Thursday, February 18, 2016

trigonometry - Resolve A=cos(pi/7)+cos(3pi/7)+cos(5pi/7) using u=A+iB



With these two sums:
A=cos(π/7)+cos(3π/7)+cos(5π/7)


B=sin(π/7)+sin(3π/7)+sin(5π/7)



How to find the explicit value of A using:





  • u=A+iB

  • the sum of n terms in a geometric sequence: u01qn+11q



I know the answer is 12 from this post, but there is no mention of this method.


Answer



Using Euler formula,



setting 2y=iπ7e14y=1




A+iB=2r=0e(2r+1)2y=e2y1(e4y)31e4y=e2y+11e4y=11e2y



Now, 11ei2u=eiueiueiu=cosuisinu2isinu=12+icotu2



Now equate the real parts.


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