3. -
thales
tangent

4. Trigonometric
Ratio)

5. ?
?
?

6. ABC
B
A
B
C B
B

8. P
Q
R

9. sine
=
cosine
=
tangent
=

10. ABC
A B
A
C A
A
sine A B
A
C
sin A = B
cosine A A
A
C
cos A = B
tangent A B
A
C
tan A = C

11. P
Q
R
P
P

12. P
Q
R
Q
P
P Q
sin P R cos P Q
tan P R
P P P
= R = R = Q

13. 1
cotangent = tan
= A
secant 1
=
= cos
A
cosine = 1
= sin
A

14. ABC
A B
A
C A
A
cotangent A A
B
C
cot A = C
secant A A
A
B
sec A = C
cosecant A A
B
B
csc A = C

15. P
Q
R

16. P
P
Q
R
P
P P P
cot P Q sec P P
R
csc P Q
R
= R = Q = R

21. 20 tan A = 21
sin A – cos A
20 tan A = 21 tan A 2
2
1 2
= 0 2 9
1 A
2
0
sin A – = 2 2
2 1 2
0
cos A 9 9
sin A – 1
= 2
cos A 9

22. o , 45o , 60o
3
0
o
4
5 6
o
0
o

23. ABC
A = B =
C = 60o

24. A
BC A
A
2
2 2
D C
B C 1
2 A
2
B D
1

25. ADC
A AD
A
2 D =
90o =
C
D C 60o =
A
1
AC2 = AD2 30o
+ DC2 AC2 –
AD2 =
DC2
= 2 –
2
12 = 4–
1
AC = 3

26. ADC
A
1
sin A sin30o
2 2
= =
3 cos A cos 30o 2
D C 3
1 = = 1
tan A tan = 3
o 3
= 30 3
cot A cot 30 o
=
= = 3
sec 2 2
sec A = 33
= 30o 3
csc A c
=sc 2
= 30o
=

27. ADC
A
sin C sin 60o 2
2 3
= =
cos C cos 60 o 1
3 2
D C = =
1 tan C tan
= 60o 3
1
cot C =ot 60
c o =
33
= = 3
sec C sec 2
= 60o
= 2 2
csc C csc = 33
= 60o
3
=

28. ABC
A
1
4
B C
1 5
o
A =
C = 45o

29. ABC
A AC
1 D =
90o =
C
4
B C 45o =
A
1 5
o 45o
AC2 = AB2
+ BC2 2
= 1 +
12 = 1+
1
AC = 2

30. ABC
1
sin A sin C sin45o =2
2
A = = = 2
cos A cos C cos 45o 1 = 2
1 = = = 2
2 21
tan A tan C tan
4 = = 45o
B C
1 5 cot A cot C cot 45o 1
=
o = = =
sec A sec C sec
= = 45o 2
csc A csc C csc
= o 2
= = 45
=

31. o 45o
60o
30o 45o 60o
1 1
sin =
2 22 32
21 1
cos 2
= 2
2 2
3
1 2
tan = 1
33
3 1 3
cot 1 =
33
2 32 3
sec = 2
3 2
3 2
csc 2 = 23
3
2

32. o
45o 60o
2 sin o
cos o tan o
2 sin o cos
o tan o
=2 1 1
2 32
3
= 1
2

33. 2 ( sin2 o +
cos2 o ) + tan 45o
2 ( sin2 o + cos2
o ) + tan 45o 2
2
=2 1
2
+ + 1
32
1 3
= 2 4
+4 + 1
= 3

34. sin cos tan

36. sin cos tan sin cos tan
0.35 0.93 0.38 0.90 0.42 2.14
21o 65o
8 4 4 6 3 5
0.37 0.92 0.40 0.91 0.40 2.24
22o 66o
5 7 4 4 7 6
0.39 0.92 0.42 0.92 0.39 2.35
23o 67o
1 1 4 1 1 6
0.40 0.91 0.44 0.92 0.37 2.47
24o 68o
7 4 5 7 5 5
0.42 0.90 0.46 0.93 0.35 2.60
25o 69o
3 6 6 4 8 5
sin 0.3 cos 0.3 tan 0.4
21o = 58 67o = 91 25o = 66

37. 1
cot A
tan
= A1 1
cot = = 0.46 = 2.1
25o tan 46
25o 6
1
sec
cos
A = A1 1
sec = = 0.39 = 2.5
67o cos 58
67o 1
1
csc
sin
A = A1 1
csc = = = 2.7
21o sin 0.35 93
21o 8

38. sin cos tan
0.74 0.66 1.11
48o
3 9 1
0.75 0.65 1.15
49o
5 6 0
0.76 0.64 1.19
50o
cos 496 = 3
o 2 tan 50o =
sin 48o =
0.743 1 0.656 1 1.192 1
csc sec cot
48o = sin1 49o = cos
1 50o =tan
1
48o = 49o =50o
= 0.74 0.65 1.19
3
= 1.3 6 2
= 0.8
= 1.5
46 24 39

39. sin cos tan sin cos tan
0.35 0.93 0.38 0.90 0.42 2.14
21o 65o
8 4 4 6 3 5
0.37 0.92 0.40 0.91 0.40 2.24
22o 66o
5 7 4 4 7 6
0.39 0.92 0.42 0.92 0.39 2.35
23o 67o
1 1 4 1 1 6
0.40 0.91 0.44 0.92 0.37 2.47
24o 68o
7 4 5 7 5 5
0.42 0.90 0.46 0.93 0.35 2.60
25o 69o
3 6 6 4 8 5
sin 0.3 sin 0.3
22o = 75
sin sin + sin o = 0.375 +
23
= 91 = 0.3
22.5 o 22o 2 23o 0.3912 83
= sin 0.3
22.5o 83

40. sin cos tan
0.54 0.83 0.64
33o
5 9 9
0.55 0.82 0.67
34o
9 9 5
0.57 0.81 0.70
35o 1
sin sin34o + 4 9 0
csc sin
34.5o sin 35o
2
0.559 + 34.5o 34.5o
1
= = = =
2
0.574 0.5
= 0.5 = 67
1.7
67 64

41. sin cos tan
0.97 0.22 4.33
77o
4 5 1
0.97 0.20 4.70
78o
8 8 5
0.98 0.19 5.14
79o 1
cos cos 77o + 2 1 5
sec cos
77.5o cos 78o
2
0.225 + 34.5o 77.5o
1
= = = =
2
0.208 0.2
= 0.2 = 17
4.6
17 08

42. sin cos tan
0.54 0.83 0.64
33o
5 9 9
0.55 0.82 0.67
34o
9 9 5
o 0.57 0.81 0.70
tan tan 33o + 35 4 cot9 0 1
tan 34o
2 tan
33.5o 0.649 + 33.5o
= = 33.5o
1
= 2
0.675 = 0.6
= 0.6 = 62
1.5
62 11

43. sin = cos ( 90o ?
A –A)
cos = sin ( 90o ?
A –A)
tan = cot ( 90o ?
A –A)
? ? ?

44. sin cos tan sin cos tan
0.35 0.93 0.38 0.90 0.42 2.14
21o 65o
8 4 4 6 3 5
0.37 0.92 0.40 0.91 0.40 2.24
22o 66o
5 7 4 4 7 6
0.39 0.92 0.42 0.92 0.39 2.35
23o 67o
1 1 4 1 1 6
0.40 0.91 0.44 0.92 0.37 2.47
24o 68o
7 4 5 7 5 5
0.42 0.90 0.46 0.93 0.35 2.60
25o 69o
3 6 6 4 8
sin 0.3 cos 0.3 sin 5 o
21 =
21o = 58 69o = 58 cos 69o
sin 0.9 cos 0.9 sin 65o =
65o = 06 25o = 06 cos 25o

45. sin 0.3 cos 0.3 sin 21o =
21o = 58 69o = 58 cos 69o
sin 0.9 cos 0.9 sin 65o =
65o = 06 25o = 06 cos 25o
21o + 69o sin A = cos+ 65o
25o B
= 90o = 90o
sin B = cos A
A + B = 90o
B = 90o – A
A + B = 90o
sin A = cos
(90o – A)
cos A = sin (90o
– A)

46. sin 53o = 0.799 ,
cos 53o =
37o
sin A = cos
(90o o =
sin 37– A) cos (90o – 37o ) =
cos 53o = 0.602
sin 37o =
0.602
cos A = sin
(90o – =
cos 37o A) sin (90o – 37o ) =
sin 53o = 0.799
cos 37o =
0.799

47. tan A sin
cos
A
= sin
A 0.6
tan = 37o = 0.79 = 0.7
02
37o cos 53
tan 37o = 9
37o
0.753
cos
cot A
Asin
= cos
A 0.7
cot = 37 o = 0.60 = 1.3
99
37o sin 27
37o o 2
cot 37 =
1.327

48. 1
sec
cos
A = A1 1
sec = = 0.79 = 1.2
37o cos 52
sec 37o = 9
37o
1.252
1
csc
sin
A = 1
A 1
csc = = 0.60 = 1.6
37o sin 61
37o o 2
csc 37 =
1.661

49. sin 73o = 0.956 ,
cos 73o =
37o
sin A = cos
(90o o =
sin 17– A) cos (90o – 17o ) =
cos 73o = 0.292
sin 17o =
0.292
cos A = sin
(90o – =
cos 17o A) sin (90o – 17o ) =
sin 73o = 0.956
cos 17o =
0.956

50. tan A sin
cos
A
= sin
A 0.2
tan = 17o = 0.95 = 0.3
92
17o cos 05
tan 17o = 6
17o
0.305
cos
cot A
Asin
= cos
A 0.9
cot = 17 o = 0.29 = 3.2
56
17o sin 74
17o o 2
cot 17 =
3.274

51. 1
sec
cos
A = A1 1
sec = = 0.95 = 1.0
17o cos 46
sec 17o = 6
17o
1.252
1
csc
sin
A = 1
A 1
csc = = 0.29 = 3.4
17o sin 25
17o o 2
csc 17 =
3.425

52. ? ?
? ?
? ?
?
?
?
?

54. ABC
B
1
a 2
3
C 0o A
a

55. B BC
1
a 2 AB A = a
=
3 A =
C 0o A 30o
a
B
C = sin
A
B A
a sin
1
=
30o 1
2
a = 1 2
2
a = 6

56. B AC
1 AB A = a
6 =
A =
3
C a 7o A 37o
B a
C = cos
A
B A
a cos
1
=
37o
6
a 16 cos
=
37o
a = 16 (
cos 37o 0.799 )
a = 12.7
84

57. o 45o 60o

58. ABC
B
1
6 2
?
C A
A

59. B BC
1 AB A = a
6
2 =
?
C A A
B
C = sin
A
B A
6 sin
1
=
A
1
2
sin =
2
A 1
sin 30o
= 2
A =
30o

60. B BC
? 1 AB A = a
5
3 =
C A B
B
C = cos
A
B B
5
1
= cos
B
3
cos = 0.3
B o
85
cos 67.5 =
0.383
B
67.5o

61. ? ?
? ? ?
?

66. B
?
3
A 6o 1 C
5
0

67. B BC
AC
3 36o
A 6o 1 C
B
5 C = tan
A
0 C A
B
C
15
= tan
36o
0B
= 150 o
C tan 36
o
tan 36 =
0.727 = 150 (
B
C 0.727 )

68. B
5
6o
1
2
A C 0
?

69. B
BC
5 AC
6o
1
2 56o
A C0
?
B
C = tan
A
C A
12
A
0
= tan
C
56o
AC 12 = AC tan
0 o 56
o
tan 56 =
1.483 12
A =
C 1.4
0
83

70. A x tan 60o
x
B
o x =
( ) tan o
x =
=