- The trigonometric ratios of sine, cosine, and tangent can be used to find the measures of angles in right triangles. - For a given angle, the trig ratios are defined as the ratio of the length of the side opposite to the angle over the length of the hypotenuse (sine), the ratio of the length of the side adjacent to the angle over the hypotenuse (cosine), and the ratio of the opposite side over the adjacent side (tangent). - The inverse trig functions (cotangent, secant, cosecant) are defined in terms of the tangent, cosine, and sine ratios respectively.

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 =
=