# Tagged Questions

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### Closed form for a trigonometric partial sum

I know that: $$\sum_{k=1}^n\arctan(2k^2)=\frac{\pi n}{2}-\frac{1}{2}\arctan(\frac{2n(n+1)}{2n+1})$$ Can a similar closed form expressions be given for $\sum_{k=1}^n \arctan(k^2)$? I was able to ...
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### Prove that $2\sum_{k=1}^n \cos(kθ) = \frac{\sin[\left(n+1/2\right)θ]}{\sin(θ/2)}-1$ [closed]

Prove that $$2\sum_{k=1}^n \cos(kθ) = \frac{\sin[\left(n+1/2\right)θ]}{\sin(θ/2)}-1$$ By using $$e^{iθ}+e^{2iθ}+\cdots+e^{niθ}=\frac{e^{iθ}(1-e^{inθ})}{1-e^{iθ}}$$
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### Basic Fourier Series Question

Let $f$ be a $2π$ periodic function where $$f(x) = \frac{π - x}2$$ over $[0, π]$. It is known that the Fourier series of $f$ is $$\sum_{n=1}^{\infty}\frac{\sin nx}n$$ At which points in $[-π, π]$ ...
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### $\sum_{k=0}^{n-1}\sin(k\frac{\pi}{n}+\theta)$

I'm trying to find the closed form of the above formula. This link shows the solution of tan version. Sum of tangent functions where arguments are in specific arithmetic series Though I'm trying to ...
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### $\sum_{n=0}^{14}\tan(12n+1^\circ)$

I often fail to find trigonometric sums such as the one in the question shown in the following. When I tried the question, I first led $z=e^{i\pi/180}$. After simple calculations, I obtained ...
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### Is $\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}$ true for $m\in\mathbb N$?

Question : Is the following true for any $m\in\mathbb N$? \begin{align}\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}\qquad(\star)\end{align} Motivation : I reached ...
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### $\frac{\cos x}{1}+\frac{\cos(2x)}{2}+\cdots+\frac{\cos (nx)}{n}\gt -1$ is true for $n\in\mathbb N, 0\lt x\lt \pi$?

Let $n$ be a natural number and let $0\lt x\lt{\pi}$. Then, here are my questions. Question 1 : Is the following true? $$\sum_{k=1}^{n}\frac{\cos(kx)}{k}\gt -1$$ Question 2 : Is the ...
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### A finite sum with cosines

I'm not able to compute the following sum : $$\sum_{k=0}^n\frac{\cos(kx)}{(\cos(x))^k}$$
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### Summation of trigonometric functions

So consider a summation of ai cos (x + phi_i) where i ranges from 1 to N. Could we describe this summation as a single cosine function? Or the sum of two cosine or sine functions? How would we do this ...
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### trigonometric summation

Taking into consideration the functions $$\sum_{t=0}^{n} \sin{(\theta + t \phi)}=\frac{\sin({\frac{(n+1)\phi}2})\sin{(\theta+\frac{n \phi}2)}}{\sin{(\frac{\phi}2)}}$$ and ...
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### Help in manipulating Integrals

I try to express : $\displaystyle 1+2\sum _{ k=1 }^n \cos(2k\theta )$ as : $\dfrac { \sin\left( \theta +2\theta n \right) }{ \sin\left( \theta \right) }$ I tried to use the exponential function ...
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### Finding the sum of a trigonometric series [duplicate]

Find the sum of the series $$\cos x + \cos 2x + \cdots + \cos (n-1)x.$$ You must calculate the sum of this series only by multiplying through by $2\sin\left(\frac{x}{2}\right)$. Now I've heard of ...
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### Proving $\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$

I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$ I have some progress made, but I am stuck and could use some help. What I did: ...
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### find the multiplicative factor for get a specific amount of sum on sin

i am not a math guru so please sorry if this is a silly question. i'm not sure on how to latexize this question so i've done a spreadsheets with openoffice (and i'm interest also in the best way to ...
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### Evaluation of a trigonometric partial sum

I just wanted to evaluate $$\sum_{k=0}^n \cos k\theta$$ and I know that it should give $$\cos\left(\frac{n\theta}{2}\right)\frac{\sin\left(\frac{(n+1)\theta}{2}\right)}{\sin(\theta / 2)}$$ ...
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### How to prove Lagrange trigonometric identity [duplicate]

I would to prove that $$1+\cos \theta+\cos 2\theta+\ldots+\cos n\theta =\displaystyle\frac{1}{2}+ \frac{\sin\left[(2n+1)\frac{\theta}{2}\right]}{2\sin\left(\frac{\theta}{2}\right)}$$ given that ...
### Show $1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}}$ for $x \neq 0$ [duplicate]
For $x \neq 0$, $$1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}}$$
### Sum of $\cos(k x)$ [duplicate]
I'm trying to calculate the trigonometric sum : $$\sum\limits_{k=1}^{n}\cos(k x)$$ This is what I've tried so far : \renewcommand\Re{\operatorname{Re}} \begin{align*} \sum\limits_{k=1}^{n}\cos(k x) ...