1. P.5 Trigonometric Function.

2. A ray, or half-line, is that portion of a line that starts at a point V on the line and extends indefinitely in one direction. The starting point V of a ray is called its vertex. VRay

3. If two lines are drawn with a common vertex, they form an angle. One of the rays of an angle is called the initial side and the other the terminal side. Vertex Initial Side Terminal side Counterclockwise rotation Positive Angle

4. Vertex Initial Side Terminal side Clockwise rotation Negative Angle Vertex Initial Side Terminal side Counterclockwise rotation Positive Angle

5. Initial side Vertex Terminal side x y

6. Angles are commonly measured in either Degrees or Radians The angle formed by rotating the initial side exactly once in the counterclockwise direction until it coincides with itself (1 revolution) is said to measure 360 degrees, abbreviated Initial side Terminal side Vertex

7. Initial side Terminal side Vertex

8. Initial sideTerminal sideVertex

9. x y Initial sideVertex Terminal side

11. 180° 90°   (a) Straight angle (b) Right angle (c) Acute angle (d) Obtuse angle Angles

12. Consider a circle of radius r. Construct an angle whose vertex is at the center of this circle, called the central angle, and whose rays subtend an arc on the circle whose length is r. The measure of such an angle is 1 radian. r 1 radian

13. For a circle of radius r, a central angle of radians subtends an arc whose length s is Find the length of the arc of a circle of radius 4 meters subtended by a central angle of 2 radians.

16. The unit circle is a circle whose radius is 1 and whose center is at the origin. Since r = 1: becomes

17. (0, 1) (-1, 0) (0, -1) (1, 0) y x

18. (0, 1) (-1, 0) (0, -1) (1, 0) y x P = (a, b)

19. r O  r Radian Measure 5-3-47 s

20. 1.To form a reference triangle for , draw a perpendicular from a point P(a, b) on the terminal side of  to the horizontal axis. 2. The reference angle  is the acute angle (always taken positive) between the terminal side of  and the horizontal axis. a b   a b P(a,b) Reference Triangle and Reference Angle 5-4-50

21. 3 1 2 30° 60° (  /6) (  /3) 2 1 1 45° ° (  /4) (  30  — 60  and 45  Special Triangles 5-4-51

22.  a b a b r a b  r a b r  a b a b If  is an arbitrary angle in standard position in a rectangular coordinate system and P(a, b) is a point r units from the origin on the terminal side of , then: Trigonometric Functions with Angle Domains Alternate Form 5-4-49 tan  =, a  0 P(a, b)

23. If x is a real number and (a, b) are the coordinates of the circular point W(x), then: Circular Functions 5-2-45

24. (a,b) W(x) (1, 0) a b x units arc length xrad  Trigonometric Circular Function sin  = x cos  = x tan  = tan x csc  = x sec  = x cot  = cot x If  is an angle with radian measure x, then the value of each trigonometric function at    is given by its value at the real number x. Trigonometric Functions with Angle Domains 5-4-48

25. Right Triangle Ratios Hypotenuse Opposite Adjacent  0° <  < 90° 5-5-52

26. Reciprocal Identities Quotient Identities

27. x y (a, b) a 0 r a > 0, b > 0, r > 0 a 0, r > 0 a > 0, b 0

28. I (+, +) All positive x y Cast Rule (C ) (A ) ( T ) (S )

29. Example :Find the value of each of the six trigonometric functions of the angle Adjacent 12 13 c = Hypotenuse = 13 b = Opposite = 12

33. (0, 1) (-1, 0) (0, -1) (1, 0) y x P = (a, b)

34. Find the exact value of the six trigonometric functions of 45 degrees. b = 1 a = 1

36. Find the exact value of the six trigonometric functions of 30 and 60 degrees. 2 a b a 2 2a = 2 so a = 1

40. If there is a smallest such number p, this smallest value is called the (fundamental) period of f.

41. The Graph of y = sin x x y

42. (0, 0)

43. Characteristics of the Sine Function 1. The domain is the set of all real numbers. 2. The range consists of all real numbers from -1 to 1, inclusive. 3. The sine function is an odd function (symmetric with respect to the origin).

44. Characteristics of the Sine Function

45. Begin with the basic sine function: (0, 0)

49. The Graph of y = cos x x y

50. (0, 1)

51. Characteristics of the Cosine Function 1. The domain is the set of all real numbers. 2. The range consists of all real numbers from -1 to 1, inclusive. 3. The cosine function is an even function (symmetric with respect to the y-axis).

52. Characteristics of the Cosine Function

54. The graphs of the sine and cosine functions are called sinusoidal graphs.

56. 1

57. 2 -2

58. Find an equation for the graph. 4 -4 1 2 3

60. The Graph of y = tan x x y

61. (0, 0)

64. Characteristics of the Tangent Function 1. The domain is the set of all real numbers, except odd multiples of 2. The range consists of all real numbers. 3. The tangent function is an odd function (symmetric with respect to the origin).

65. Characteristics of the Tangent Function

66. The graphs of the other three trig functions can be obtained from the graphs of their respective reciprocal functions. For example:

70. or

72. (0,0)

73. 1 2 3

74. Figure 39: Graphs of the (a) cosine, (b) sine, (c) tangent, (d) secant, (e) cosecant, (f) cotangent functions using and radian measure.

75. Continued.

77. Figure 45: The general sine curve y = A sin [(2  /B)(x – C)]+D, shown for A, B, C, and D positive. (Example 3)

78. Notes: The domain of the sine function is the set of all real numbers. 1- The domain of the cosine function is the set of all real numbers. 2-The domain of the tangent function is the set of all real numbers except odd multiples of 3-The domain of the secant function is the set of all real numbers except odd multiples of

79. 4-The domain of the cotangent function is the set of all real numbers except integral multiples of 5- The domain of the cosecant function is the set of all real numbers except integral multiples of

80. Let P = (a, b) be the point on the unit circle that corresponds to the angle. Then, -1 < a < 1 and -1 < b < 1. Range of the Trigonometric Functions

82. Periodic Properties

84. Even-Odd Properties

86. c b a

89. Sum and Difference Formulas for Cosines

91. Cofunction Identities Sum and Difference Formulas for Sines

94. 3

96. Sum and Difference Formulas for Tangents

98. 5 12 13 4 -3 5

100. Double-Angle Formulas

101. 13 5 -12

103. Double-angle Formula for Tangent Variations of the Double Angle Formula ’ s

105. Half-Angle Formulas

108. Half-Angle Formulas for

109. Product-to-Sum Formulas

112. Sum-to-Product Formulas

114. Establish the identity:

115. The End Good Luck

116. Summery 6-1-64

117. Sum Identities sin( x + y ) = sin x cos y + x sin y cos( x + y ) =cos x y – sin x sin y tan( x + y ) = tan x + y 1 – x y Difference Identities sin( x – y ) = sin x cos y – x sin y cos( x – y ) =cos x y + sin x sin y tan( x – y ) = tan x – y 1 + x y Cofunction Identities       Replace  2 with 90° if x is in degrees. cos        2 – x = sin x sin        2 – x =cos x tan        2 – x = cot x Sum, Difference, and Cofunction Identities 6-2-66

118. Double-Angle and Half-Angle Identities 6-3-67

119. Product-Sum Identities Sum-Product Identities Copyright © 2000 by the McGraw-Hill Companies, Inc. 6-4-68