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Trigonometric Functions of Real Numbers - Masteral
- 1. Trigonometric Functions of RealNumbers Prepared by: Jeanny Rose M. Urbina Tarlac State University GRADUATE SCHOOL MATH 502
- 2. A. The UnitCircle B. Trigonometric Functions of Real Numbers C. Trigonometric Graphs of Sine, Cosine, and Tangent Function Topic Outline
- 3. “ A Look Backon H I S T O R Y
- 4. APOLLONIUS OF PERGA ◎He was known as the Great Geometer. ◎ One of his famous book is entitled Conics. ◎ He had taken geometric construction beyond that of Euclid’s Elements.
- 5. FIGURE 1 Excerpt fromEuclid’s Elements (Book III)
- 6. FIGURE 2 Excerpt fromApollonius’Tangencies
- 7. “ Since we cannotknow all that there is to be known, we ought to know a little about everything. - Apollonius of Perga
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- 9. A.The Unit Circle ◎Circleof radius 1 ◎Centered at (0,0) ◎Equation: + = 1
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- 14. + counterclockwise - clockwise UnitCircle: Degree
- 15. + counterclockwise - clockwise UnitCircle: Radian
- 16. + counterclockwise - clockwise UnitCircle: Coordinates
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- 18. B.Trigonometric Functions of RealNumbers sin ( cos ( x tan ( csc ( (y ≠ 0) cos ( (x tan ( (y
- 19. B.Trigonometric Functions of RealNumbers Here are some identities you need to know: Reciprocal Identities tan ( cot ( sec ( csc ( cot ( Pythagorean Identities
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- 22. Even and Oddfunctions The cos and sec functions are even functions; the rest other functions are odd functions. sin(-x) = -sin x cos(-x) = cos x tan(-x) = – tan x cot(-x) = -cot x csc(-x) = -csc x sec(-x) = sec x
- 23. Example 3: a. Sin(- b. Cos (-
- 24. C.Trigonometric Graphs ofSine and Cosine Function
- 25. Graphing the SineFunction y = sin Properties: 1. Domain (- 2. Range {y| -1 3. The maximum value is 1 and the minimum value is -1. 4. The sine function is a continuous function. It has no break in its graph 5. The sine function is periodic. Its period is 2 6. The amplitude is 1.
- 26. Graphing the CosineFunction y = cos Properties: 1. Domain (- 2. Range {y| -1 3. The maximum value is 1 and the minimum value is -1. 4. The cos function is a continuous function. It has no break in its graph 5. The cos function is periodic. Its period is 2 6. The amplitude is 1.
- 27. y = sin y= cos
- 28. y = sin y= 2sin
- 29. Example 1: y= 2sin 1. The period of 2sin is 2 2. The maximum value is 2 and the minimum value is -2. 3. The amplitude is 2.
- 30. Example 3: Sketch thegraph of y = coson the same cartesian coordinate plane. State the period, maximum and minimum values. Period: 2 Maximum value is 3 Minimum value is 1
- 31. Phase Shift andVertical Shift y = a sin (b ( + c)) + d Amplitude is a. Period is Phase Shift is c Vertical Shift is d. 1. y = cos( The mother graph y = cos must be shifted to units to the right
- 32. Phase Shift andVertical Shift y = a sin (b ( + c)) + d Amplitude is a. Period is Phase Shift is c Vertical Shift is d. 2. y = 2cos( Period: Phase shift: Amplitude: Vertical Shift: