# Questions tagged [trigonometric-sums]

The trigonometric-sums tag has no usage guidance.

73
questions

17
votes

2
answers

2k
views

### Boundedness of sum of sin(sin(n))

Playing with desmos I have accidentally noticed that the sequence of partial sums
$$\left\{ \sum_{n=1}^{N}\sin(\sin(n)) : N\geq 1 \right\}$$
is bounded.
However, I did not succeed in proving this ...

33
votes

3
answers

4k
views

### A trigonometric equation: how hard could it be?

The following problem started out with a formulation in terms of complex numbers: let $\epsilon=e^{\frac{\pi i}3}$ and $z=e^{\frac{2\pi i}{3(2n-1)}}$. It's rather amusing that the following appears to ...

2
votes

0
answers

108
views

### Upper and lower bounds on the number of solutions to the equation $\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}} \right) $

Background
The Norwegian mathematician and astronomer Carl Størmer did important work on the equation
$$\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}}\right), \label{1}\tag{1} $$
...

3
votes

0
answers

202
views

### A problem about the series $\sin(n^p)$ [closed]

Prove that when $p>0,$ the series $$\sum_{n=1}^\infty \sin(n^p)$$
is divergent

1
vote

3
answers

179
views

### Evaluating a sinusoidal series

Define the sequence of functions
$$f_n(x)=\sum_{m=n}^\infty(-1)^m\frac{x^{2m}}{(2m+1)!} {m \choose n} $$
Is there a closed form expression for arbitrary $n$? It is clear that the result should assume ...

8
votes

1
answer

504
views

### Integer solutions of $2\cos\left(\frac{p\pi}n\right)+2\cos\left(\frac{q\pi}n\right)+4\cos\left(\frac{p\pi}n\right)\cos\left(\frac{q\pi}n\right)=1$

I asked this question on MSE yesterday. Due to observations that I have made only after that, I feel that it may be a much harder problem than it looks like. So, I am posting the question here as well....

9
votes

2
answers

428
views

### How to prove this sum involving powers of cosec is an integer?

It is claimed that the following function produces only integer values for all integer $m \geq 1$, $N \geq 2$.
$F(m,N)=\frac{N^m}{2^m}\displaystyle \sum_{j=1}^{N-1} \operatorname{cosec} ^{2m}\left(\...

1
vote

1
answer

154
views

### CDF of sum of independent cosines?

Consider the random variable
$$X=\frac{1}{d}\sum_{k=1}^d\cos X_k$$
where $X_k$ are each drawn uniformly i.i.d. from $[0,2\pi]$. What is the CDF of X?
It seems that a relatively direct way could be to ...

1
vote

1
answer

130
views

### Is it possible to sum this analytically in any way?

The sum I am looking for is the following sum as $M \to \infty$:
$$ L(\omega) = \sum_{m = 1}^{M} \frac{\sin\left( N \frac{\omega_m - \omega}{2} \right)}{\sin\left( \frac{\omega_m - \omega}{2} \...

3
votes

0
answers

105
views

### A new arranging of discrete sine transform

Let $n$ be even and consider the discrete sine transform of type 5 which is the matrix
$$S=\left(\sin(k+1)(l+1)\frac{\pi}{n+\frac12}\right)_{k,l=0}^{n-1}$$
Let us denote by $s_{-,l}$ the $l^{\text{...

4
votes

0
answers

149
views

### Trigonometric sum and residues

I am interested in the sum
$$
\sum_{n=1}^k 2\biggl[\sin\biggl(\frac{n\pi}{2k+2}\biggr)\biggr]^{-2g}
$$
where $k$, $g$ are integers. It is not too hard to show that this can also be expressed as
$$
-1-...

2
votes

1
answer

193
views

### When does a random trigonometric sum approximate $1$?

I am looking for an upper bound $R=R_{n,\varepsilon}$ such that for given $\varepsilon>0$ and real numbers $\alpha_1, \dotsc, \alpha_n$ in, say, $[1,2]$, there is $x\in [1,R]$ such that
$$
\frac 1n\...

7
votes

0
answers

316
views

### Multiple Fourier series

In the book by Elias M.Stein and Guido Weiss "Introduction to Fourier Analysis on Euclidean Spaces" one states in page 268 the following theorem:
Theorem 1: The trigonometric series
$$\...

9
votes

1
answer

610
views

### Infinite series with inverse trigonometric functions

Consider the infinite series
$$
F(s)=\sum_{k=1}^\infty \frac{1}{k^{2s+1} \sin(k \pi \sqrt{2})}
$$
Prudnikov Vol.1 , gives a result for this sum in 5.4.16.13 for $s=1.$
$$
F(1)=-\frac{13 \pi^3}{360 \...

5
votes

0
answers

400
views

### Generalization of Pompeiu's theorem

Let us recall the statement of
Pompeiu's theorem. Let $A_1A_2A_3$ be a regular triangle inscribed in a circle $\omega$. Let $X$ be an arbitrary point on the arc $A_1A_3$. Then $$|XA_1|-|XA_2|+|XA_3|=...

5
votes

1
answer

241
views

### Smallest regular $m$-gon covering a regular $n$-gon

I start by stating the problem, which is already hinted in the title of the question. I do believe it is a research-level question.
Let us fix a regular $n$-gon with area $1$. What is the smallest ...

2
votes

1
answer

227
views

### $L_p$ norms of $0-1$ exponential sums

Consider $f_n(t)=\sum_{i=1}^{n}e^{ik_{i}t}$ with all $k_i$ some distinct integers for $t\in [-\pi,\pi)$. For $p>2$ I am interested in the maximum possible value of $$||f_n||_p,$$
where $f_n$ runs ...

7
votes

2
answers

319
views

### Sum of $\sin$ when angles shrink by $1/n$

There are many identities known like
$$\sum_{k=0}^{n-1} \sin (k \cdot \theta + \varphi) = \frac{\sin\left(n \cdot \frac{\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)} \cdot \sin \left(\frac{2 \...

1
vote

0
answers

637
views

### What is the Jacobi-Anger expansion of $\sin^{[k]} (\theta) $?

Cross-post from MSE.
The Jacobi-Anger expansion gives expressions for functional iterates of trigonometric functions in terms of Bessel functions. For instance, we have $$\sin(z \sin(\theta)) = 2 \...

4
votes

1
answer

1k
views

### The cotangent sum $\sum_{k=0}^{n-1}(-1)^k\cot\Big(\frac{\pi}{4n}(2k+1)\Big)=n$

On the Wolfram Research Reference page for the cotangent function (https://functions.wolfram.com/ElementaryFunctions/Cot/23/01/), I saw the following partial sum formula
$$\sum_{k=0}^{n-1}(-1)^k\cot\...

4
votes

2
answers

331
views

### The complex trigonometric function degenerates to the positive integer

For any integer $N \geq 2$, we have the identity:
$$\frac{\ \prod _{n=1}^{N-1}\ \left(2+2\sum _{m=1}^{n\ }\cos \frac{\ m\pi \ }{N}\ \right)\ }{\prod _{n=1}^{N-1}\ \left(1+2\sum _{m=1}^{n\ }\cos \frac{\...

11
votes

2
answers

1k
views

### A problem in additive combinatorics

$\color{red}{\mathrm{Problem:}}$
$n\geq3$ is a given positive integer, and $a_1 ,a_2, a_3, \ldots ,a_n$ are all given integers that aren't multiples of $n$ and $a_1 + \cdots + a_n$ is also not a ...

3
votes

0
answers

377
views

### For which values of $ x$ we have $\sum\limits_{n=0}^{\infty}x^{\tan(n!)-n!}$ converges?

The copy of this question is posted here
I have tried to to determine values of real $x$ for which $\sum\limits_{n=0}^{\infty}x^{\tan(n!)-n!}$ converges but I can't , Presumably the set of values $\{ ...

2
votes

0
answers

88
views

### (Dis)continuity of periodic functions with non-summable Fourier series

Let $f : [0,2 \pi)^d \rightarrow \mathbb{R}$ be a square-integrable periodic function in $L^2( [0,2 \pi)^d )$ with $d \geq 1$.
We assume moreover that the square-summable Fourier coefficients of $f$, ...

0
votes

1
answer

471
views

### coordinate free foundations of trigonometry [closed]

What axioms for geometry and trigonometry would I have to chose in order to completely avoid coordinates in defining trig functions and showing the equivalence of their geometric (unit circle) and ...

17
votes

3
answers

1k
views

### Show that $(\sum_{k=1}^{n}x_{k}\cos{k})^2+(\sum_{k=1}^{n}x_{k}\sin{k})^2\le (2+\frac{n}{4})\sum_{k=1}^{n}x^2_{k}$

Let $x_{1},x_{2},\cdots,x_{n}>0$, show that
$$\left(\sum_{k=1}^{n}x_{k}\cos{k}\right)^2+\left(\sum_{k=1}^{n}x_{k}\sin{k}\right)^2\le \left(2+\dfrac{n}{4}\right)\sum_{k=1}^{n}x^2_{k}$$
This ...

11
votes

0
answers

2k
views

### A question on trig series

Assume $\{a_k\}_{k\ge1}$ is a real sequence such that $u(x) = \sum_{k\ge 1}a_k\sin(kx)$ is a smooth function, and for every $x \in [-\pi, \pi]$
$$\left(\sum_{k\ge 1}\frac{a_k}{k}\sin(kx)\right)\left(\...

2
votes

0
answers

509
views

### Conjecture about $\prod_{n=1}^{\infty}\left|1+\frac{\tan\left(n\right)+1}{n\tan\left(n+1\right)+n}\right|=0$

Ps: I have reached the number of question I can propose so let this here please .And it's a work on open questions .
Does we have in radians :
$$\prod_{n=1}^{\infty}\left|1+\frac{\tan\left(n\right)+1}{...

8
votes

2
answers

946
views

### Better trigonometrical inequalities for $\zeta(s)$?

The inequality $$3 + 4 \cos \theta + \cos 2 \theta \geq 0$$ plays a key role in the proof of the classical zero-free region of the Riemann zeta function. Are there other inequalities of the form
$$\...

1
vote

2
answers

409
views

### Order of magnitude for trigonometric sum

Consider the following sum:
$$ S_N = \sum_{ \substack{ k_1 + k_2 + k_3 =N \\ -(N-2) \leq k_1, k_2 , k_3 \leq N \\ k_1, k_2 , k_3 \neq 0 } } \frac{1}{k_1 k_2 k_3} \sin\left( \frac{k_1\pi}{N} \right) \...

0
votes

0
answers

125
views

### $\sin(\mu A)\cos(\mu B) + \sin(\mu C)\cos(\mu D) - \sin(\mu E)\cos(\mu F) \geq 0$ - Version 2.0

I recently asked this question here Inequality involving sine and cosine
It turned out that with the conditions I required there, the inequality does not hold. I tried to add extra conditions now, ...

3
votes

1
answer

744
views

### Inequality involving sine and cosine

I am trying to prove that given $A,B,C,D,E,F \in \big]0,\frac{\pi}{2}\big]$ fixed and $A+C \geq E$ and the following equation holds for $\mu = 1$:
$$\sin(\mu A)\cos(\mu B) + \sin(\mu C)\cos(\mu D) - \...

32
votes

3
answers

3k
views

### A conjectural trigonometric identity

Recently, I formulated the following conjecture which seems novel.
Conjecture. For any positive odd integer $n$, we have the identity
$$\sum_{j,k=0}^{n-1}\frac1{\cos 2\pi j/n+\cos 2\pi k/n}=\frac{n^2}...

3
votes

1
answer

147
views

### Trigonometric cancellation on the unit circle

Let $z \in \mathbb{C}$ with $|z|=1$ and $z\ne 1$. Now consider the sum
$$S(N,p)=\sum_{k=0}^N k^p z^k,$$
for some positive integers $N,p$.
An immediate upper bound on $|S(N,p)|$ is
$$|S(N,p)|\le C_1(...

2
votes

0
answers

233
views

### Finite sum involving root of unity

I have the following sum:
$$\sum_{\sigma=1,\text{odd}}^{\frac{p}{2}-2}\frac{\sin \frac{r \sigma \pi}{p}}{\sin \frac{q \sigma \pi}{p}}$$
where $p\equiv2\pmod4$, $p$ and $q$ are coprime numbers such ...

6
votes

3
answers

438
views

### The first zero-crossing of a combination of sines

Let $\{c_i\}_{i=1}^n$ be a sequence of real numbers such that $c_i \geq 0$ for each $i$ and $\sum_{i=1}^n c_i = 1$. Let $\omega_i \in [\delta, \Delta]$ for each $i$, where $\delta$ and $\Delta$ are ...

6
votes

1
answer

373
views

### Asymptotic behavior of a certain trigonometric partial sum

Let $a<0$ and $b>0$ be real numbers such that $a<-2b$. Let $n>1$ be a positive integer and consider the following partial sum:
$$
f(n) = \frac{1}{(n+1)^2}\sum_{i=1}^{n}\sum_{j=1}^{n} (-1)^{...

1
vote

1
answer

135
views

### Proof of an inequality $s_m(n) \le f_m(n)$

For fixed $m = 0, 1, 2, ...$
$$f_m(k) = \prod_{j=1}^{m}(k+j).$$
Some examples of $f_m(k)$ are as following:
$$f_0(k) = 1, \quad f_1(k) = (k+1), \quad f_2(k) = (k+1)(k+2).$$
The $s_m(n)$ is defined as ...

3
votes

2
answers

229
views

### Inductive proof of $s(n)≤n+1$

I was able to conclude, numerically, the following:
$$s(n) = \sum_{j=0}^n\frac{(-4)^j}{(2j+1)!}\left(\sum_{k=j}^n\frac{(k+1)(2k+1)(k+j)!}{(k-j)!}\right)x^{2j+2}\le n+1$$
for $x\in[0,1]$. For example
...

1
vote

0
answers

145
views

### Fejer-Jackson-like inequality with divisor sum

A question was recently asked about a generalization of the Fejer-Jackson inequality $$\sum_{k=1}^n \frac{\sin kx}{k}\gt 0 \quad \forall\: n\in\mathbb{Z}^+\: \text{and}\: 0\lt x\lt\pi$$
to ...

4
votes

0
answers

281
views

### A uniform Riemann sum approximation of the integral of the Fejer kernels

Let $F_N(t)$ denote the Fejer kernel
$$F_N(t):={1\over N+1}{\sin^2\big({(N+1)}{t\over2}\big)\over \sin^2\big( {t\over2}\big)}\ .$$
Consider Riemann sums approximation for $\int_{-\pi}^\pi F_N(t) dt$ ...

0
votes

1
answer

212
views

### Finding closed form expression for the roots of $f(x) = \sum_{i=1}^K \frac{\alpha_i \gamma_i \sin(x-\theta_i)}{1+\gamma_i[1+\cos(x-\theta_i) ]}$

Let us define function $f:[0~ 2\pi] \rightarrow R$ as follows:
\begin{align}
f(x)\triangleq \sum_{i=1}^K \frac{\alpha_i \gamma_i \sin(x-\theta_i)}{1+\gamma_i[1+\cos(x-\theta_i) ]},
\end{align}
...

2
votes

0
answers

110
views

### Lower $L^1$ norm estimates of null average trigonometric polynomials depending on the order of the polynomial

Let $p(x)=\sum_{k=1}^m [a_k\cos(n_kx)+b_k\sin(n_kx)]$ be a null average trigonometric polynomial (null average means that is $\int_\mathbb T p =0$ or, equivalently, there are no $a_0$ and $b_0$). ...

4
votes

1
answer

359
views

### Identities for Chebyshev polynomials of the second kind

While calculating an integral in a quantum mechanical problem by two different methods, I came across the following identity
$$\sum_{k=0}^n\sum_{m=0}^{2k}(-2)^m\binom{2(n-k)}{n-k}\binom{2k}{k}\binom{...

5
votes

3
answers

773
views

### positive sum of sines

This was asked but never answered at MSE.
Let $f(x) = \sin(a_1x) + \sin(a_2x) + \cdots + \sin(a_nx)$, where the $a_i$'s
represent distinct positive integers. Suppose also that $f(x)$ satisfies the ...

1
vote

0
answers

130
views

### How to evaluate this sum of roots of unity with condition to zero

In evaluating the sum:
$$\tag{1}\label{e1}\sum\limits_{j = 1 < h}^N {\left( {{e^{\frac{{i\pi }}{N}\left( {{s_1}j + {s_2}h} \right)}} - {e^{\frac{{i\pi }}{N}\left( {{s_1}h + {s_2}j} \right)}}} \...

6
votes

0
answers

78
views

### Delaying the first zero of a trigonometric series

Let $a_1,\ldots,a_n \in [0,1], \omega_1,\ldots,\omega_n \in (0,+\infty)$, and define, for every $t \ge 0$,
$$
f(t) := \sum_{i = 1}^n a_i \sin(\omega_i t).
$$
I'm interested in trying to optimize the ...

6
votes

2
answers

2k
views

### Distribution of distances of successive zeros of $f(x)={\rm cos}(x)+{\rm cos}(\alpha x)+{\rm cos}(\beta x)$

Let $\alpha$ and $\beta$ be incommensurate real numbers.
Consider the function
$f(x)={\rm cos}(x)+{\rm cos}(\alpha x)+{\rm cos}(\beta x)$
and its positive zeros $x_k(\alpha,\beta)$.
Fix $\alpha$ and ...

7
votes

2
answers

849
views

### Closed formula for sine powers

I am looking for a closed formula for the expressions
$$ \sum_{k=1}^{n-1} \sin\left(\pi \frac{k}{n}\right)^m,$$
with $n \in \mathbb{N}$ and $m \in \mathbb{N}$ odd.
Playing with these sums a bit, I ...

7
votes

3
answers

766
views

### Something interesting about the quintic $x^5 + x^4 - 4 x^3 - 3 x^2 + 3 x + 1=0$ and its cousins

(Update):
Courtesy of Myerson's and Elkies' answers, we find a second cyclic quintic for $\cos\frac{2\pi}{p}$ with $p=\text{1 mod 10}$ as,
$$\frac{z^5}{\beta} = 10 z^3 - 20 n^2 z^2 + 5 (3 n^4 - 25 n^...