Appendix B: Trigonometric Functions

Trigonometric functions are based on relationships present between the sides and angles of triangles. Many functions are derived from a right triangle—a triangle containing a right (90˚) angle. Consider the right triangle below with sides A, B, and C, and angles α, β, and γ.

Side C, which is the longest side and the side opposite the right angle, is known as the hypotenuse of the triangle.

A commonly used trigonometric relationship for right triangles is the Pythagorean theorem. The Pythagorean theorem is an expression of the relationship between the hypotenuse and the other two sides of a right triangle:

The sum of the squares of the lengths of the two sides of a right triangle is equal to the square of the length of the hypotenuse.

Using the sides of the labeled triangle yields the following:

Suppose that sides A and B are 3 and 4 units long, respectively. The Pythagorean theorem can be used to solve for the length of side C:

Three trigonometric relationships are based on the ratios of the lengths of the sides of a right triangle. The sine (abbreviated sin) of an angle is defined as the ratio of the length of the side of the triangle opposite the angle to the length of the hypotenuse. Using the labeled triangle yields the following:

With A = 3, B = 4, and C = 5:

The cosine (abbreviated cos) of an angle is defined as the ratio of the length of the side of the triangle adjacent to the angle to the length of the hypotenuse. Using the labeled triangle yields the following:

With A = 3, B = 4, and C = 5:

The third function, the tangent (abbreviated tan) of an angle, is defined as the ratio of the length of the side of the triangle opposite the angle to that of the side adjacent to the angle. Using the labeled triangle yields the following:

With A = 3, B = 4, and C = 5:

Two useful trigonometric relationships are applicable to all triangles. The first is known as the law of sines:

The ratio between the length of any side of a triangle and the angle opposite that side is equal to the ratio between the length of any other side of the triangle and the angle opposite that side.

With respect to the labeled triangle, this may be stated as the following:

A second trigonometric relationship applicable to all triangles is the law of cosines:

The square of the length of any side of a triangle is equal to the sum of the squares of the lengths of the other two sides of the triangle minus two times the product of the lengths of the other two sides and the cosine of the angle opposite the original side.

This relationship yields the following for each of the sides of the labeled triangle:

A table of the values of the basic trigonometric functions follows.

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Table of Basic Trigonometric Function Values

Deg

Sin

Cos

Tan

Deg

Sin

Cos

Tan

00

.0000

1.0000

.0000

—

01

.0175

.9998

.0175

46

.7193

.6947

1.0355

02

.0349

.9994

.0349

47

.7314

.6820

1.0723

03

.0523

.9986

.0524

48

.7431

.6691

1.1106

04

.0698

.9976

.0699

49

.7547

.6561

1.1504

05

.0872

.9962

.0875

50

.7660

.6428

1.1918

06

.1045

.9945

.1051

51

.7771

.6293

1.2349

07

.1219

.9925

.1228

52

.7880

.6157

1.2799

08

.1392

.9903

.1405

53

.7986

.6018

1.3270

09

.1564

.9877

.1584

54

.8090

.5878

1.3764

10

.1736

.9848

.1763

55

.8192

.5736

1.4281

11

.1908

.9816

.1944

56

.8290

.5592

1.4826

12

.2079

.9781

.2126

57

.8387

.5446

1.5399

13

.2250

.9744

.2309

58

.8480

.5299

1.6003

14

.2419

.9703

.2493

59

.8572

.5150

1.6643

15

.2588

.9659

.2679

60

.8660

.5000

1.7321

16

.2756

.9613

.2867

61

.8746

.4848

1.8040

17

.2924

.9563

.3057

62

.8829

.4695

1.8807

18

.3090

.9511

.3249

63

.8910

.4540

1.9626

19

.3256

.9455

.3443

64

.8988

.4384

2.0503

20

.3420

.9397

.3640

65

.9063

.4226

2.1445

21

.3584

.9336

.3839

66

.9135

.4067

2.2460

22

.3746

.9272

.4040

67

.9205

.3907

2.3559

23

.3907

.9205

.4245

68

.9279

.3746

2.4751

24

.4067

.9135

.4452

69

.9336

.3584

2.6051

25

.4226

.9063

.4663

70

.9397

.3420

2.7475

26

.4384

.8988

.4877

71

.9456

.3256

2.9042

27

.4540

.8910

.5095

72

.9511

.3090

3.0779

28

.4695

.8829

.5317

73

.9563

.2924

3.2709

29

.4848

.8746

.5543

74

.9613

.2756

3.4874

30

.5000

.8660

.5774

75

.96593

.2588

3.7321

31

.5150

.8572

.6009

76

.9703

.2419

4.0108

32

.5299

.8480

.6249

77

.9744

.2250

4.3315

33

.5446

.8387

.6494

78

.9781

.2079

4.7046

34

.5592

.8290

.6745

79

.9816

.1908

5.1446

35

.5736

.8192

.7002

80

.9848

.1736

5.6713

36

.5878

.8090

.7265

81

.9877

.1564

6.3138

37

.6018

.7986

.7536

82

.9903

.1391

7.1154

38

.6157

.7880

.7813

83

.9925

.1219

8.1443

39

.6293

.7771

.8098

84

.9945

.1045

9.5144

40

.6428

.7660

.8391

85

.99625

.0872

11.4301

41

.6561

.7547

.8693

86

.9976

.0698

14.3007

42

.6691

.7431

.9004

87

.99866

.05239

19.0811

43

.6820

.7314

.9325

88

.9994

.0349

28.6363

44

.6947

.7193

.9657

89

.9998

.0175

57.2900

45

.7071

1.0000

90

1.0000

.0000

Infinity

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