Show that 1+cos(theta)+cos(2 theta)+cos(3 theta)+cos(4 theta)+cos(5 theta)+cos(6 theta)=0

Ricky Arias

Ricky Arias

Answered question

2022-11-03

Show that 1 + cos ( θ ) + cos ( 2 θ ) + cos ( 3 θ ) + cos ( 4 θ ) + cos ( 5 θ ) + cos ( 6 θ ) = 0

Answer & Explanation

lesinetzgl5

lesinetzgl5

Beginner2022-11-04Added 18 answers

If you can use the property that cos ( x ) = R e ( e i x ), where R e ( z ) is the real part of z, then:
k = 0 6 cos ( k θ ) = R e ( k = 0 6 e i k θ ) = R e ( 1 e i 7 θ 1 e i θ ) = R e ( 1 e i 2 π 1 e i 2 π 7 ) = 0
as e i 2 π = 1

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