Trigonometric Summation <munderover> &#x2211;<!-- ∑ --> <mrow class="MJX-TeXAtom-ORD">

desertiev5

desertiev5

Answered question

2022-07-07

Trigonometric Summation r = 0 sin ( r θ ) 3 r

Answer & Explanation

gutinyalk

gutinyalk

Beginner2022-07-08Added 11 answers

The sum is
r = 0 ( 1 3 exp i θ ) r = 1 1 1 3 exp i θ = 1 1 3 exp i θ 10 9 2 3 cos θ = 3 sin θ 10 6 cos θ .
For sin θ = 1 3 , cos θ = ± 2 2 3 , so the sum simplifies to 1 10 4 2 = 5 ± 2 2 34 .
Riya Hansen

Riya Hansen

Beginner2022-07-09Added 4 answers

Use geometric series. We have
r = 0 sin ( r θ ) 3 r = r = 0 I m ( e i r θ 3 r ) = I m ( r = 0 ( e i θ 3 ) r ) = I m ( 1 1 e i θ / 3 ) = I m ( 1 + e i θ / 3 1 + ( e i θ e i θ ) / 3 1 / 9 ) = I m ( 1 + cos ( θ ) / 3 i sin ( θ ) / 3 8 / 9 + 2 sin ( θ ) / 3 ) = 1 / 9 8 / 9 + 2 / 9 = 1 10
if I didn't make a silly mistake somewhere.

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