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: u0∗1−qn+11−q
I know the answer is 12 from this post, but there is no mention of this method.
Answer
Using Euler formula,
setting 2y=iπ7⟹e14y=−1
A+iB=2∑r=0e(2r+1)2y=e2y⋅1−(e4y)31−e4y=e2y+11−e4y=11−e2y
Now, 11−ei2u=−e−iueiu−e−iu=−cosu−isinu2isinu=12+i⋅cotu2
Now equate the real parts.
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