How to sum up these sine functions? \(\displaystyle{A}={\sin{{2}}}{a}+{\sin{{4}}}{a}+\ldots+{\sin{{10}}}{a}\)

afasiask7xg

afasiask7xg

Answered question

2022-03-30

How to sum up these sine functions?
A=sin2a+sin4a++sin10a

Answer & Explanation

Yaritza Phillips

Yaritza Phillips

Beginner2022-03-31Added 12 answers

You may write
k=1nsin(2ka)=I(k=1ne2ika)
=I(e2iae2a1e2ia1)
=I(e2iaea(eaea)eia(eiaeia))
=I(ei(n+1)asin(na)sin(a))
=I((cos(n+1)a)+isin((n+1)a))sin(na)sin(a))
=sin(na)sin(a)sin((n+1)a)

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