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  1. Path: sparky!uunet!gatech!destroyer!caen!uflorida!mailer.cc.fsu.edu!fsu1.cc.fsu.edu!rose
  2. From: rose@fsu1.cc.fsu.edu (Kermit Rose)
  3. Newsgroups: sci.math
  4. Subject: Re: "proof" that sin(x) = -sin(x)
  5. Message-ID: <1992Jul19.232938.21247@mailer.cc.fsu.edu>
  6. Date: 24 Jul 92 15:38:10 GMT
  7. References: <1992Jul19.220735.10289@leland.Stanford.EDU>
  8. Reply-To: rose@fsu1.cc.fsu.edu
  9. Organization: Florida State University
  10. Lines: 56
  11. News-Software: VAX/VMS VNEWS 1.3-4
  12.  
  13. In article <1992Jul19.220735.10289@leland.Stanford.EDU>, ledwards@leland.Stanford.EDU (Laurence James Edwards) writes...
  14. > 
  15. >d sin(-x)             d (-x)
  16. >--------   = cos(-x)  ------- = -cos(-x)
  17. >   dx                   dx
  18. > 
  19. >and
  20. > 
  21. >d -cos(-x)            d (-x)
  22. >---------- = sin(-x)  ------ = -sin(-x) = sin(x)
  23. >   dx                   dx
  24. > 
  25. >so
  26. > 
  27. > 2
  28. >d sin(-x)
  29. >--------   = sin(x)                  (1)
  30. >     2
  31. >   dx
  32. > 
  33. > 
  34. >but we also have:
  35. > 
  36. >d sin(-x)             d (-x)
  37. >--------   = cos(-x)  ------ = -cos(-x) = -cos(x)
  38. >   dx                   dx
  39. > 
  40. >and
  41. > 
  42. >d -cos(x)
  43. >--------- = -sin(x)
  44. >   dx
  45. > 
  46.  
  47. ^^^^^^^^^^^^^^^^^^^^^^^^^^    error is above.  d[-cos(x)]/dx = sin(x).
  48.                                                d[cos(x)]dx   = - sin(x)
  49.  
  50. >so
  51. > 
  52. > 2
  53. >d sin(-x)
  54. >---------  = -sin(x)                   (2)
  55. >     2
  56. >   dx
  57. > 
  58. >from (1) and (2) we therefore have: sin(x) = -sin(x)!
  59. > 
  60. >Whats wrong with the above "proof"?
  61. > 
  62. >Larry Edwards
  63. >edwards@sunrise.stanford.edu
  64.  
  65.  
  66. rose@fsu1.cc.fsu.edu          To be sure I see your response, use e-mail.
  67. -----------------------------------------------------------------------
  68. Be of good cheer, for it is much more fun than being depressed.
  69.