 4
2
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
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
119
BC
AB2
= AC2 + BC
2
152
= 92 + BC
2
BC2
= 152 – 9
2
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
A = G = 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
120
4) AB = 210
AC = 56
ABC A =
cos = 210
56 = 0.949
cos 18 = 0.951
18
ABC = 180
CBA 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
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 A = 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
BC10
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 CAB ABC
AD CAB ABC AD
BC
A
CB D
18CB
14 14
A
123
BD = DC = 9
B = C =
cos B =AB
BD
cos =14
9
cos = 0.643
cos 50 = 0.643
= 50
180 = BACACBCBA
180 = 50 + 50 + BAC
BAC = 180 – 100
BAC = 80
50CBA , 50ACB 80BAC
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
125
2. 6.5
4
AB
AC 6.5
BC 4
BCA BCA =
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
C4
A
B
6.5
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
55A B
D C
6
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
C39
16 A
B
A
B
C7
2 .
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
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
C1316
40
55
C
A
B .
.
x
5
D4055
A
B
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
1112
AO
9 .
B
131
12 3 15 1
15.07 . 7
715
90 42
cos =OB
OA
OA = OB cos 42
cos 42 = 0.743
OA = 9 0.743
= 6.687
6.7
12
3
