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Trigonometric Fun (Posted on 2004-12-06) Difficulty: 4 of 5
Show that cos(π/7) - cos(2π/7) + cos(3π/7) = 1/2.

  Submitted by SilverKnight    
Rating: 3.6667 (3 votes)
Solution: (Hide)
Subtracting the identities sin( x + y ) = sin(x) cos (y) + cos(x) sin(y),
and sin(x-y) = sin(x) cos(y) - cos(x) sin(y) gives us the identity:
  2cos(x)sin(y) = sin(x+y) - sin(x-y).
So, we have:
  2cos(p/7)sin(p/7)=sin(2p/7)-sin(0)   2cos(3p/7)sin(p/7)=sin(4p/7)-sin(2p/7) 2cos(5p/7)sin(p/7)=sin(6p/7)-sin(4p/7)

Adding these three equations gives:
  2sin(p/7)[cos(p/7) + cos(3p/7) + cos(5p/7)] = sin(6p/7),
which gives the result.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionSolutionK Sengupta2007-05-28 13:33:23
I dont know why?Pemmadu Raghu Ramaiah2005-01-24 19:36:11
re(2): Arbitrary EvaluationCeeAnne2004-12-06 23:34:53
re: SolutionCharlie2004-12-06 21:50:53
re: Arbitrary EvaluationCharlie2004-12-06 21:34:51
Arbitrary EvaluationCeeAnne2004-12-06 21:20:43
SolutionSolutionNick Hobson2004-12-06 21:10:32
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