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Exact Trigonometric Values

Following is the list of Sine of integer angles (A). All of the result of Sin A has divided by 2. Double of Sin A is chord (a) of double angle (2A).

For example, Sin 1° is ½ √(2 – √(2 + √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – + √(2]

[Please care the sign of repetition has underlined. Closing brackets are collapsed for easy.]

Double angle for 1° is 2° and chord (a) for 2° is √(2 – √(2 + √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – + √(2].

What’s about supplementary chord (b) for 2°?

This is simple, just change the first sign (which is –ve in all case) into +ve. Therefore, supplementary chord for 2° is √(2 + √(2 + √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2].

From these ‘a’ and ‘b’, we can compute any trigonometric ratios as given in the formulae list above.

What’s about the complementary angle? The answer is simple; it is based on supplementary chord (b). Therefore, Sin 89° is ½ √(2 + √(2 + √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2].

Sin 1°= ½ √(2 – √(2 + √(2

+ √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 ++ √(2 – + √(2]

[Please care the sign of repetition has underlined.]

All integer angles Sin 1° – Sin 45°

[Please care the sign of repetition has underlined. Closing brackets have collapsed by ‘]’ for easy.]

A° Sin A°

1° ½ √(2 – √(2 + √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 ]

2° ½ √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 ]

3° ½ √(2 – √(2 + √(2 + √(2 + √(2 – √(2 – √(2 ]

4° √(2 – √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2]

5° ½ √(2 – √(2 + √(2 + √(2 + √(2 – √(2]

6° ½ √(2 – √(2 + √(2 + √(2 – √(2 – √(2]

7° ½ √(2 – √(2 + √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2]

8° ½ √(2 – √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 ]

9 ½ √(2 – √(2 + √(2 + √(2 – √(2 ]

10 ½ + √(2 – √(2 + √(2 ]

11 √(2 – √(2 + √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 ]

12 ½ √(2 – √(2 + √(2 – √(2 – √(2 + √(2 ]

13 ½ √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 ]

14 ½ √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 ]

15 ½ √(2 – √(2 + √(2 – √(2]

16 ½ √(2 – √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 ]

17 ½ √(2 – √(2 + √(2 – √(2+ √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 ]

18 ½ + √(2 – √(2 ]

19 ½ √(2 – √(2 + √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 ]

20 ½ √(2 – √(2 + √(2 – √(2 + √(2 ]

21 ½ √(2 – √(2 + √(2 – √(2 + √(2 + √(2 – √(2 – √(2 + √(2 + √(2 – √(2 ]

Easy alternative is ½ √(2 – √(2 + √(2 – √(2 + √(2 + √(2 – √(2 – √(2 ]

22 ½ √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 ]

23 ½ √(2 – √(2 – √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 ]

24 ½ √(2 – √(2 – √(2 – √(2 + √(2 + √(2 ]

25 ½ √(2 – √(2 – √(2 – √(2 + √(2 + √(2 ]

26 ½ √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 ]

27 ½ √(2 – √(2 – √(2 – √(2 + √(2 ]

28 ½ √(2 – √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 ]

29 ½ √(2 – √(2 – √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 ]

30 ½ √(2 – √(2]

31 ½ √(2 – √(2 – √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 ]

32 ½ √(2 – √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 ]

33 ½ √(2 – √(2 – √(2 – √(2 – √(2 + √(2 + √(2 – √(2 – √(2 + √(2 + √(2]

34 ½ √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 ]

35 ½ √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 – √(2 + √(2 ]

36 ½ √(2 – √(2 – √(2 + √(2 – √(2 ]

37 ½ √(2 – √(2 – √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 ]

38 ½ √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 ]

39 ½ √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 ]

40 ½ √(2 – √(2 – √(2 + √(2 + √(2 ]

41 ½ √(2 – √(2 – √(2 + √(2+ √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 ]

42 ½ √(2 – √(2 – √(2 + √(2 + √(2– √(2 ]

43 ½ √(2 – √(2 – √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 ]

44 ½ √(2 – √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 ]

45 ½ √2

46 ½ √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 ]

47 ½ √(2 + √(2 – √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 ]

48 ½ √(2 + √(2 – √(2 + √(2 + √(2– √(2 ]

49 ½ √(2 + √(2 – √(2 + √(2+ √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 ]

50 ½ √(2 + √(2 – √(2 + √(2 + √(2 ]

51 ½ √(2 + √(2 – √(2 + √(2 – √(2 – √(2 + √(2 ]

52 ½ √(2 + √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 ]

53 ½ √(2 + √(2 – √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 ]

54 ½ √(2 + √(2 – √(2 + √(2 – √(2 ]

55 ½ √(2 + √(2 – √(2 + √(2 – √(2 + √(2 + √(2 – √(2 + √(2 ]

56 ½ √(2 + √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 ]

57 ½ √(2 + √(2 – √(2 – √(2 – √(2 + √(2 + √(2 – √(2 – √(2 + √(2 + √(2]

58 ½ √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 ]

59 ½ √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 ]

60 ½ √(2 + √(2 – √(2]

61 ½ √(2 + √(2 – √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 ]

62 ½ √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 ]

63 ½ √(2 + √(2 – √(2 – √(2 + √(2 ]

64 ½ √(2 + √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 ]

65 ½ √(2 + √(2 – √(2 – √(2 + √(2 + √(2 ]

66 ½ √(2 + √(2 – √(2 – √(2 + √(2 + √(2 ]

67 ½ √(2 + √(2 – √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 ]

68 ½ √(2 + √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 ]

69 ½ √(2 + √(2 + √(2 – √(2 + √(2 + √(2 – √(2 – √(2 + √(2 + √(2 – √(2 ]

Easy alternative is ½ √(2 + √(2 + √(2 – √(2 + √(2 + √(2 – √(2 – √(2 ]

70 ½ √(2 + √(2 + √(2 – √(2 + √(2 ]

71 ½ √(2 + √(2 + √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 ]

72 ½ + √(2 – √(2 ]

73 ½ √(2 + √(2 + √(2 – √(2+ √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 ]

74 ½ √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 ]

75 ½ √(2 + √(2 + √(2 – √(2]

76 ½ √(2 + √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 ]

77 ½ √(2 + √(2 + √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 ]

78 ½ √(2 + √(2 + √(2 – √(2 – √(2 + √(2 ]

79 √(2 + √(2 + √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 ]

80 ½ + √(2 + √(2 + √(2+ √(2 – √(2 ]

81 ½ √(2 + √(2 + √(2 + √(2 – √(2 ]

82 ½ √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 ]

83 ½ √(2 + √(2 + √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2]

84 ½ √(2 + √(2 + √(2 + √(2 – √(2 – √(2]

85 ½ √(2 + √(2 + √(2 + √(2 + √(2 – √(2]

86 √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2]

87 ½ √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 ]

88 ½ √(2 + √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 + √(2 ]

89 ½ √(2 + √(2 + √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 ]

90° ½ 2 = 1

[Please care the sign of repetition has underlined. Closing brackets have collapsed by ‘]’ for easy.]

Other trigonometric ratios of angle A

To determination of Cosine of an angle (Cos A), we can either use the formula given above or just reversing the order of angle. Relation between Sine and Cosine is reversal in first quadrant of a circle. For example, Sin 1° is ½ √(2 - √(2 + √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 ] and this is Cos 89°. Similar the case for Sin 89°is ½ √(2 + √(2 + √(2 + √(2 + √(2 + √(2 + √(2 – √(2 – √(2 – √(2 + √(2 – √(2 – √(2 + √(2 – √(2 ], which is Cos 1°.

We can express above differently as if we know the chord of double angle (a),

half of a (a/2) is Sin A; otherwise, compute the chord ‘a’ and supplementary chord ‘b’ of respective angle (not double angle), Sin A is half of product of ‘a’ and ‘b’. The resulting solution is same in both cases.
half of b (b/2) is Cos A; which is just changes in the first sign from –ve to +ve and vice versa.

Using either above formula depending upon ‘a’ or ‘b’ or using classical trigonometric relations, we can compute other trigonometric relations for any integer angle (A).