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Principles of trigonometry animated

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Trigonometry
History of Trigonometry
Principles of Trigonometry
Classical Trigonometry
Modern Trigonometry
Trigonometric Functions

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Principles of trigonometry animated

1. 1. Trigonometry History of Trigonometry Principles of Trigonometry
2. 2. Principles of Trigonometry Trigonometric Function Plane Trigonometry Spherical Trigonometry Analytic Trigonometry Coordinates and transformation of coordinates
3. 3. Trigonometric Functions • Trigonometric Function of an Angle • Tables of Natural Function PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
4. 4. Trigonometric functions • A somewhat more general concept of angle is required for trigonometry than for geometry. • An angle A with vertex at V, the initial side of which is VP and the terminal side of which is VQ, is indicated in the figure by the solid circular arc. PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY •
5. 5. Trigonometric functions • This angle is generated by the continuous counterclockwise rotation of a line segment about the point V from the position VP to the position VQ. • A second angle A′ with the same initial and terminal sides, indicated in the figure by the broken circular arc, is generated by the clockwise rotation of the line segment from the position VP to the position VQ. PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
6. 6. Trigonometric functions • Angles are considered positive when generated by counterclockwise rotations, negative when generated by clockwise rotations. • The positive angle A and the negative angle A′ in the figure are generated by less than one complete rotation of the line segment about the point V. PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
7. 7. Trigonometric functions • All other positive and negative angles with the same initial and terminal sides are obtained by rotating the line segment one or more complete turns before coming to rest at VQ. PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
8. 8. Trigonometric functions • Numerical values can be assigned to angles by selecting a unit of measure. • The most common units are the degree and the radian. • There are 360° in a complete revolution, with each degree further divided into 60′ (minutes) and each minute divided into 60″ (seconds). PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
9. 9. Trigonometric functions • In theoretical work, the radian is the most convenient unit. • It is the angle at the centre of a circle that intercepts an arc equal in length to the radius; simply put, there are 2π radians in one complete revolution. • From these definitions, it follows that 1° = π/180 radians. PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
10. 10. Trigonometric functions • Equal angles are angles with the same measure; i.e., they have the same sign and the same number of degrees. • Any angle −A has the same number of degrees as A but is of opposite sign. • Its measure, therefore, is the negative of the measure of A. PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
11. 11. Trigonometric functions • If two angles, A and B, have the initial sides VP and VQ and the terminal sides VQ and VR, respectively, then the angle A + B has the initial and terminal sides VP and VR (see the figure). • The angle A + B is called the sum of the angles A and B, and its relation to A and B when A is positive and B is positive or negative is illustrated in the figure. PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY •
12. 12. Trigonometric functions • The sum A + B is the angle the measure of which is the algebraic sum of the measures of A and B. The difference A − B is the sum of A and −B. • Thus, all angles coterminal with angle A (i.e., with the same initial and terminal sides as angle A) are given by A ± 360n, in which 360n is an angle of n complete revolutions. • The angles (180 − A) and (90 − A) are the supplement and complement of angle A, respectively. PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
13. 13. Trigonometric functions of an angle • PRINCIPLES OF TRIGONOMETRY TRIGONOMETRY FUNCTIONS •
14. 14. Trigonometric functions of an angle • • PRINCIPLES OF TRIGONOMETRY TRIGONOMETRY FUNCTIONS •
15. 15. Trigonometric functions of an angle • PRINCIPLES OF TRIGONOMETRY TRIGONOMETRY FUNCTIONS
16. 16. Trigonometric functions of an angle • • PRINCIPLES OF TRIGONOMETRY TRIGONOMETRY FUNCTIONS
17. 17. Trigonometric functions of an angle • PRINCIPLES OF TRIGONOMETRY TRIGONOMETRY FUNCTIONS
18. 18. Trigonometric functions of an angle • • • PRINCIPLES OF TRIGONOMETRY TRIGONOMETRY FUNCTIONS
19. 19. Trigonometric functions of an angle • • PRINCIPLES OF TRIGONOMETRY TRIGONOMETRY FUNCTIONS
20. 20. Trigonometric functions of an angle • • • PRINCIPLES OF TRIGONOMETRY TRIGONOMETRY FUNCTIONS
21. 21. Trigonometric Functions • Trigonometric Function of an Angle • Tables of Natural Function PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
22. 22. Tables of Natural Functions • • PRINCIPLES OF TRIGONOMETRY TRIGONOMETRY FUNCTIONS
23. 23. Tables of Natural Functions • ′ • PRINCIPLES OF TRIGONOMETRY TRIGONOMETRY FUNCTIONS
24. 24. Tables of Natural Functions • • PRINCIPLES OF TRIGONOMETRY TRIGONOMETRY FUNCTIONS
25. 25. Tables of Natural Functions • • PRINCIPLES OF TRIGONOMETRY TRIGONOMETRY FUNCTIONS
26. 26. Principles of Trigonometry Trigonometric Function Plane Trigonometry Spherical Trigonometry Analytic Trigonometry Coordinates and transformation of coordinates
27. 27. Plane Trigonometry • • • PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
28. 28. Plane Trigonometry • • PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY • In addition to the angles (A, B, C) and sides (a, b, c), one of the three heights of the triangle (h) is included by drawing the line segment from one of the triangle's vertices (in this case C) that is Standard lettering of a triangle
29. 29. Plane Trigonometry • PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
30. 30. Plane Trigonometry • • PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
31. 31. Plane Trigonometry • • PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
32. 32. Plane Trigonometry • • PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
33. 33. Principles of Trigonometry Trigonometric Function Plane Trigonometry Spherical Trigonometry Analytic Trigonometry Coordinates and transformation of coordinates
34. 34. Spherical Trigonometry • • PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
35. 35. Spherical Trigonometry • • PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
36. 36. Spherical Trigonometry • • PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
37. 37. Spherical Trigonometry • • α β γ PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
38. 38. Spherical Trigonometry • α β γ PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
39. 39. Spherical Trigonometry • • PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
40. 40. Spherical Trigonometry • PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
41. 41. Spherical Trigonometry • • PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
42. 42. Principles of Trigonometry Trigonometric Function Plane Trigonometry Spherical Trigonometry Analytic Trigonometry Coordinates and transformation of coordinates
43. 43. Analytic Trigonometry • PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
44. 44. Analytic Trigonometry • • • PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
45. 45. Analytic Trigonometry • PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
46. 46. Analytic Trigonometry • • PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
47. 47. Analytic Trigonometry • PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
48. 48. Analytic Trigonometry • • PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
49. 49. Analytic Trigonometry • • • PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
50. 50. Principles of Trigonometry Trigonometric Function Plane Trigonometry Spherical Trigonometry Analytic Trigonometry Coordinates and transformation of coordinates
51. 51. Coordinates and Transformation of Coordinates • Polar Coordinates • Transformation of Coordinates PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
52. 52. Polar Coordinates • θ θ • PRINCIPLES OF TRIGONOMETRY COORDINATE AND TRANSFORMATION OF COORDINATE • θ
53. 53. Polar Coordinates • θ θ θ θ • θ θ θ PRINCIPLES OF TRIGONOMETRY COORDINATE AND TRANSFORMATION OF COORDINATE
54. 54. Polar Coordinates • θ θ • θ PRINCIPLES OF TRIGONOMETRY COORDINATE AND TRANSFORMATION OF COORDINATE
55. 55. Polar Coordinates • θ ϕ θ θ ϕ θ ϕ • θ ϕ PRINCIPLES OF TRIGONOMETRY COORDINATE AND TRANSFORMATION OF COORDINATE
56. 56. Coordinates and Transformation of Coordinates • Polar Coordinates • Transformation of Coordinates PRINCIPLES OF TRIGONOMETRYTRIGONOMETRY
57. 57. Transformation of Coordinates • • PRINCIPLES OF TRIGONOMETRY COORDINATE AND TRANSFORMATION OF COORDINATE
58. 58. Transformation of Coordinates • θ θ PRINCIPLES OF TRIGONOMETRY COORDINATE AND TRANSFORMATION OF COORDINATE
59. 59. Transformation of Coordinates • • PRINCIPLES OF TRIGONOMETRY COORDINATE AND TRANSFORMATION OF COORDINATE
60. 60. Transformation of Coordinates • ′ ′ ′ ′ • PRINCIPLES OF TRIGONOMETRY COORDINATE AND TRANSFORMATION OF COORDINATE
61. 61. Transformation of Coordinates • ′ ′ ′ ′ PRINCIPLES OF TRIGONOMETRY COORDINATE AND TRANSFORMATION OF COORDINATE
62. 62. Transformation of Coordinates • ϕ ′ ′ • ′ ϕ ′ ϕ ′ ϕ ′ ϕ PRINCIPLES OF TRIGONOMETRY COORDINATE AND TRANSFORMATION OF COORDINATE • ϕ
63. 63. Transformation of Coordinates • ϕ ′ ′ • PRINCIPLES OF TRIGONOMETRY COORDINATE AND TRANSFORMATION OF COORDINATE