For questions about or related to trigonometric series.

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3
votes
1answer
49 views

Why do the averages of $\sin (p+\cos p )$ approach a positive limit?

$$S_1 = \sum_{k=1}^n (\sin \left[~ p(k) + \cos p(k)~ \right])$$ I wonder why this appears to give $$\frac{1}{n}S_2\sim 1/2 $$ Thanks for any insights or references. Edited in light of ...
0
votes
2answers
14 views

Use of small approximations to get the following formula

I cannot see how the following formula has been found $\displaystyle \cos(\theta + \frac{v\epsilon^2 \Omega}{L}+O(\epsilon^4))-\cos(\theta) = -\epsilon^2\frac{\Omega v}{L}\sin(\theta)+O(\epsilon^4)$ ...
-1
votes
2answers
46 views

How can I find fifth root of unity?

I have no idea to do this question, how can I find the fifth root of unity? Question : Find all the distinct fifth root of unity. Let $\alpha$ be a fifth root of unity such that $\alpha \ne 1$. ...
1
vote
0answers
28 views

Estimation of trignometric polynomial and lipshitz estimation

Let $\mathcal{T}_n$ denote the linear space of trigonometric polynomials of degree up to $n$ and $$E_n(f)=\inf_{P\in\mathcal{T}_n} ...
0
votes
1answer
24 views

MacLaurin of the Third-degree in sin(a*x)*cos(b*x) at given values

Alright so from my understanding MacLaurin is a special case of Taylor Series but at f(0). However my question involves solving the third degree of MacLaurin for $$f(x) = sin(a \times x)\times ...
0
votes
1answer
22 views

Is it possible to rearrange this for x?

Is it possible to rearrange $\tan(y) = \frac{\sin(x)}{\cos(x)+C}$ for x, where C is a constant? Thank you for any suggestions!
0
votes
0answers
13 views

Understanding this question in regards to Taylor Polynomials

I am looking at the following question, and I simply don't understand it. I've calculated the Taylor series for $f(x) = \arcsin(x)$ centered around $0$ of order $n=3$. Evaluating this series at $x = ...
1
vote
1answer
21 views

Cosine of the Natural Logarithm - Series Expansion

I am interested in a computable series expansion of the following equation: $f(n) = \cos(\log(n))$ Specifically, I am interested in real values of $n$ where $n>1$. From basic series definitions ...
6
votes
0answers
77 views

Solving a problem of Ramanujan's interest

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} ...
6
votes
2answers
132 views

How do I show that the series: $\sin(m) + \sin(\sin(m)) + \sin(\sin(\sin(m))) + \cdots$ converges for all real numbers $m$?

I'm trying to show that the series: $\sin(m) + \sin(\sin(m)) + \sin(\sin(\sin(m)) + \cdots$ converges for all real numbers $m$. To be specific, the series is defined as follows: $\sum_1^\infty{a_k}$ ...
0
votes
0answers
26 views

How to calculate the sample autocorrelation coefficient of this time series?

Given : $\hat{\gamma}(h)=n^{-1}\Sigma_{t=1} ^{n-|h|}(x_{t+|h|}-\bar{x})(x_{t}-\bar{x})$ $\bar{x}=\frac{1}{n}\Sigma_{t=1} ^n x_{t}$ $\hat{\rho}(h)=\frac{\hat{\rho}(h)}{\hat{\rho}(0)}$ $x_{t}=c \cdot ...
3
votes
1answer
91 views

Measure of the set where a trigonometric polynomial with zero mean is non-negative

Suppose $f$ is a real trigonometric polynomial of degree $N$ with constant term $0$. What lower bounds can we place on the measure $\mu$ of the set $\{ t \in S^1 : f(t) \geq 0 \}$, independent of the ...
1
vote
2answers
47 views

Sum of Complex series

Let $\theta\in\mathbb{R}$ and $\theta \neq k\pi$ for $k\in\Bbb Z$. By summing a geometric progression show that $$1 + e^{2i\theta} + e^{4i\theta}+e^{6i\theta} + e^{8i\theta}= ...
1
vote
1answer
61 views

Sum of the trigonometric series

I'm studying de Moivre's theorem's application on the summation of trigonometric series. Here's what I have so far: \begin{align*} \sum_{k=0}^n \cos(k\theta)&= \text{Re}\sum_{k=0}^n e^{ki\theta} ...
0
votes
1answer
39 views

Trigonometric series problem: finding a second valid solution.

Given that I can do part of this question so here goes: Substituting $\theta=\frac{1\pi}{11}$ into LHS of given expression gives $$\cos\frac{1\pi}{11} + \cos\frac{2\pi}{11} + \cos\frac{3\pi}{11} ...
0
votes
2answers
34 views

Problem involving summing exponential series:

I can show the first part (i) (a), but the second part (b) i think it should be $S=\infty$ since the denominator is zero with that value of $\theta$. However, this is not the answer, any ideas? ...
2
votes
1answer
44 views

Sequence of trigonometric polynomials which converges to an integrable function

A function $f:\mathbb{R}\to \mathbb{C}$ is said to be a trigonometric polynomial if it has the form $$f(x)=\sum_{k=-N}^Na_ke^{ib_kx},$$ where $a_k\in \mathbb{C}$ and $b_k\in \mathbb{R}$. Can we find ...
1
vote
1answer
65 views

Hard Trigonometric Equation

its possible to solve a equation like $$\prod^{45}_{k = 0} \left( 1 + \tan \frac{k \pi }{180} \right) = \left[ \log_{\frac{\sqrt{6}}{3}} \big| \sin(2x)\big| \right]^{\frac{9}{2}\sin(2x) + 20}$$ ...
2
votes
2answers
123 views

Differentiability of the sum of the series $\sum_k \sin(kx)/k^2$

How to show the following: If $ f(x) = \displaystyle\sum_{k=1}^{\infty} \dfrac {\sin(kx)}{k^2} $, then show that $f(x)$ is differentiable on $(0,1)$ I guess it should be related to uniform ...
0
votes
1answer
54 views

Solve for x: sin2 x − cos2 x = sin x, −π ≤ x ≤π

I have to solve for x using the domain of −π ≤ x ≤π sin^2 x − cos^2 x = sin x I tried changing cos^2 x to 1 - sin^2 x so it would be sin^2 x - 1 + sin^2 x = sin x making it, 2sin^2 x - 1 = sin x ...
0
votes
0answers
36 views

Using Algebra with Trig Functions

Using Algebra with Trig Functions I'm trying to find the correct 1 second audio signal I would need to apply to a 1 second known noise signal to have the output signal be a sin wave. The basic ...
1
vote
3answers
72 views

Evaluation of a trigonometric series

$$ \mbox{Question: Evaluate}\quad \tan^{2}\left(\pi \over 16\right) + \tan^{2}\left(2\pi \over 16\right) + \tan^{2}\left(3\pi \over 16\right) + \cdots + \tan^{2}\left(7\pi \over 16\right) $$ What I ...
0
votes
0answers
50 views

Trigonometry - proving a rule

I was given that $\sin a + \sin b + \sin c \cdots$ is equal to: (where $a,b,c$ are in arithmetic progression) $$\frac{\sin\frac{a + c}{2}\sin\frac{nb}{2}}{\sin{b/2}}$$ Here $a$ is the first term of ...
0
votes
0answers
27 views

Help in finding a book

That is my problem: i have this series: $\sum_{k=0}^{\infty} \psi_{k}(h)\cos(k\theta)$. I need to see to what it converges assuming something on the functions $\psi_{k}(h)$. There is a book or a site ...
0
votes
2answers
53 views

Proving that $\sum^n_{k=1} e^{ik\theta}=\sum^n_{i=1}\cos k\theta +i\sum^n_{k=1}\sin k\theta$.

Prove: $$\sum^n_{k=1} e^{ik\theta}=\sum^n_{i=1}\cos k\theta +i\sum^n_{k=1}\sin k\theta$$ Thanks a lot!! I tried: With Euler's identity I can get $\sin x= \dfrac{e^{ix} - e^{-ix}}{2i}$ and the ...
1
vote
0answers
105 views

Is there a relationship between a function's period and number of roots?

Let: $f(x,a,l)=\prod _{k=a}^l\sin \left(\frac{\pi x}{k}\right)$ and $f(x,k)=\sin \left(\frac{\pi x}{k}\right)$ I came up with this equation to find the period $T(f(x,a,l))=2\,{{\pi }^{l-a+1} ...
3
votes
2answers
77 views

$\int _0^{\pi }\:\sum _{n=0}^{\infty \:}\frac{n\cdot \sin \left(nx\right)}{e^n}dx=\frac{2e}{e^2-1}$

It is asked to prove: $\int _0^{\pi }\:\sum _{n=0}^{\infty \:}\frac{n\cdot \sin \left(nx\right)}{e^n}dx=\frac{2e}{e^2-1}$ I have tried to search for convergence and it gave me 0 so i can't solve it. ...
1
vote
1answer
25 views

Closed-form expressions for coefficients $a_k$ in $2^{2n-1}\sinh^{2n}(x)=\sum_{k=0}^n a_k \cosh(2kx)$

It is known that $$2^{2-1}\sinh^{2}(x)=\cosh(2x)-1$$ $$2^{4-1}\sinh^{4}(x)=\cosh(4x)-4\cosh(2x)+3$$ What are the closed-form expression for coefficients $a_k$ ($k=0,1,\cdots,n$) in the expression ...
4
votes
2answers
178 views

Find the limit of $\prod_{k = 4}^{\infty}\cos\left(\pi \over k\right)$

Find the limit of $$\prod_{k = 4}^{\infty}\cos\left(\pi \over k\right)$$ The limit does exist, but I can not get it. Thanks Willie-Wong & Lee Mosher for correcting the expression.
6
votes
4answers
492 views

A sine integral

The integral \begin{align} \int_{0}^{\pi/2} \frac{ \sin(n\theta) }{ \sin(\theta) } \ d\theta \end{align} is claimed to not have a closed form expression. In this view find the series solution of the ...
1
vote
2answers
43 views

$\frac{1}{\sin\theta\cdot \sin2\theta} + \frac{1}{\sin2\theta\cdot \sin3\theta} + \cdots + \frac{1}{\sin n \theta \sin (n+1)\theta}$ [closed]

$$\sum_{k=1}^n \frac{1}{\sin k\theta \sin (k+1)\theta} = \dfrac{1}{\sin\theta\cdot \sin2\theta} + \dfrac{1}{\sin2\theta\cdot \sin3\theta} + \cdots + \frac{1}{\sin n \theta \sin (n+1)\theta}$$ up to ...
2
votes
4answers
59 views

Can somebody explain to me why these terms are equal?

I just read a proof on ProofWiki that proves Euler's formula, but I can't seem to understand what is done in this following step: ...
0
votes
0answers
61 views

Find a short formula for $\sin x+\sin (x+y)+\sin (x+2y)+. . .+\sin (x+(n-1)y)$

The answer is : $$\sin(\frac {x+x+(n-1)y}{2}) \dfrac {\sin \frac{ny}{2}}{\sin \frac {y}{2}}$$ I could've written the question as: Show that..., but then people would try induction. What I did: ...
2
votes
0answers
72 views

Proving that sin(x)/x=(1-x^2/pi^2)(1-x^2/4pi^2)(…)

I am faced with explaining to a bunch of people who have taken a course in real analysis, but no course in complex analysis, why $\sin(x)=x\prod_{n\geq1}(1-x^2/\pi^2n^2)$. I vaguely remember being ...
0
votes
0answers
44 views

trigonometric polynomial using FFT

I am trying to use the FFT to approximate a given function. So i have 10 points xk that are given for example, if i use the FFT that will give me Xk. So now using the inverse FFT we can get the ...
0
votes
1answer
94 views

Integral is equal to $0$

Let be $f \in L^1[0,1]$, then it applies $ \int_0^1 \exp(2i\pi xk)f(x n)\,dx=0$ for $n,k\in \mathbb{N}$ with $0<k<n$. Ideas: f can be extended to a function on $\mathbb{R}$ with period $1$, ...
4
votes
3answers
103 views

How does WolframAlpha simplify sine and cosine?

When I feed WolframAlpha an expression like $\sin({\pi\frac{2}{3}})$, it correctly prints that this is equal to $\frac{\sqrt3}{2}$, instead of the decimal expansion $0.866025403\ldots$. Perhaps it ...
1
vote
1answer
26 views

Determine the maximal compact interval such that $\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1} = \arctan(x)$ holds true

The Assignment: Determine the maximal compact interval, such that the following identity holds true:$$\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1} = \arctan(x)$$ Explain your answer and show ...
1
vote
1answer
34 views

Finding the limit using McLaurin series and/or “little o” method.

Another mathematical analysis question to do with limits. Tried a few approaches, but I'm not getting the correct answer. Question:Find the limit of: $\lim_{x\rightarrow0}\frac{\sqrt{\cos ...
1
vote
0answers
108 views

Fourier Series; odd and even half-range expansion

I have some standard Fourier series questions which I cannot solve. My fourier series is defined like this: $$s(x)=\frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos (nx) + b_n \sin (nx))$$ For $f(t) = ...
3
votes
0answers
65 views

Finite Trigonometric Sum

I have a dynamical system model whose equilibria depend on the solution of the following finite sum: \begin{align} \sum_{j\neq ...
3
votes
2answers
185 views

Taylor Series of $\tan x$

I found a nice general formula for the Taylor series of $\tan x$: $$\tan x = \sum_{n\,=\,1}^\infty \frac {(-1)^{n-1}2^{2n} (2^{2n}-1) B_{2n}} {(2n)!} x^{2n - 1} $$ where $B_n$ are the Bernoulli ...
0
votes
0answers
38 views

Binomial series for $2^{n-1}\cos^n\vartheta$ and $2^{n-1}(-1)^{\frac{n}{2}}\sin^n\vartheta$

Can somebody confirm for me whether the following series are correct? $$2^{n-1}\cos^n\vartheta=\cos ...
2
votes
1answer
55 views

Explain how the following is equal to $2\cos x$.

The question was Prove $$\frac{1+\sin2x+\cos2x}{\cos x+\sin x}=2\cos x$$ I simplified it using several trigonometric identities, what I got is this "$\dfrac{2\cos^2 x + 2\cos x \sin x}{\cos x + ...
2
votes
1answer
42 views

Solving $\operatorname{ctg} x=x/b$

I have no problems finding first solution (both: $b \to 0$ and $b \to \infty$). My solutions on photos. I got stuck trying to find solution when $x \to \infty$. As I think, solution for $x$ will have ...
2
votes
3answers
100 views

History of infinite series representations of $\sin(x)$ and $\cos(x)$

When did the famous infinite series representations for $\sin(x)$ and $\cos(x)$ came about? To be specific when did people realise that the ratio of the two sides of a right triangle with one angle ...
1
vote
3answers
40 views

Find the value of $\sin(B-A)$.

If $A$ is an acute angle whose tangent is $\frac{15}{8}$ and $B$ is and obtuse angle whose sine is $\frac{12}{13}$, find $\sin (B-A)$. [Without calculators] I suppose I gotta use this formula: $\sin ...
1
vote
2answers
62 views

Where is the series $\displaystyle\sum_{n=1}^{\infty}\dfrac{\cos(nx)}{n}$ pointwise convergent?

Where is the series $\displaystyle\sum_{n=1}^{\infty}\dfrac{\cos(nx)}{n}$ pointwise convergent? I tried to apply the Dirichlet's test but I couldn't.
2
votes
1answer
143 views

What is a fast method for evaluate this trigonometric series?

$$\sum_{n=1}^{11}\sin^{14}\left(\theta+\frac{2n\pi}{11}\right)=?$$ By wolfram alpha, we know that ...
8
votes
2answers
157 views

To compute $\tan1-\tan3+\tan5-\cdots+\tan89$, $\tan1+\tan3+\tan5+\cdots+\tan89$

How do we compute : $$i)\ S_1 = \tan1-\tan3+\tan5-\cdots+\tan89$$ and $$ii)\ S_2 = \tan1+\tan3+\tan5+\cdots+\tan89$$ all the angles are in degrees. Thanks