1) This document contains examples of trigonometric functions involving angles and side lengths of right triangles. 2) It provides the values of trig functions like sine, cosine, and tangent for specific angles like 30°, 45°, and 60°. 3) Several problems apply trigonometric definitions and properties to solve for unknown side lengths or angles in right triangles.

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