t5 graphs of trig functions and inverse trig functions

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t5 graphs of trig functions and inverse trig functions

  1. 1. Graphs of Trig. Functions<br />
  2. 2. Graphs of Trig. Functions<br />The graph of y=sin(x)<br />
  3. 3. Graphs of Trig. Functions<br />The graph of y=sin(x)<br />
  4. 4. Graphs of Trig. Functions<br />The graph of y=sin(x)<br />
  5. 5. Graphs of Trig. Functions<br />The graph of y=sin(x)<br />
  6. 6. Graphs of Trig. Functions<br />The graph of y=sin(x)<br />
  7. 7. Graphs of Trig. Functions<br />The graph of y=sin(x)<br />
  8. 8. Graphs of Trig. Functions<br />The graph of y=sin(x)<br />
  9. 9. Graphs of Trig. Functions<br />The graph of y=sin(x)<br />
  10. 10. Graphs of Trig. Functions<br />The graph of y=cos(x)<br />
  11. 11. Graphs of Trig. Functions<br />The graph of y=cos(x)<br />
  12. 12. Graphs of Trig. Functions<br />The graph of y=cos(x)<br />90o<br />180o<br />0o<br />
  13. 13. Graphs of Trig. Functions<br />The graph of y=cos(x)<br />90o<br />180o<br />270o<br />360o<br />0o<br />
  14. 14. Graphs of Trig. Functions<br />The graph of y=cos(x)<br />
  15. 15. Graphs of Trig. Functions<br />The graph of y=cos(x)<br />The graph of y=sin(x)<br />
  16. 16. Periodic Functions<br />
  17. 17. Periodic Functions<br />Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that <br />f(x) = f(x+b) for all x.<br />
  18. 18. Periodic Functions<br />Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that <br />f(x) = f(x+b) for all x.<br />The graph of a periodic function:<br /> Frank Ma<br />2006<br />p<br />
  19. 19. Periodic Functions<br />Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that <br />f(x) = f(x+b) for all x.<br />The smallest number p>0 such that f(x) = f(x+p) <br />is called the period of f(x). <br />The graph of a periodic function:<br /> Frank Ma<br />2006<br />p<br />
  20. 20. Periodic Functions<br />Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that <br />f(x) = f(x+b) for all x.<br />The smallest number p>0 such that f(x) = f(x+p) <br />is called the period of f(x). <br />The graph of a periodic function:<br /> Frank Ma<br />2006<br />p<br />one period<br />
  21. 21. Periodic Functions<br />Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that <br />f(x) = f(x+b) for all x.<br />The smallest number p>0 such that f(x) = f(x+p) <br />is called the period of f(x). <br />Over every interval of length p, the graph of a periodic function repeats itself. <br />The graph of a periodic function:<br /> Frank Ma<br />2006<br />p<br />one period<br />
  22. 22. Periodic Functions<br />Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that <br />f(x) = f(x+b) for all x.<br />The smallest number p>0 such that f(x) = f(x+p) <br />is called the period of f(x). <br />Over every interval of length p, the graph of a periodic function repeats itself. <br />The graph of a periodic function:<br /> Frank Ma<br />2006<br />p<br />x<br />x+p<br />p<br />one period<br />
  23. 23. Periodic Functions<br />Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that <br />f(x) = f(x+b) for all x.<br />The smallest number p>0 such that f(x) = f(x+p) <br />is called the period of f(x). <br />Over every interval of length p, the graph of a periodic function repeats itself. <br />The graph of a periodic function:<br /> Frank Ma<br />2006<br />p<br />x<br />x+p<br />p<br />one period<br />f(x) = f(x+p) for all x<br />
  24. 24. Periodic Functions<br />sin(x) and cos(x) are periodic with period p=2π. <br />
  25. 25. Periodic Functions<br />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.<br />For y=cos(x):<br />
  26. 26. Periodic Functions<br />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.<br />For y=cos(x):<br />
  27. 27. Periodic Functions<br />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.<br />For y=cos(x):<br />
  28. 28. Periodic Functions<br />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.<br />For y=cos(x):<br />
  29. 29. Periodic Functions<br />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.<br />For y=cos(x):<br />
  30. 30. Periodic Functions<br />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.<br />For y=sin(x):<br />0<br />
  31. 31. Periodic Functions<br />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.<br />For y=sin(x):<br />0<br />
  32. 32. Periodic Functions<br />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.<br />For y=sin(x):<br />0<br />
  33. 33. Periodic Functions<br />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.<br />For y=sin(x):<br />0<br />
  34. 34. Periodic Functions<br />The basic period for:<br /> y=sin(x) <br />
  35. 35. Periodic Functions<br />The basic period for:<br /> y=sin(x) y=cos(x)<br />1<br />-1<br />
  36. 36. Periodic Functions<br />The basic period for:<br /> y=sin(x) y=cos(x)<br />1<br />-1<br />The Graph of Tangent<br />
  37. 37. Periodic Functions<br />The basic period for:<br /> y=sin(x) y=cos(x)<br />1<br />-1<br />The Graph of Tangent<br />The function tan(x) is not defined when cos(x) is 0,<br />i.e. when x = ±π/2, ±3π/2, ±5π/2, ..<br />
  38. 38. Periodic Functions<br />The basic period for:<br /> y=sin(x) y=cos(x)<br />1<br />-1<br />The Graph of Tangent<br />The function tan(x) is not defined when cos(x) is 0,<br />i.e. when x = ±π/2, ±3π/2, ±5π/2, ..<br /> Frank Ma<br />2006<br />As the values of x goes from 0 to π/2 the values of sin(x) goes from 0 to 1, <br />
  39. 39. Periodic Functions<br />The basic period for:<br /> y=sin(x) y=cos(x)<br />1<br />-1<br />The Graph of Tangent<br />The function tan(x) is not defined when cos(x) is 0,<br />i.e. when x = ±π/2, ±3π/2, ±5π/2, ..<br /> Frank Ma<br />2006<br />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 +∞. <br />
  40. 40. The Graph of Tangent<br />Since tan(x) is odd, so as the values of x goes from <br />0 to -π/2 we get the corresonding negative outputs <br />for tan(x). <br />
  41. 41. The Graph of Tangent<br />Since tan(x) is odd, so as the values of x goes from <br />0 to -π/2 we get the corresonding negative outputs <br />for tan(x). Specifically,<br />x<br />π/6 <br />0<br />π/4 <br />π/3 <br />-π/2 <br />-π/6 <br />-π/4 <br />-π/3 <br />π/2 <br />tan(x)<br />
  42. 42. The Graph of Tangent<br />Since tan(x) is odd, so as the values of x goes from <br />0 to -π/2 we get the corresonding negative outputs <br />for tan(x). Specifically,<br />x<br />π/6 <br />0<br />π/4 <br />π/3 <br />-π/2 <br />-π/6 <br />-π/4 <br />-π/3 <br />π/2 <br />∞<br />0<br />1/3<br />1<br />3<br />tan(x)<br />
  43. 43. The Graph of Tangent<br />Since tan(x) is odd, so as the values of x goes from <br />0 to -π/2 we get the corresonding negative outputs <br />for tan(x). Specifically,<br />x<br />π/6 <br />0<br />π/4 <br />π/3 <br />-π/2 <br />-π/6 <br />-π/4 <br />-π/3 <br />π/2 <br />∞<br />0<br />1/3<br />1<br />3<br />-1/3<br />-1<br />-3<br />-∞<br />tan(x)<br />
  44. 44. The Graph of Tangent<br />Since tan(x) is odd, so as the values of x goes from <br />0 to -π/2 we get the corresonding negative outputs <br />for tan(x). Specifically,<br />x<br />π/6 <br />0<br />π/4 <br />π/3 <br />-π/2 <br />-π/6 <br />-π/4 <br />-π/3 <br />π/2 <br />∞<br />0<br />1/3<br />1<br />3<br />-1/3<br />-1<br />-3<br />-∞<br />tan(x)<br />0<br />π/2 <br />-π/2 <br />
  45. 45. The Graph of Tangent<br />Since tan(x) is odd, so as the values of x goes from <br />0 to -π/2 we get the corresonding negative outputs <br />for tan(x). Specifically,<br />x<br />π/6 <br />0<br />π/4 <br />π/3 <br />-π/2 <br />-π/6 <br />-π/4 <br />-π/3 <br />π/2 <br />∞<br />0<br />1/3<br />1<br />3<br />-1/3<br />-1<br />-3<br />-∞<br />tan(x)<br />The same pattern repeats itself every πinterval. <br />0<br />π/2 <br />-π/2 <br />
  46. 46. The Graph of Tangent<br />Since tan(x) is odd, so as the values of x goes from <br />0 to -π/2 we get the corresonding negative outputs <br />for tan(x). Specifically,<br />x<br />π/6 <br />0<br />π/4 <br />π/3 <br />-π/2 <br />-π/6 <br />-π/4 <br />-π/3 <br />π/2 <br />∞<br />0<br />1/3<br />1<br />3<br />-1/3<br />-1<br />-3<br />-∞<br />tan(x)<br />The same pattern repeats itself every πinterval. <br />In other words, <br />y = tan(x) is a periodic function with period πas shown in the graph.<br />0<br />π/2 <br />-π/2 <br />
  47. 47. The Graph of Tangent<br />Since tan(x) is odd, so as the values of x goes from <br />0 to -π/2 we get the corresonding negative outputs <br />for tan(x). Specifically,<br />x<br />π/6 <br />0<br />π/4 <br />π/3 <br />-π/2 <br />-π/6 <br />-π/4 <br />-π/3 <br />π/2 <br />∞<br />0<br />1/3<br />1<br />3<br />-1/3<br />-1<br />-3<br />-∞<br />tan(x)<br />The same pattern repeats itself every πinterval. <br />In other words, <br />y = tan(x) is a periodic function with period πas shown in the graph.<br />π<br />-π<br />0<br />π/2 <br />-π/2 <br />3π/2 <br />-3π/2 <br />y = tan(x)<br />
  48. 48. The Graph of Inverse Trig-Functions<br />Recalll that for y = cos-1(x), then 0 < y < π. <br />
  49. 49. The Graph of Inverse Trig-Functions<br />Recalll that for y = cos-1(x), then 0 < y < π. It's graph may be plotted in the similar manner.<br />
  50. 50. The Graph of Inverse Trig-Functions<br />Recalll that for y = cos-1(x), then 0 < y < π. It's graph may be plotted in the similar manner.<br />(-1, π)<br />(0, π/2)<br />(1, 0)<br />-1<br />1<br />The graph of y = cos-1(x)<br />
  51. 51. The Graph of Inverse Trig-Functions<br />Recalll that for y = cos-1(x), then 0 < y < π. It's graph may be plotted in the similar manner.<br />(-1, π)<br />(0, π/2)<br />(1, 0)<br />-1<br />1<br />The graph of y = cos-1(x)<br />Remark: The above graphs of y = sin-1(x) and <br />y = cos-1(x) are the complete graphs (i.e. that's all there is).<br />
  52. 52. The Graph of Inverse Trig-Functions<br />The domain of y = tan-1(x) is all real numbers and <br />the output y is restricted to -π/2 < y < π.<br />
  53. 53. The Graph of Inverse Trig-Functions<br />The domain of y = tan-1(x) is all real numbers and <br />the output y is restricted to -π/2 < y < π.<br />x<br />∞<br />0<br />1/3<br />1<br />3<br />-1/3<br />-1<br />-3<br />-∞<br />tan-1(x)<br />
  54. 54. The Graph of Inverse Trig-Functions<br />The domain of y = tan-1(x) is all real numbers and <br />the output y is restricted to -π/2 < y < π.<br />x<br />∞<br />0<br />1/3<br />1<br />3<br />-1/3<br />-1<br />-3<br />-∞<br />tan-1(x)<br />π/6 <br />0<br />π/4 <br />π/3 <br />π/2 <br />
  55. 55. The Graph of Inverse Trig-Functions<br />The domain of y = tan-1(x) is all real numbers and <br />the output y is restricted to -π/2 < y < π.<br />x<br />∞<br />0<br />1/3<br />1<br />3<br />-1/3<br />-1<br />-3<br />-∞<br />tan-1(x)<br />π/6 <br />0<br />π/4 <br />π/3 <br />-π/2 <br />-π/6 <br />-π/4 <br />-π/3 <br />π/2 <br />
  56. 56. The Graph of Inverse Trig-Functions<br />The domain of y = tan-1(x) is all real numbers and <br />the output y is restricted to -π/2 < y < π.<br />x<br />∞<br />0<br />1/3<br />1<br />3<br />-1/3<br />-1<br />-3<br />-∞<br />tan-1(x)<br />π/6 <br />0<br />π/4 <br />π/3 <br />-π/2 <br />-π/6 <br />-π/4 <br />-π/3 <br />π/2 <br />Here is the graph of y = tan-1(x)<br />y = π/2 <br />(1,π/4)<br />(0,0)<br />(-1,-π/4)<br />y = -π/2 <br />
  57. 57. The Graph of Inverse Trig-Functions<br />The domain of y = tan-1(x) is all real numbers and <br />the output y is restricted to -π/2 < y < π.<br />x<br />∞<br />0<br />1/3<br />1<br />3<br />-1/3<br />-1<br />-3<br />-∞<br />tan-1(x)<br />π/6 <br />0<br />π/4 <br />π/3 <br />-π/2 <br />-π/6 <br />-π/4 <br />-π/3 <br />π/2 <br />Here is the graph of y = tan-1(x)<br />y = π/2 <br />(1,π/4)<br />(0,0)<br />(-1,-π/4)<br />y = -π/2 <br />Remark: y =tan-1(x) has two horizontal asymptoes.<br />

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