Linked Questions

9 votes
5 answers
1k views

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 $\cos$...
sidt36's user avatar
  • 261
8 votes
5 answers
3k views

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 ...
user avatar
5 votes
3 answers
6k views

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\...
Itsnhantransitive's user avatar
4 votes
1 answer
10k views

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 ...
Benji's user avatar
  • 43
0 votes
1 answer
3k views

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{...
user5389726598465's user avatar
3 votes
2 answers
224 views

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 ...
Empty's user avatar
  • 13.1k
0 votes
1 answer
729 views

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+......
Daniel's user avatar
  • 187
0 votes
1 answer
216 views

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 ...
Jake Evergreen's user avatar
0 votes
2 answers
150 views

$Re(e^{i\theta}+e^{i\theta}+...+e^{ni\theta})=\frac{\cos(n+1)\vartheta\sin n\varphi}{\sin\varphi}$

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 ...
Pedro Gomes's user avatar
  • 3,961
2 votes
1 answer
105 views

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 ...
Omar Shehab's user avatar