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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 $\cos$...
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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 ... 5k views

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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 ...
Attempt at part b: $\sin(\theta)+\sin(2\theta)+... = \frac {(e^{i\theta}-e^{-i\theta})+(e^{2i\theta}-e^{-2i\theta})+...+(e^{ni\theta}-e^{-ni\theta})} {2i} \\ = \frac {(1+e^{i\theta}+(e^{i\theta})^2+...... 0 votes 2 answers 141 views ###$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 ... 2 votes 1 answer 100 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 ... 0 votes 1 answer 131 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 ...