For questions about or related to trigonometric series.

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0answers
57 views

Fourier series of half of $\sin(\pi x)$

So my question is: Find the Fourier series (using integrals) for the half wave rectified sine function: $$f(x)= \begin{cases}0&-1<x<0\\ \sin(\pi x)& 0<x<1\end{cases}$$ ...
1
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2answers
327 views

Solving for linear velocity

A wheel is rotating at 3 radians per second and the wheel has an 80 inch diameter To the nearest foot per minute, what is the linear velocity?
5
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3answers
83 views

Solving a Complicated Trig Problem

I am preparing for AIME and I came across this problem which I need help solving: $$\begin{eqnarray} 10^{10^{10}} \sin\left( \frac{109}{10^{10^{10}}} \right) - 9^{9^{9}} \sin\left( ...
6
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1answer
717 views

Sum of tangent functions where arguments are in specific arithmetic series

By looking through an book, I found this interesting series To prove that: $$\tan(\theta)+\tan \left(\theta+ \frac{\pi}{n} \right) + \tan(\theta + \frac{2\pi}{n}) + \dots + \tan \left (\theta + ...
5
votes
1answer
116 views

$\sin x$ as a sum involving fractional parts

Does there exist a formula giving a sense to the formal equation $$ \sin x=-\pi\sum_{n=1}^{+\infty}\frac{\mu(n)}{n}\left\{\frac{nx}{2\pi}\right\}, $$ where $\mu$ is the Möbius function, $\{\cdot\}$ ...
2
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2answers
79 views

Computing the limit of a series involving trigonometric identities

Let $x_n\in(n\pi,(n+1)\pi)$ with $\tan(x_n)=x_n$. Can $$s:=\sum_{n=1}^\infty \frac{1}{x_n^2}$$ be determined explicitly? My ideas so far: First, I wrote $x_n$ as $$x_n= \pi (n+z_n)$$ with ...
5
votes
7answers
309 views

Sine/cosine series

$$\frac{\sin²(1°) + \sin²(2°) + \sin²(3°) + .. + \sin²(90°)}{\cos²(1°) + \cos²(2°) + \cos²(3°) + .. + \cos²(90°)} = ?$$ I tried to use multiple identities but I couldn't simplify the expression. ...
3
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3answers
80 views

Values of the sums $\sum\limits_{k=1}^{n}\cos^4(πk/(2n+1))$

I have been given a question which asks you to prove that $$ \sum_{k=1}^{n}\cos^4\left(\frac{πk}{2n+1}\right)=\frac{6n-5}{16} $$ The main problem I have with solving this is that since the summands ...
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1answer
25 views

When does $-\frac{\pi z}{2}\cot(\pi z)+\frac{1}{2}=0$ where $z$ is a complex variable?

Let $z$ be a complex variable. Is there someone who can show me when does :$$-\frac{\pi z}{2}\cot(\pi z)+\frac{1}{2}=0$$ Note: I have tried using trigonometric formulas but it didn't work. Maybe I ...
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2answers
56 views

Is there any trigonometric function that cannot be written as an infinite series?

Let $p_n(x)=x^n$ for $x\in \Bbb{R}$ and let P=span$\{p_0,p_1,p_2,p_3\dots\}$. Then- P is the vector space of all real valued continuous functions on R P is a subspace of all real ...
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2answers
70 views

Do they have a mistake in this heat equation?

I need to know if there is a mistake in these notes: In the second page we have a representation of a function $f(x)$ as a $\sin$ series. Dont we need to have $f(0)=0=f'(l)$ for such a ...
0
votes
1answer
30 views

What are the unknown lengths are angles for the roof?

Sharma is building a shed and wants to determine the measurements for the roof. The span of the roof will be 10 feet and she plans to use a 5:12 roof pitch. This means that the roof will rise 5 ...
1
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2answers
124 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 ...
8
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0answers
88 views

Which Fourier series are “legal”?

Let's consider a continuous function $f(x)$ and real numbers $\lambda_n=(\alpha+\beta n)\pi$ where both $\alpha$ and $\beta$ are integers. In any interval $I$, is it true that $$ f(x)=\sum_{n\geq ...
2
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2answers
44 views

Cosine and Sine of Sums

What's a good way to simplify $\sin( \sum\nolimits_{i=1}^{\infty} x_i)$ as the product and sum of $\sin(x_i)$ and $\cos(x_i)$ alone? And the same for $\cos( \sum\nolimits_{i=1}^{\infty} x_i)$?
4
votes
1answer
57 views

trigonometric finite series equals to polynomial function [duplicate]

I am interest to prove the equation below : $$ \sum_{k=1}^m \tan^2\left(\frac{k\pi}{2m+1}\right) = m(2m+1) $$ you can understand better the first member of the equation here: WolframAlpha (mark ...
2
votes
2answers
365 views

Algebraic solution to find circle radius given distance of three external points from perimeter

I have an engineering problem, which involves math. The reason it's "engineering" is that I don't need a pure mathematical solution, but a good-enough approximation could work - the only constraint is ...
2
votes
2answers
36 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 ...
8
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2answers
132 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 ...
2
votes
5answers
102 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
46 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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votes
2answers
50 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 ...
2
votes
1answer
65 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
votes
3answers
73 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 ...
5
votes
2answers
112 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
vote
1answer
29 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 ...
0
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1answer
32 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
47 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
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2answers
659 views

Does the sum $\sum^{\infty}_{k=1} \frac{\sin(kx)}{k^{\alpha}}$ converge for $\alpha > \frac{1}{2}$ and $x \in [0,2 \pi]$?

Does the series $$ \sum^{\infty}_{k=1} \frac{\sin(kx)}{k^{\alpha}}, $$ converge for all $\alpha > \frac{1}{2}$ and for all $x \in [0,2 \pi]$? It is obvious that it does when $\alpha > 1$, ...
3
votes
1answer
55 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 ...
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 ...
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2answers
52 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 ...
2
votes
2answers
60 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 $\tan^{-1}$?
0
votes
1answer
36 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 ...
5
votes
0answers
39 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
120 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
32 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
vote
2answers
51 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 ...
1
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1answer
55 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
votes
1answer
32 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
53 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 ...
5
votes
1answer
86 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
43 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, ...
7
votes
1answer
219 views

A problem of Ramanujan's interest: closed form of $1 + 2\sum_{n=1}^{\infty} \frac{\cosh(n\theta)}{\cosh(n\pi)} $

I am Brian Diaz, and I am new to the math.stackexchange community. I have been struggling with attempting to find a closed form of the following series: $$ \varphi(\theta) = 1 + 2\sum_{n=1}^{\infty} ...
2
votes
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
2answers
142 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, $f(n) = \sum_{k=1}^n \csc \frac{\pi k}{2n+1}$ $g(n) = \sum_{k=1}^n [(-1)^{k+1}(\csc \frac{\pi k}{2n+1})]$ ...
4
votes
3answers
159 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
votes
2answers
25 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 ...