1. Section 3.5 Trigonometric Functions Section 3.5 Trigonometric Functions

2. Objectives: 1.To define the trigonometric functions. 2.To determine periods, amplitudes, phase shifts, domains, and ranges for trigonometric functions. 3.To graph trigonometric functions. Objectives: 1.To define the trigonometric functions. 2.To determine periods, amplitudes, phase shifts, domains, and ranges for trigonometric functions. 3.To graph trigonometric functions.

3. Sine function f(x) = sin x where x is an angle measured in radians. Cosine function f(x) = cos x where x is an angle measured in radians. Sine function f(x) = sin x where x is an angle measured in radians. Cosine function f(x) = cos x where x is an angle measured in radians. DefinitionDefinition

4. Tangent function f(x) = tan x where x is an angle measured in radians. DefinitionDefinition

5. EXAMPLE 1 Find the set of ordered pairs described by y = sin x when the domain is {0.1745, 0.3840, 1.2392}. y = sin x y = sin 0.1745 ≈ 0.1736 y = sin 0.3840 ≈ 0.3746 y = sin 1.2392 ≈ 0.9455 {(0.1745, 0.1736), (0.3840, 0.3746), (1.2392, 0.9455)} y = sin x y = sin 0.1745 ≈ 0.1736 y = sin 0.3840 ≈ 0.3746 y = sin 1.2392 ≈ 0.9455 {(0.1745, 0.1736), (0.3840, 0.3746), (1.2392, 0.9455)}

6. 22 2 33 2  1 Sine function

7. Domain = {all real numbers} Range = [-1, 1] Period = 2  Sine function Domain = {all real numbers} Range = [-1, 1] Period = 2 

8. 22 2 33 2  1 Cosine function

9. Domain = {all real numbers} Range = [-1, 1] Period = 2  Cosine function Domain = {all real numbers} Range = [-1, 1] Period = 2 

10. Tangent function 1 vertical asymptotes vertical asymptotes 22 2 33 2 

11. Tangent function Domain = {  |   real numbers but    / 2 + k , k  integers} Range = {all real numbers} Period =  Tangent function Domain = {  |   real numbers but    / 2 + k , k  integers} Range = {all real numbers} Period = 

12. Amplitude The amplitude of a function is the absolute value of half the difference between the maximum and minimum values of the function. 2 2 m m m m A A 2 2 1 1 - - = = DefinitionDefinition

13. Sine function f(x) = sin x Range = [-1, 1] 2 2 m m m m A A 2 2 1 1 - - = = 2 2 (-1) 1 1 - - = = 2 2 2 2 = = = 1 f(x) = A sin x Amplitude = A A

14. EXAMPLE 2 Graph y = 2 sin .  22 -- -2  2 -2 y = sin 

15. EXAMPLE 2 Graph y = 2 sin . y = 2 sin  Amplitude = |2| = 2 y = 2 sin  Amplitude = |2| = 2  22 -- -2  2 -2 y = 2 sin 

16. The general form of the sine function is y = A sin (n  – b). The general form of the cosine function is y = A cos (n  – b). The general form of the tangent function is y = A tan (n  – b).

17. The period of a function is the length of a cycle for the values of a periodic function. The period of a sine or cosine function will be where n is the coefficient of the  in the equation. The period of a function is the length of a cycle for the values of a periodic function. The period of a sine or cosine function will be where n is the coefficient of the  in the equation. n n 22 22 DefinitionDefinition

18. DefinitionDefinition The period of a function is the length of a cycle for the values of a periodic function. The period of a tangent function will be where n is the coefficient of the  in the equation. The period of a function is the length of a cycle for the values of a periodic function. The period of a tangent function will be where n is the coefficient of the  in the equation.  |n|  |n|

19. EXAMPLE 3 Graph y = cos 2 .  22 -- -2  2 -2 y = cos 

20. EXAMPLE 3 Graph y = cos 2 . y = cos 2  period: 2  /2 =  y = cos 2  period: 2  /2 =   22 -- -2  2 -2 y = cos 2 

21. The phase shift is the horizontal translation of a trigonometric function. The phase shift of a trigonometric function in general form is. The phase shift is the horizontal translation of a trigonometric function. The phase shift of a trigonometric function in general form is. n n b b DefinitionDefinition

22. EXAMPLE 3 Graph y = cos (  –  ).  22 -- -2  2 -2 y = cos 

23. EXAMPLE 3 Graph y = cos (  –  ). y = cos (  –  ) phase shift:  /1 units to the right y = cos (  –  ) phase shift:  /1 units to the right  22 -- -2  2 -2 y = cos (  -  )

24. Amplitude stretches or shrinks the graph vertically (Amp. = |A|). Period stretches or shrinks the graph horizontally (Per. = 2  /n). Phase Shift moves the graph right or left (Phase shift = b/n). When y = A sin (n  - b) or y = A cos (n  - b) When y = A sin (n  - b) or y = A cos (n  - b)

25. 22 2 33 2  1 ) ) 2 2 sin( 2 2 1 1 y y Graph   - -   - - = = ) ) 2 2 sin( 2 2 1 1 y y Graph   - -   - - = =

26. 22 2 33 2  1 ) ) 2 2 sin( 2 2 1 1 y y Graph   - -   - - = =

27. 22 2 33 2  1 ) ) 2 2 sin( 2 2 1 1 y y Graph   - -   - - = = ) ) 2 2 sin( 2 2 1 1 y y Graph   - -   - - = =

28. 22 2 33 2  1 ) ) 2 2 sin( 2 2 1 1 y y Graph   - -   - - = = ) ) 2 2 sin( 2 2 1 1 y y Graph   - -   - - = =

29. 22 2 33 2  1 ) ) 2 2 sin( 2 2 1 1 y y Graph   - -   - - = = ) ) 2 2 sin( 2 2 1 1 y y Graph   - -   - - = =

30. Homework: pp. 140-141 Homework: pp. 140-141

31. ►A. Exercises 1.Find the set of ordered pairs described by y = cos x when the domain is {0.1745, 0.3840, 1.239, 0.7854, 1.396}. ►A. Exercises 1.Find the set of ordered pairs described by y = cos x when the domain is {0.1745, 0.3840, 1.239, 0.7854, 1.396}.

32. ►A. Exercises 3.Without graphing, give the amplitude, period, and phase shift for y = 3 cos . ►A. Exercises 3.Without graphing, give the amplitude, period, and phase shift for y = 3 cos .

33. ►A. Exercises 9.Without graphing, give the amplitude, period, and phase shift for y = 2 cos(2  +  ). ►A. Exercises 9.Without graphing, give the amplitude, period, and phase shift for y = 2 cos(2  +  ).

34. ►B. Exercises 11.Write the equation of the sine function with amplitude = 2, period = 4 , and phase shift =  / 2. ►B. Exercises 11.Write the equation of the sine function with amplitude = 2, period = 4 , and phase shift =  / 2.

35. ►B. Exercises 16.Give the zeros of f(x) = sin x. ►B. Exercises 16.Give the zeros of f(x) = sin x.

36. ►B. Exercises 17.Give the zeros of g(x) = cos x. ►B. Exercises 17.Give the zeros of g(x) = cos x.

37. ►B. Exercises 18.Give the zeros of h(x) = tan x. ►B. Exercises 18.Give the zeros of h(x) = tan x.

38. ►B. Exercises 19.Give the domain and range for the general function f(x) = A sin (nx - b). Assume A > 0. ►B. Exercises 19.Give the domain and range for the general function f(x) = A sin (nx - b). Assume A > 0.

39. ►B. Exercises 20.Give the domain and range for the general function h(x) = A tan (nx - b). Assume A > 0. ►B. Exercises 20.Give the domain and range for the general function h(x) = A tan (nx - b). Assume A > 0.

40. ►B. Exercises Graph. 22.y = 3 sin 2  ►B. Exercises Graph. 22.y = 3 sin 2 

41. ►B. Exercises Graph. 24.y = -tan ½  ►B. Exercises Graph. 24.y = -tan ½ 

42. ►B. Exercises Graph. 26.y = 4 cos (  +  ) ►B. Exercises Graph. 26.y = 4 cos (  +  )

43. ■ Cumulative Review 36.Write a polynomial function with roots of multiplicity 2 at -3 and at 4. ■ Cumulative Review 36.Write a polynomial function with roots of multiplicity 2 at -3 and at 4.

44. ■ Cumulative Review If P ( x ) is a quadratic function, and Q ( x ) is a cubic polynomial function, find the following. 37. The degree of P ( x ) + Q ( x ). ■ Cumulative Review If P ( x ) is a quadratic function, and Q ( x ) is a cubic polynomial function, find the following. 37. The degree of P ( x ) + Q ( x ).

45. ■ Cumulative Review If P ( x ) is a quadratic function, and Q ( x ) is a cubic polynomial function, find the following. 38. The degree of P ( x ) Q ( x ). ■ Cumulative Review If P ( x ) is a quadratic function, and Q ( x ) is a cubic polynomial function, find the following. 38. The degree of P ( x ) Q ( x ).

46. ■ Cumulative Review If P ( x ) is a quadratic function, and Q ( x ) is a cubic polynomial function, find the following. 39. The degree of Q ( x )/ P ( x ) if the remainder is zero and P ( x ) ≠ 0. ■ Cumulative Review If P ( x ) is a quadratic function, and Q ( x ) is a cubic polynomial function, find the following. 39. The degree of Q ( x )/ P ( x ) if the remainder is zero and P ( x ) ≠ 0.

47. ■ Cumulative Review 40. When is the sum of two quadratic functions not quadratic? ■ Cumulative Review 40. When is the sum of two quadratic functions not quadratic?