For questions about or related to trigonometric series.

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2answers
90 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?
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1answer
39 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 ...
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1answer
49 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} ...
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1answer
302 views

How do I find the direction angle of a vector?

v= -3i-10j I found the dot product to be 109 and the magnitude to be the square root of 11881. I divided it out and it came out to be one. I don't know what I'm doing wrong. Please help.
4
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0answers
94 views

Has this function a name?

Has this function a name? $$f(x) = \prod_{i=2}^{\lceil \frac{x}{2} \rceil} \sin\left( \frac {\pi x}{i}\right)$$ (the product of $\sin( \frac {\pi x}{i})$ for $i=2,3,...,\lceil x/2 \rceil$)
4
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0answers
172 views

On the series $ \displaystyle \sum_{n=1}^{\infty} \frac{\Lambda(n)}{\sqrt{n}} \cos(x \log(n) + a) $.

Is the following series ‘summable’ in the sense that it may be divergent but we can attach a meaning to it? $$ \sum_{n=1}^{\infty} \frac{\Lambda(n)}{\sqrt{n}} \cos(x \log(n) + a) = (\text{reg}) ...
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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 ...
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0answers
65 views

Proof of a series containing normal as well as hyperbolic functions

Show that $$\sum _{n=0}^{\infty}\frac{\sinh(\pi n \sqrt 2)- \sin(\pi n \sqrt 2)}{n^3{\cosh(\pi n \sqrt 2})- \cos( \pi n \sqrt 2)}=\frac{ \pi^3}{18 \sqrt 2}$$ I have no hint as to how to even start.
2
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0answers
63 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 ...
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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} ...
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0answers
66 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) = ...
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0answers
81 views

Extensions of Ramanujan's Ramanujan Cos/Cosh Identity

The Ramanujan Cos/Cosh Identity is as stated here . Then there is a line . Equating coefficients of $\theta^0$, $\theta^4$, and $\theta^8$ gives some amazing identities for the hyperbolic secant. ...
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0answers
25 views

integrate trigonometric function

I have to determine the value of $\int_{-\pi }^{\pi } \left(\cos (3\cdot x)\cdot \sin (x)^2\right)^2 \, dx$. What I did was rewriting the integrand as $$\frac{1}{16}(6\cdot \sin (x)-\sin (3\cdot ...
1
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0answers
35 views

Understanding the indices in a Fourier series

Sometimes the truncated Fourier series of a function with Fourier coefficients $\hat{u}_k$ is written $$\sum_{k=-N}^N\hat{u}_ke^{ikx}$$ which is a linear combination of $\cos(nx) +i\sin(nx)$ for ...
0
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0answers
32 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 ...
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0answers
49 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 ...
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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
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0answers
49 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: ...
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0answers
23 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 ...
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0answers
33 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 ...
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0answers
9 views

Is there a specific name for Fourier cosine series divided by its input?

I am thinking about a univariate model $$ y=a_{0}x^{-1}+\sum_{k=1}^{n}a_{k}\cos(kx)x^{-1}. $$ It seems that this form looks like a Fourier cosine series w.r.t. $x$ divided by $x$. Could you tell me ...
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0answers
46 views

Proving a Certain Inequality that Involves the Sinc Function

Could someone kindly show me how to rigorously prove that there exists a constant $ C > 0 $ such that $$ \forall N \in \mathbb{N}: \quad \sup_{x \in \mathbb{R}} \sum_{\substack{k \in \mathbb{Z} \\ ...
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0answers
125 views

Cosine series as a Fourier series.

Theorem: Let $a_{n}\downarrow 0$ and suppose that $\left(a_{n}\right)$ is a quasi-convex. Then $\displaystyle \frac{a_{0}}{2}+\sum a_{n}\cos nx$ is the Fourier series of the $L^{1}$ function ...
0
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0answers
147 views

Trigonometric series as a Fourier series of essentially bounded function.

A trigonometric series $ \displaystyle \frac{a_0}{2}+\sum_{n=1}^{\infty}(a_{n}\cos nx +b_{n}\sin nx)$ is a Fourier series of a essentially bounded function if and only if there exists a constant $K$ ...