For questions about or related to trigonometric series.

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0answers
9 views

Positiveness of partial sums of type $ \psi * D_N $

In his paper about Extremal Functions for the Fourier Transform (see, for example, here? https://projecteuclid.org/download/pdf_1/euclid.bams/1183552525), Jeffrey Vaaler, while trying to build ...
7
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1answer
76 views

Does $\tan (x)$ equal $\frac{-1}{x-\frac{\pi}{2}}+\frac{-1}{x+\frac{\pi}{2}}+\frac{-1}{x-\frac{3\pi}{2}}+\frac{-1}{x+\frac{3\pi}{2}}+…$?

I set my Year 12 students a question involving the sums of rational functions $\frac{1}{x-n}$. The graph of a sum of these functions looks an awful lot like a tan graph. This led me to ask: Does ...
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5answers
77 views

Can anyone prove the identity $\sum_{m=-\infty}^\infty (z+\pi m)^{-2} = (\sin z)^ {-2} $

I came across this identity in a paper on elliptic curves, and the proof wasn't provided. It really irked me, and I couldn't find an explanation anywhere else. Can anyone shed some light? ...
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2answers
36 views

What angle does the board need to be cut at?

If someone has a 2'' wide board and a 1 1/2'' wide board, and they want to cut the narrower board at an angle so the cut is 2'' long and the boards will fit together, what angle do they need to cut ...
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2answers
38 views

If a 16' ladder is placed correctly on a level surface, how high up will the ladder reach?

So i have just began learning about sin cos and tan, and i came across this problem and for some reason I'm having trouble figuring it out. *** When using a straight ladder, it is recommended that ...
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1answer
74 views

Sum of trigonometric infinite series

I am trying to prove that for any $x\geq 1$ we have: $$ \sum_{m=1}^{\infty} \frac{\sin\frac{(2m-1)\pi}{x}}{\left(\frac{(2m-1)\pi}{x}\right)^3} = \frac{x}{8}(x-1). $$ Could I have some help please? I ...
0
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0answers
41 views

Solving a specific equation involving cos and sin

Here is the equation: $a\sin(\alpha+2\theta)+b\sin(\beta+\theta)=0$, where $\theta$ is the variable, $a$ and $b$ are positive, $\alpha$ and $\beta$ are constant. Please help and thank you very ...
2
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1answer
64 views

How to solve this seemingly easy problem?

In short, I need to prove that: $\sin 2nx\not\to-\sin 2x\quad x\ne\frac{k\pi}2,k\in\Bbb Z,n\in\Bbb N\quad \text{as}\quad n\to\infty$ The biggest trouble is that I know little about $x$, not even ...
0
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3answers
71 views

Prove sum of $\sin$ of angles is greater than $\sin$ of sum of angles

It seems that $\displaystyle \sum_{x_i \in X} \sin\left(x_i\right) \geq \sin\left(\sum_{x_i \in X} x_i\right)$ where $X$ is a set of angles where $\displaystyle \sum_{x_i \in X} x_i \leq \pi$ radians ...
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2answers
95 views

Prove that $\tan^6 20°+\tan^6 40°+\tan^6 80°$ is an integer

Prove that $\tan^6 20°+\tan^6 40°+\tan^6 80°$ is an integer. Doesn't this problem seem a little out of the box? It seems beautiful, but I don't have an idea on how to start. Calculating the value does ...
1
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1answer
23 views

Question regarding an unsolved problem involving a trigonometric sequence

On our last Complex analysis course, our professor announced his retirement. Upon ending the class, he mentioned that he has an interesting problem he wants to leave me with, given my interest in ...
5
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2answers
125 views

Simplify $\prod_{k=1}^5\tan\frac{k\pi}{11}$ and $\sum_{k=1}^5\tan^2\frac{k\pi}{11}$

My question is: If $\tan\frac{\pi}{11}\cdot \tan\frac{2\pi}{11}\cdot \tan\frac{3\pi}{11}\cdot \tan\frac{4\pi}{11}\cdot \tan\frac{5\pi}{11} = X$ and ...
0
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1answer
27 views

Summing series of cosines with binomial coefficients

One part of a STEP-question from 1991 is Prove that $$1 + m \cos 2\theta + \binom {m} {2}\cos 4\theta + \cdots + \binom {m}{r}\cos 2r\theta + \cdots + \cos 2m\theta ~=~ 2^m \cos^m \theta ...
1
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1answer
52 views

Simple Trig Identity?

I have the equation $\sum_1^N \cos\omega_p t\cos\omega_qt$ Where N is an even number representing the number of time steps $\omega_p=\frac{2\pi p}{N}$ p=1,...,$\frac{N}{2}$ I need to prove the ...
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2answers
43 views

Quick Method to Calculate the Maclaurin Series of $\frac{1}{\sqrt{\cos{x}}} $

I am supposed to calculate the maclaurin series for $\frac{1}{\sqrt{\cos{x}}} $ but I can't seem to figure out an efficient way to go about doing this.
3
votes
1answer
51 views

weird trig problem $\tan(\theta)=-\sqrt{2}\sin(\theta)$ on the interval $0 \leq \theta \leq 2\pi$

$\tan(\theta)=-\sqrt{2}\sin(\theta)$ on the interval $0 \leq \theta \lt 2\pi$ I started off with $[(\sin(\theta)/\cos(\theta)] \times (1/\sin(\theta) )= - \sqrt 2$, then after simplification i got ...
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2answers
42 views

Prove trigonometric identity, hence or otherwise find the general solution

The following question requires one to prove the below trigonometric identity $$\cos 3x = 4\cos ^3 x - 3\cos x$$ Hence, or otherwise, find the general solution of the following equation $$(4\cos ^2 x ...
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2answers
58 views

Sum of the series $\tan^{-1}\frac{4}{4n^2+3}$

Find the value of $$\sum^{n=k}_{n=1}\tan^{-1}\frac{4}{4n^2+3}$$ I tried multiplying numerator and denominator by $n^2$, but got nothing. How do I split the term inside $\tan^{-1}$?
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1answer
31 views

Partial sum of Fourier series of square wave

Let $f$ be a $2π$ -periodic square wave function so that $$f\, = -1 \quad -π \le x<0$$ $$f=1 \qquad 0 \le x< π$$ $S_{2n-1}(x)$ is the $(2n-1)st$ Fourier polynomial of $f$. Prove ...
4
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0answers
38 views

query about the cosine of an irrational multiple of an angle?

de Moivre's identity $$ (\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta $$ only applies as written when $n \in \mathbb{Z}$. if the exponent is a fraction $\frac{m}{n}$ then there will ...
0
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1answer
99 views

Chirp with linearly changing frequency and amplitude?

A linear chirp or linearly swept sine is a signal in which the frequency changes linearly with time: the starting frequency changes into the ending frequency over time at a rate of: and ...
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2answers
27 views

Differential Equations: Recursive Functions

Functions I have some familiarity with look so, $y^\prime(x) = \tan(x+2)$: straightforward expression of the first derivative of y as a function of x. But say I have a function, $y^\prime(x) = ...
1
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2answers
46 views

Find the value of the following series.

The expression $\tan\theta+2\tan(2\theta)+2^2\tan(2^2\theta)+\dots+2^{14}\tan(2^{14}\theta)+2^{15}\cot(2^{15}\theta)$ equals to : The answer in the answer book is given to be $\cot\theta$. I am ...
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1answer
38 views

Trigonometric Identities//Fourier Series

Basically I have to find the value of a constant $M$ from this equation: $$l(x)=0=\sum M\Big(\frac{n\pi}{L}\Big)\sin(n\pi x) $$ using the Fourier Series. However the usual Fourier Series formula is: ...
2
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1answer
28 views

Problem with a trigonometric function: $\arctan ( \sin x /(1-\cos x))$

I am studying Abel summability right now, and at a certain point I obtain the following identity: $$ \sum_{k=1}^{\infty}\frac{\sin kx}{k} r^k = \arctan \frac{r\sin x}{1-r\cos x} $$ By previous ...
1
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0answers
52 views

Summation of trigonometric series

The first part of the question requires me to show that the sum of $$cos(2n - 1)x = \frac{sin(2Nx)}{2sin(x)}$$ from $n = 1$ to $N$. This I have done by considering the real part of the geometric ...
2
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1answer
34 views

Generalization of $\sup \limits_{\theta} (a \sin \theta + b \cos \theta) = \sqrt{a^2 + b^2}$

I'm looking for a generalization of the following statement $\sup \limits_{\theta} (a \sin \theta + b \cos \theta) = \sqrt{a^2 + b^2}$ In particular, I want to find $\sup \limits_{\theta} (a \sin ...
2
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1answer
37 views

What is the sum over a shifted sinc function?

What is the sum of a shifted sinc function: $$g(y) \equiv \sum_{n=-\infty}^\infty \frac{\sin(\pi(n - y))}{\pi(n-y)} \, ?$$
1
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1answer
42 views

Is $H(\theta) = \sum \limits_{k=1}^{\infty} \frac{1}{k} \cos (2\pi n_k \theta)$ for a given sequence $n_k$ equal a.e. to a continuous function?

I am studying Furstenberg's article Strict ergodicty and transformation of the torus and I'm stuck with the following construction. Define sequence $(v_k)_{k \in \mathbb{N}}$ as $v_1 =1, ...
2
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1answer
63 views

Bound for the sum of a finite sequence

Consider ${\bf c} = (a,b) \in \mathbb{R}^2$ with $0< \|{\bf c}\| < 1.$ Let $n \in \mathbb{N} $ and define \begin{align*} F_{n}(k) & := \frac{ [a + x_{n}(k)]^2}{ [a + x_{n}(k)]^2 + [b + ...
4
votes
3answers
148 views

To prove $\prod\limits_{n=1}^\infty\cos\frac{x}{2^n}=\frac{\sin x}{x},x\neq0$

Prove $$\prod_{n=1}^\infty\cos\frac{x}{2^n}=\frac{\sin x}{x},x\neq0$$ This equation may be famous, but I have no idea how to start. I suppose it is related to another eqution: (Euler)And how can I ...
0
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2answers
24 views

Using the denseness of trigonometric polynomials to prove the following

$f:[0,2\pi] \to \mathbb R$ is a continuous function. For every trigonometric function $T(x)=\sum_{k=0}^n a_k\cos(kx)+b_k\sin(bx)$, we have $\int_0^{2\pi}f(x)T(x)dx=0$. We need to prove $f=0$(and I ...
4
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2answers
137 views

Is there a way to simplify a sum of cosecants?

A problem I have been working on recently results in a sum of cosecant terms. Specifically, $f(n) = \sum_{k=1}^n \csc \frac{\pi k}{2n+1}$ $g(n) = \sum_{k=1}^n [(-1)^{k+1}(\csc \frac{\pi k}{2n+1})]$ ...
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2answers
60 views

Sum of the area of infinite similar equilateral triangles

How would I solve for the side depicted in the picture?
0
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1answer
26 views

Cauchy product when indices don't match

I want to find the first $4$ coefficients of the Maclaurin series of $\tan(z)$ by multiplying by $\cos(z)$ and using a Cauchy product. Letting $\tan(z)=\sum\limits_{k=0}^\infty c_kz^k$ and ...
0
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0answers
10 views

Upper bound $f(t,d)$ such that $\sum_{j=1}^{d} cos(2 \pi j \; t) \leq f(t,d)$. [duplicate]

I have a sum of a series of trig functions as follows: $\sum_{j=1}^{d} cos(2 \pi j \; t)$ where t is just a constant. Here, we can assume $t$ is a small number and $t \neq 0$. what is the upper ...
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0answers
31 views

Lip $\alpha$ trigonometric series

Assume we have a trigonometric series $$ f(x)=\sum_{n=1}^{\infty} a_n\sin nx \in \text{Lip }\alpha, \, 0<\alpha <1. $$ Is there anything we can say about the series $$ g(x)=\sum_{n=1}^{\infty} ...
1
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2answers
44 views

Upper bound $f(t,d)$ such that $\sum_{j=1}^{d} cos(2 \pi j \; t) \leq f(t,d)$?

I have a sum of a series of trig function as follows: $\sum_{j=1}^{d} cos(2 \pi j \; t)$ where t is just a constant. I am looking for the upper bound $f(t,d)$ such that $\sum_{j=1}^{d} cos(2 \pi j ...
1
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0answers
62 views

Is $\sum_{k=2}^{\infty}{\sin(k!)}/{(k\log(k))}$ convergent or divergent? How can I prove it?

How can I prove that $~\displaystyle\sum_{k=2}^{\infty}\frac{\sin(k!)}{k\log(k)}~$ converges ? Both Leibniz's criterion and Dirichlet's test seem rather inadequate for handling this particular task. ...
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0answers
42 views

Trigonometry - word problem - functions cosθ = adjacent/hypotenuse

I'm not sure what trigonometric equation I should use for this problem: At a certain instant, a ship was 5km south of a lighthouse. The ship was travelling westward and after 30 minutes it's bearing ...
2
votes
3answers
64 views

Can we subtract a trigonometric term from a polynomial?

Can we find the root of a function like $f(x) = x^2-\cos(x)$ using accurate algebra or do we need to resort to numerical methods approximations? thanks.
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0answers
25 views

Approximation of functions by trigonometric polynomials?

I'm seemingly not understanding the fact that a continuous function $f$ that is periodic on [0, 1) can be approximated by a trigonometric polynomial of the form $$g(x) = \sum_{n = 0}^k c_n e^{2\pi ...
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1answer
33 views

Trigonometry system - complex conjugate

I have the following function $$e_n(t) = e^{2\pi int}, t \in R, n \in Z$$ Could anyone explain how one can go from this: $$e_m(t) \bar e_n(t) $$ to $$e^{-2\pi i(m - n)t}$$ Shouldn't it be $e^{2\pi ...
0
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2answers
34 views

Inverse Sine and cosine

$\arcsin(\cos(x))=1/2$ Find $x$. I got $-1/2$ or $2\pi-1/2$, but I don't know the correct answer. I tried graphing unit circle.
2
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2answers
84 views

Evaluating the sum $ \sum_{n = 1}^{44} {\sin^{2}}(n^{\circ}) ~ {\cos^{2}}(n^{\circ}) $.

I want to find the sum $$ {\sin^{2}}(1^{\circ}) ~ {\cos^{2}}(1^{\circ}) + {\sin^{2}}(2^{\circ}) ~ {\cos^{2}}(2^{\circ}) + {\sin^{2}}(3^{\circ}) ~ {\cos^{2}}(3^{\circ}) + \cdots + ...
1
vote
1answer
74 views

Summation of $\tan^{-1}$ series

I am given $$S=\sum\limits_{n=1}^{23}\cot^{-1}\left(1+ \sum\limits_{k=1}^n 2k\right)$$ On expanding the sigma series becomes $$S= 23\cot^{-1}(3)+22\cot^{-1}(5) + \cdots + \cot^{-1}(47)$$ And in tan ...
4
votes
1answer
55 views

Why does a fourier series have a 1/2 in front of the a_0 coefficient

I am reading up on the fourier series, and I keep seeing it as being defined as: $$ f(\theta)= \frac{1}{2}a_0 + \sum_{n=1}^{\infty}(a_n \cos(n\theta) + b_n \sin(n\theta)) $$ where $$ a_n = ...
2
votes
1answer
185 views

How to find the sum?

How to find $$\cot^2\frac{\pi}{2m+1}+\cot^2\frac{2\pi}{2m+1}+\cot^2\frac{3\pi}{2m+1}+\ldots+\cot^2\frac{m\pi}{2m+1}?$$ Of course, the number $m$ is assumed to be a positive integer.
0
votes
1answer
45 views

General form of $\sum_{i=1}^{k} \frac{1}{k}\tan(\frac{i\theta}{k})$

I have tried calculating this equation by setting a large value of $k$. given $\theta$, $$S = \sum_{i=1}^{k} \frac{1}{k}\tan\bigg(\frac{i\theta}{k}\bigg)$$ When increasing $k$, $S$ seems converging ...
1
vote
1answer
24 views

Trigonometical sum from Fourier analysis

(Edit) Note: $a\in \mathbb{R},0<a\leq\pi$. Also, the sum skips $n=0$ (that's where the other term comes from). Working through a Fourier analysis exercise I've got stuck in a clearly ...