1. Graphs of Trig. Functions

2. Graphs of Trig. Functions The graph of y=sin(x)

3. Graphs of Trig. Functions The graph of y=sin(x)

4. Graphs of Trig. Functions The graph of y=sin(x)

5. Graphs of Trig. Functions The graph of y=sin(x)

6. Graphs of Trig. Functions The graph of y=sin(x)

7. Graphs of Trig. Functions The graph of y=sin(x)

8. Graphs of Trig. Functions The graph of y=sin(x)

9. Graphs of Trig. Functions The graph of y=sin(x)

10. Graphs of Trig. Functions The graph of y=cos(x)

11. Graphs of Trig. Functions The graph of y=cos(x)

12. Graphs of Trig. Functions The graph of y=cos(x) 90o 180o 0o

13. Graphs of Trig. Functions The graph of y=cos(x) 90o 180o 270o 360o 0o

14. Graphs of Trig. Functions The graph of y=cos(x)

15. Graphs of Trig. Functions The graph of y=cos(x) The graph of y=sin(x)

16. Periodic Functions

17. Periodic Functions Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x.

18. Periodic Functions Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x. The graph of a periodic function:  Frank Ma 2006 p

19. Periodic Functions Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x. The smallest number p>0 such that f(x) = f(x+p) is called the period of f(x). The graph of a periodic function:  Frank Ma 2006 p

20. Periodic Functions Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x. The smallest number p>0 such that f(x) = f(x+p) is called the period of f(x). The graph of a periodic function:  Frank Ma 2006 p one period

21. Periodic Functions Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x. The smallest number p>0 such that f(x) = f(x+p) is called the period of f(x). Over every interval of length p, the graph of a periodic function repeats itself. The graph of a periodic function:  Frank Ma 2006 p one period

22. Periodic Functions Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x. The smallest number p>0 such that f(x) = f(x+p) is called the period of f(x). Over every interval of length p, the graph of a periodic function repeats itself. The graph of a periodic function:  Frank Ma 2006 p x x+p p one period

23. Periodic Functions Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x. The smallest number p>0 such that f(x) = f(x+p) is called the period of f(x). Over every interval of length p, the graph of a periodic function repeats itself. The graph of a periodic function:  Frank Ma 2006 p x x+p p one period f(x) = f(x+p) for all x

24. Periodic Functions sin(x) and cos(x) are periodic with period p=2π.

25. Periodic Functions sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period. For y=cos(x):

26. Periodic Functions sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period. For y=cos(x):

27. Periodic Functions sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period. For y=cos(x):

28. Periodic Functions sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period. For y=cos(x):

29. Periodic Functions sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period. For y=cos(x):

30. Periodic Functions sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period. For y=sin(x): 0

31. Periodic Functions sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period. For y=sin(x): 0

32. Periodic Functions sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period. For y=sin(x): 0

33. Periodic Functions sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period. For y=sin(x): 0

34. Periodic Functions The basic period for: y=sin(x)

35. Periodic Functions The basic period for: y=sin(x) y=cos(x) 1 -1

36. Periodic Functions The basic period for: y=sin(x) y=cos(x) 1 -1 The Graph of Tangent

37. Periodic Functions The basic period for: y=sin(x) y=cos(x) 1 -1 The Graph of Tangent The function tan(x) is not defined when cos(x) is 0, i.e. when x = ±π/2, ±3π/2, ±5π/2, ..

38. Periodic Functions The basic period for: y=sin(x) y=cos(x) 1 -1 The Graph of Tangent The function tan(x) is not defined when cos(x) is 0, i.e. when x = ±π/2, ±3π/2, ±5π/2, ..  Frank Ma 2006 As the values of x goes from 0 to π/2 the values of sin(x) goes from 0 to 1,

39. Periodic Functions The basic period for: y=sin(x) y=cos(x) 1 -1 The Graph of Tangent The function tan(x) is not defined when cos(x) is 0, i.e. when x = ±π/2, ±3π/2, ±5π/2, ..  Frank Ma 2006 As the values of x goes from 0 to π/2 the values of sin(x) goes from 0 to 1, but the values of cos(x) goes from 1 to 0. So tan(x) goes from 0 to +∞.

40. The Graph of Tangent Since tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x).

41. The Graph of Tangent Since tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically, x π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 tan(x)

42. The Graph of Tangent Since tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically, x π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 ∞ 0 1/3 1 3 tan(x)

43. The Graph of Tangent Since tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically, x π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan(x)

44. The Graph of Tangent Since tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically, x π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan(x) 0 π/2 -π/2

45. The Graph of Tangent Since tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically, x π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan(x) The same pattern repeats itself every πinterval. 0 π/2 -π/2

46. The Graph of Tangent Since tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically, x π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan(x) The same pattern repeats itself every πinterval. In other words, y = tan(x) is a periodic function with period πas shown in the graph. 0 π/2 -π/2

47. The Graph of Tangent Since tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically, x π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan(x) The same pattern repeats itself every πinterval. In other words, y = tan(x) is a periodic function with period πas shown in the graph. π -π 0 π/2 -π/2 3π/2 -3π/2 y = tan(x)

48. The Graph of Inverse Trig-Functions Recalll that for y = cos-1(x), then 0 < y < π.

49. The Graph of Inverse Trig-Functions Recalll that for y = cos-1(x), then 0 < y < π. It's graph may be plotted in the similar manner.

50. The Graph of Inverse Trig-Functions Recalll that for y = cos-1(x), then 0 < y < π. It's graph may be plotted in the similar manner. (-1, π) (0, π/2) (1, 0) -1 1 The graph of y = cos-1(x)

51. The Graph of Inverse Trig-Functions Recalll that for y = cos-1(x), then 0 < y < π. It's graph may be plotted in the similar manner. (-1, π) (0, π/2) (1, 0) -1 1 The graph of y = cos-1(x) Remark: The above graphs of y = sin-1(x) and y = cos-1(x) are the complete graphs (i.e. that's all there is).

52. The Graph of Inverse Trig-Functions The domain of y = tan-1(x) is all real numbers and the output y is restricted to -π/2 < y < π.

53. The Graph of Inverse Trig-Functions The domain of y = tan-1(x) is all real numbers and the output y is restricted to -π/2 < y < π. x ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan-1(x)

54. The Graph of Inverse Trig-Functions The domain of y = tan-1(x) is all real numbers and the output y is restricted to -π/2 < y < π. x ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan-1(x) π/6 0 π/4 π/3 π/2

55. The Graph of Inverse Trig-Functions The domain of y = tan-1(x) is all real numbers and the output y is restricted to -π/2 < y < π. x ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan-1(x) π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2

56. The Graph of Inverse Trig-Functions The domain of y = tan-1(x) is all real numbers and the output y is restricted to -π/2 < y < π. x ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan-1(x) π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 Here is the graph of y = tan-1(x) y = π/2 (1,π/4) (0,0) (-1,-π/4) y = -π/2

57. The Graph of Inverse Trig-Functions The domain of y = tan-1(x) is all real numbers and the output y is restricted to -π/2 < y < π. x ∞ 0 1/3 1 3 -1/3 -1 -3 -∞ tan-1(x) π/6 0 π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 Here is the graph of y = tan-1(x) y = π/2 (1,π/4) (0,0) (-1,-π/4) y = -π/2 Remark: y =tan-1(x) has two horizontal asymptoes.