Linked Questions
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Show $\sin(\theta) + \sin(2\theta) + ... +\sin(n\theta) = \frac {\sin(n\theta/2)\sin((n+1)\theta/2)} {\sin(\theta/2)}$ using De Moivre's formula [duplicate]
Attempt at part b:
$\sin(\theta)+\sin(2\theta)+...
= \frac {(e^{i\theta}-e^{-i\theta})+(e^{2i\theta}-e^{-2i\theta})+...+(e^{ni\theta}-e^{-ni\theta})} {2i} \\
= \frac {(1+e^{i\theta}+(e^{i\theta})^2+......
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How to proof that $\sum_{k=-N}^{N} e^{2 \pi i k t}=\frac{\sin [(2 N+1) \pi t]}{\sin (\pi t)}$?
$$\sum_{k=-N}^{N} e^{2 \pi i k t}=\frac{\sin [(2 N+1) \pi t]}{\sin (\pi t)}$$
I am trying to solve the above question. But I have literally no idea to where to start. How can a logarithmic expression ...
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$Re(e^{i\theta}+e^{i\theta}+...+e^{ni\theta})=\frac{\cos(n+1)\vartheta\sin n\varphi}{\sin\varphi}$
For any $0<\theta<\pi$ and the integer $n\geqslant 1$ show that:
$$\sin\theta+\frac{\sin 2\theta}{2}+...+\frac{\sin n\theta}{n}>0$$
Denote by $s_n(\theta)$ the left-hand side of the ...
- 3,875
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3
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Proving complex series $1 + \cos\theta + \cos2\theta +... + \cos n\theta $
So I have this result
$1 + z + z^2 + ... + z^n = \frac{z^{n+1}-1}{z-1}$
which I proved already. Now I am supposed to use that result and De Moivre's formula to establish this identity
$1 + \cos\...
- 1,542
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Trigonometry Olympiad problem: Evaluate $1\sin 2^{\circ} +2\sin 4^{\circ} + 3\sin 6^{\circ}+\cdots+ 90\sin180^{\circ}$
Find the value of
$$1\sin 2^{\circ} +2\sin 4^{\circ} + 3\sin 6^{\circ}+\cdots+ 90\sin180^{\circ}$$
My attempt
I converted the $\sin$ functions which have arguments greater than $90^\circ$ to $\cos$...
- 261
0
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Help simplifying $\sum_{k=0}^n\cos(k\theta)=\frac{1}{2}+\frac{\sin[(n+\frac{1}{2})\theta]}{2\sin(\theta/2)}$
In a proof of $\sum_{k=0}^n\cos(k\theta)=\frac{1}{2}+\frac{\sin[(n+\frac{1}{2})\theta]}{2\sin(\theta/2)}$
I need help figuring out the identity used to simplify from red $ \color{red}{1}$ to $\color{...
- 1,171
2
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Sum of cosines with a multiplicative factor in the angle and different interval
I have found the following formula for the sum of cosines in both here and here.
\begin{align}
\sum^n_{l=1} \cos \left(\frac{2 \pi l}{n}\right) = 0
\end{align}
I would like to know what the sum ...
- 519
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Geometric series and complex numbers [duplicate]
I'm new to this site, english is not my mother tongue, and I'm just learning LaTeX. I'm basically a noob, so please be indulgent if I break any rule or habits.
I'm stuck at proving the following ...
- 43
3
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2
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Evaluate the following limit of finite sum
Evaluate the limit $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=0}^{[n/2]}\cos \left(\frac{k\pi}{n}\right)$$
I tried through considering two cases : (i) When $n$ is even (ii) when $n$ is odd.
When $n$ is ...
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Prove that $\sin\frac{\pi}{14}$ is a root of $8x^3 - 4x^2 - 4x + 1=0$
Prove that $\sin\frac{\pi}{14}$ is a root of $8x^3 - 4x^2 - 4x + 1=0$.
I have no clue how to proceed and tried to prove that the whole equation becomes $0$ when $\sin\frac{\pi}{14}$ is placed in ...
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