Prove That $\sin(1)+\sin(2)+...+\sin(n)=\frac{\sin(\frac{n+1}{2})\sin(\frac{n}{2})}{\sin(\frac{1}{2})}"$

Question

I need help Proving this: "For Every $n\in \mathbb{N-{0}}$ Prove the following:

$$\sin(1)+\sin(2)+...+\sin(n)=\frac{\sin(\frac{n+1}{2})\sin(\frac{n}{2})}{\sin(\frac{1}{2})}"$$

I assume we have to use Induction On this but I failed to do so

If anyone can provide a simple and clear proof I would be grateful!

Use the de Moivre's identity $e^{i k x}= \cos(k x)+i\sin(k x)$ giving $\sum_k e^{i k x}$ — Cesareo, Feb 08 '21 at 16:35
Is there no other way of proving this other than with complex numbers? — ssuag, Feb 08 '21 at 16:37
Welcome to MSE. Remember to show your work on the problem. Otherwise it looks like you are asking others to do your homework for you. You said you failed trying to prove it by induction, show what you did, where did you get stuck, so we can help you. — jjagmath, Feb 08 '21 at 16:40
@ssuag I was writing my answer when the question was closed. This is what I wrote https://imgur.com/Bm7DlEy — Raffaele, Feb 08 '21 at 16:58
@Raffaele I'm sorry for that, your image link isn't working for me — ssuag, Feb 08 '21 at 17:06
I'm very grateful, exactly how I tried proving it and got stuck, Thank you so much!@Raffaele — ssuag, Feb 08 '21 at 17:11

score 2 · Answer 1 · answered Feb 08 '21 at 16:44

2

Hint:

You don't need induction. Simply use that

$$\sin(1)+\sin(2)+\dots+\sin(n)=\operatorname{Im}\bigl(\mathrm e^{i}+\mathrm e^{2i}+\dots+\mathrm e^{ni}\bigr),$$ which is the sum of a geometric series with common ratio $\mathrm e^{i}$.

answered Feb 08 '21 at 16:44

Bernard

175,478

Prove That $\sin(1)+\sin(2)+...+\sin(n)=\frac{\sin(\frac{n+1}{2})\sin(\frac{n}{2})}{\sin(\frac{1}{2})}"$

1 Answers1