For questions about or related to trigonometric series.

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4
votes
3answers
129 views

Closed form for $\prod_{l=1}^\infty \cos\dfrac{x}{3^l}$

Is there any closed form for the infinite product $\prod_{l=1}^\infty \cos\dfrac{x}{3^l}$? I think it is convergent for any $x\in\mathbb{R}$. I think there might be one because there is a closed form ...
1
vote
0answers
28 views

Explaining a Series Expansion for the Divisor Function

The "sum-of-divisors" function, defined as $\sigma(n)=\sum_{d\,\mid\,n}d$, can be expressed as $$\sigma(n)=\sum_{i=1}^n\sum_{j=1}^i\cos\Big(2\pi n \frac{j}{i}\Big)$$This expansion makes logical sense. ...
2
votes
1answer
33 views

Sum of cosines with a multiplicative factor in the angle and different interval

I have found the following formula for the sum of cosines in both here and here. \begin{align} \sum^n_{l=1} \cos \left(\frac{2 \pi l}{n}\right) = 0 \end{align} I would like to know what the sum ...
0
votes
1answer
31 views

Writing an operator $T$ defined by $(T f)(t) = \int_{-\pi}^\pi h(t − s)f(s)ds$ as $\sum_{n \in \mathbb Z} \mu_n \langle f, \varphi_n\rangle \varphi_n$

Let $h$ be a continuous function with period $2\pi$. Define $T : L_2[−\pi, \pi] \to L_2[−\pi, \pi]$ by $(T f)(t) = \int \limits _{-\pi}^\pi h(t − s)f(s)ds$. Let $\{\varphi_n(t) =\frac{1}{\sqrt{2\pi}} ...
18
votes
3answers
343 views

A Sine and Inverse Sine integral

A demonstration of methods While reviewing an old text book an integral containing sines and sine inverse was encountered, namely, $$\int_{0}^{\pi/2} \int_{0}^{\pi/2} \sin(x) \, \sin^{-1}(\sin(x) \, \...
0
votes
0answers
49 views

Expression for $\arccos\frac2\pi$

Are there any 'nice' expressions for $$\arccos\frac2\pi$$ By 'nice', I mean involving rationals, square roots, powers of $\pi$, etc... I'm hoping there is something that doesn't involve sum or ...
1
vote
0answers
33 views

A geometric proof for the “small angle approximation” for the sine, cosine and tangent

How can be deduced the so called "small angle approximations" for sine, cosine and tangent, namely $\sin \theta \approx \theta$ $\tan\theta \approx \theta$ $\cos\theta \approx 1-\frac{\theta^2}{2}$ ...
1
vote
1answer
93 views

Solve $\sin^2 x − \cos^2 x = \sin x$, when $x\in -\pi ≤ x ≤\pi$

I have to solve for $x$ using the domain of $-\pi ≤ x ≤\pi$ $$\sin^2 x − \cos^2 x = \sin x $$ I tried changing $\cos^2 x$ to $1 - \sin^2 x$, and then getting $$\sin^2 x - 1 + \sin^2 x = \sin x \to ...
0
votes
1answer
74 views
2
votes
1answer
28 views

Using a taylor polynomial to approximate $cos(\frac14)$ with an error no more than $10^{-12}$

So the lagrange remainder is given by: $$R_n(x)=\frac{f^{n+1}(c)}{(n+1)!}(x-a)^{n+1}$$ We want $cos(\frac14)$ and we can do it around a=0. We know that $f^{n+1}$ is either $\pm$ cosx or $\pm$sinx, ...
0
votes
1answer
35 views

Trigonometric fraction Inequality question

Let $$ 1< \alpha_1<\alpha_2<......<\alpha_n< \pi/2$$ then prove that $$ tan(\alpha_1)< \frac{sin(\alpha_1)+sin(\alpha_2)+.....+sin(\alpha_n)}{cos(\alpha_1)+cos(\alpha_2)+......+...
0
votes
2answers
71 views

Other types of closed form for $\sum_{n=1}^\infty \frac{\cos(nx)}{n}$ in specific interval

Is there other types of closed form for the following sum? $$\sum_{n=1}^\infty \frac{\cos(nx)}{n}$$ needs to be valid over $ 0\le{x}\le{1}$ and need to be real numbers. The form below does not ...
1
vote
2answers
40 views

Given $\tan 3x=4$, find the value of $\tan^2 x+\tan ^2(120+x)+\tan^2 (60+x)$

Given $\tan 3x=4$, find the value of $S=\tan^2 x+\tan ^2(120+x)+\tan^2 (60+x)$ I expanded each of $\tan (120+x)$ and $\tan (60+x)$ getting as $$S=\tan^2 x+\left(\frac{\tan 120+\tan x }{1-\tan 120 \...
0
votes
1answer
38 views

How does $ \frac{1}{x}\left(\frac{\pi}{2} - \arctan\frac{1}{x}\right)$ simplify to $\frac{1}{x} \arctan x $?

Here is a solution I read when trying to solve a problem, and I can't figure out how it jumped in this step here: $$ \frac{1}{x}\left(\frac{\pi}{2} - \arctan\frac{1}{x}\right) = \frac{1}{x} \...
1
vote
0answers
23 views
3
votes
1answer
53 views

$\lim\limits_{i\mapsto \infty} \sum_{i=1}^\infty \sin(\theta_{i+1}-\theta_{i})$ = ? , where $\theta_{i}= \pi\sum_{j=0}^i \frac{1}{{(2)}^j}$

$\lim\limits_{i\mapsto \infty} \sum_{i=1}^\infty \sin(\theta_{i+1}-\theta_{i})$ = ? $\theta_{i}= \pi\sum_{j=0}^i \frac{1}{{(2)}^j} = \pi \left(2 - \frac{1}{2^{i}}\right)$ The limit $\lim\limits_{i\...
-2
votes
1answer
35 views

a Proof onInverse Trigonometry

Ff $\arcsin x + \arcsin z + \arcsin z = 1.5\pi$, Prove that $x^{2006}+y^{2007}+z^{2008}-\frac 9{x^{2006}+y^{2007}+z^{2008}}=0$.
1
vote
1answer
23 views

How does $\cos(x)\cdot\cos\left(\frac{3}{2}x\right)$ become $\frac{1}{2}\left(\cos\left(\frac{1}{2}x\right) + \cos\left(\frac{5}{2}x\right)\right)$?

How can you rewrite $\cos(x)\cdot\cos\left(\frac{3}{2}x\right)$ to $\frac{1}{2}\left(\cos\left(\frac{1}{2}x\right) + \cos\left(\frac{5}{2}x\right)\right)$? What rules have been used? I need it on ...
0
votes
2answers
115 views

Find the number of solutions of $\sin x+2 \sin 2x- \sin 3x=3$

In $(0 \:\:\pi)$Find the number of solutions of $$\sin x+2 \sin 2x- \sin 3x=3$$ The equation can be written as $$\sin x+4 \sin x \cos x=3+\sin 3x$$ i.e. $$\sin x(1+4\cos x)=3+\sin 3x$$ i.e., $$\...
1
vote
1answer
43 views

Trigonometric series convergence

I was interested in evaluating $$ \sum_{n=1}^{\infty} \frac{\cos n}{n+k} $$ I saw with computation that many of such series converge. Is there a general result? I've tried using Taylor expansion of ...
17
votes
0answers
316 views

Simplify the sum $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$? - a sum shows all primes $\le N^2$

I was looking for a closed form but it seemed too difficult. Now I'm seeking help to simplify this sum. The 50 bounty points or more will be awarded for any meaningful simplification of this sum. I ...
0
votes
2answers
43 views

Product of trigonometric polynomials is a trigonometric polynomial

A trigonometric polynomial was defined as $$f(x) = \frac{a_0}{2} + \sum_{k=1}^{n}(a_k \cos(kx) + b_k \sin(kx))$$ I heard somewhere that trigonometric polynomials have a ring structure, i.e. a product ...
0
votes
2answers
62 views

Coefficients of a cosine series

Let $u$ have the cosine series representation $$u = \sum_{k_1=0}^{\infty} \sum_{k_2=0}^{\infty} a_\underline{k} \cos\left(\frac{2\pi k_1 x }{L_1}\right) \cos\left(\frac{2\pi k_2 y }{L_2}\right) $$ ...
0
votes
0answers
24 views

What is support and spectrum of this nonnegative trigonometric function (or Finite Fourier Sum )?

This is a follow up of another question. The zeros of the following cosine sum shows the prime distribution, and the gap between the zeros can help to study the gap between prime numbers. $$ P(p_i,x)=\...
3
votes
4answers
102 views

How to evaluate this limit using Taylor expansions?

I am trying to evaluate this limit: $\lim_{x \to 0} \dfrac{(\sin x -x)(\cos x -1)}{x(e^x-1)^4}$ I know that I need to use Taylor expansions for $\sin x -x$, $\cos x -1$ and $e^x-1$. I also realise ...
2
votes
2answers
46 views

Evaluate the following trignometric sum

I am interested in the following sum $$\sum_{\text{even } n=-\infty}^{\infty}\left(-\cos^2x\delta_{n,0}+\cos x\left(\frac{1-\cos x}{\sin x}\right)^{|n|}\right).$$ Wolfram alpha returns answer $\sin^...
1
vote
2answers
55 views

Why do I get different results for the same integral?

The variables $ a, b, s, c $ are constants, so: $$ \int \left ( a \cos(s + cx) - b \sin(s + cx) \right ) dx = \frac{a\sin(s + cx) + b\cos(s + cx)}{c} +C $$ But if $c=0$ then: $$ \int \left ( a \...
4
votes
0answers
82 views

Finite Messy Trigonometric Sum

Show the following result:$$\sum_{m=1}^{99}{\frac{\sin{\left(\frac{17 m \pi}{100}\right)} \sin{\left(\frac{39 m \pi}{100}\right)}}{1+\cos{\left( \frac{m\pi}{100} \right) }}}=1037$$ The source of this ...
0
votes
0answers
23 views

On a certain series of cosines

For natural numbers $p$ and $q$, compute the value of $$\displaystyle \sum_{k=0}^{q-1} \cos^{p} \left(\dfrac{2\pi k}{q}\right).$$ I got the answer $$\dfrac{q}{2^p} \sum_{l=1}^{p} \binom{p}{l} \...
2
votes
2answers
56 views

prove $\sum_{k=1}^n(-1)^{k+1}\cos^{2n+1}\left(\dfrac{k\pi}{2n+1}\right)=\frac{1}{2}-\frac{2n+1}{2^{2n+1}}$

Today I found the identity : $$\sum_{k=1}^n(-1)^{k+1}\cos^{2n+1}\left(\dfrac{k\pi}{2n+1}\right)=\frac{1}{2}-\frac{2n+1}{2^{2n+1}}$$. How to prove or disprove this? Thank you.
1
vote
1answer
43 views

Computation of an inverse trigonometric series using complex numbers

The following is a popular question (in competitive exams) in India: Compute the value of $S=\displaystyle \sum_{k=1}^{\infty} \tan^{-1}\left( \dfrac1{2k^2}\right)$. I can compute the value by ...
2
votes
1answer
31 views

How to get from $\sum_{n=0}^N (a_n \cos{nx} + b_n \sin{nx})$ to $\sum_{-N}^{N} c_n e^{inx}$?

I'm currently reading Baby Rudin, and I'm in the section of Chapter 8 that covers Fourier series. There is one line that I just can't figure out for the life of me, and I can't find anything online ...
0
votes
1answer
32 views

How to appraoch solving this series?

I am given the following series and asked to solve it. $\sum\limits_{n=1}^\infty \dfrac{(-2)^n}{(2n+1)!}$ I recognize that this series is somewhat similar to the Taylor series for $sinx$ which is $\...
-1
votes
1answer
61 views

Prove $a_n = \sin(n\pi /2)$ does not converge to $0$

Not sure how to prove it doesn't converge to $0$. Prove by contradiction? How do I assume that the sequence converges to $0$ since if I do the partial sums it does end up adding to $0$.
0
votes
1answer
31 views

How to find the scaling with N for FWHM and MAX of $F(X) = (1+\cos((2N+1)\pi X)/(1+\cos(\pi X))$?

Given a function $F(X) = (1+\cos((2N+1)\pi X)/(1+\cos(\pi X))$. Peaks are at odd $X$ integer values. How to find the scaling with $N$ of the FWHM (full width half max) and peak max? For example for ...
1
vote
3answers
57 views

How to find $\sum_{k=1}^nk\sin\left(k\pi/n\right)$ [duplicate]

I would appreciate if somebody could help me with the following problem Q: How to find ? $$\sin\left( \frac{\pi}{n}\right)+2\sin\left( \frac{2\pi}{n}\right)+\cdots+ n\sin\left( \frac{n\pi}{n}\right)$$...
2
votes
1answer
114 views

new $\arctan$ series working for any $x$?

Using Log series, we can write: $$ \frac{1}{2 i} \left( \text{Log} \left( 1 + e^{\text{i2t}} \right) - \text{Log} \left( 1 + e^{\text{-i2t}} \right) \right) = \frac {1} {2 i}\left ( \sum _ {k = 1}^{\...
1
vote
2answers
68 views

Uniform convergence of a trigonometric series

Let $L>0$ be a constant. With what coefficients $\alpha_k$ and $\beta_k$ does the trigonometric series $$ \alpha_0 +\sum_{k=1}^{\infty} \left[\alpha_k \cos\left( \frac{k\pi x}{L} \right) ...
2
votes
4answers
74 views

How to transform the equation $\sum_{k=1}^{n}{sin(kx)}$? [duplicate]

How to prove this equation or how to transform the left part to the right: $$\sin(x)+\sin(2x)+\dots +\sin(nx)=\frac{\sin(\frac{nx}{2})\sin\left(\frac{(n+1)x}{2}\right)}{\sin\left(\frac{x}{2}\right)}$$ ...
0
votes
0answers
32 views

For a real number $r>0$ for the function $\cot(z)$ has a Laurent series where 0<|z|<r. What is the largest possible r?

Okay so I know that $\cot(z)$ expands as: $$ \cot(z) = \frac{1}{z} - \frac{z}{3} - \frac{z^3}{45} - \ldots$$ But I'm unsure how to find the largest $r$. I'm assuming this is the point where it ...
1
vote
0answers
51 views

Laurent series for $\cot(\pi z)/z^2$

I'm looking for the series for $\frac{cot(\pi z)}{z^2}$ using the residue theory, where the function denotes a circle about the origin with a radius of $k+\frac{1}{2}$ I found that the residues of $\...
1
vote
2answers
67 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.
4
votes
0answers
38 views

Sum of Squares of Tangents [duplicate]

Find $S$ if $S=\tan^21^\circ+\tan^23^\circ+\tan^25^\circ+\dots+\tan^289^\circ$. I tried converting all tangents above $45^\circ$ to cotangents and added them with tangents with the same angle, but ...
1
vote
1answer
55 views

Error of Taylor Series?

Part of my assignment is to find the third degree Taylor Series of $\tan(x)$ about $\pi/4$ and then estimate the error of this approximation when evaluated at 0.75. Finding the series was easy enough,...
0
votes
1answer
37 views

Find $\sum_{k=0}^6cot({\pi\over 21} + {k\pi\over 7})$

Find $$\sum_{k=0}^6cot({\pi\over 21} + {k\pi\over 7})$$ I don't know how to do this at all. I initially started using complex numbers but couldn't get it. Please give a simple solution.Thanks.
0
votes
0answers
26 views

Defining trigonometric functions for obtuse angle

I realized that following is apparently true: we had definitions of sine, cosine, etc. for angles of right triangle. Then, one suddenly draws X and Y axes, incorporates negative numbers, and sees that ...
0
votes
1answer
22 views

How to show: $-\sum_{k=-\infty}^\infty \frac{i*k*(-1)^k}{1+k^2}e^{-i*k*x}=2\sum_{k=1}^\infty \frac{k(-1)^{k-1}}{1+k^2}sin(kx)$

I am struggling to show below in a big question: $-\sum_{k=-\infty}^\infty \frac{i*k*(-1)^k}{1+k^2}e^{-i*k*x}=2\sum_{k=1}^\infty \frac{k(-1)^{k-1}}{1+k^2}sin(kx)$ Tried to analyse with geometric ...
0
votes
1answer
31 views

Converting certain complex exponentials to trigonometric functions

The original question is: Show that $$f(x)=\sum_{k=-\infty}^\infty c_k e^{-i kx}=\frac{2\sinh(\pi)}{\pi} \sum_{k=1}^\infty \frac{(-1)^{k-1}k}{1+k^2}\sin(kx)$$ where $\displaystyle c_k=\frac{1}{4\...
6
votes
1answer
104 views

Simplify Product of sines

Is there a way simplify this product? $$ \sin\left({n} \frac{\pi}{2}\right) \sin\left({n} \frac{\pi}{3}\right) \sin\left({n} \frac{\pi}{4}\right) ...\sin\left({n} \frac{\pi}{n-1}\right) $$ And, is ...
1
vote
0answers
85 views

Value of $\tan(1^{\circ})+\tan(2^{\circ})+\tan(3^{\circ})+…+\tan(89^{\circ})$

Is there any method to find value of $$\tan(1^{\circ})+\tan(2^{\circ})+\tan(3^{\circ})+\dots+\tan(89^{\circ})$$ without using calculator. To find the same sum for sine and cosine , I used De-Movier's ...