For questions about or related to trigonometric series.

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8
votes
2answers
114 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
0
votes
1answer
34 views

Alternating cosine series, what is the closed form?

What is the closed form for this series: $$\sum _{n=0}^{\infty } \left(\frac{\cos \left(\frac{4 \pi}{2 n+1}\right)}{2 n+1}-\frac{\cos \left(\frac{4 \pi}{2 n+2}\right)}{2 n+2}\right)$$ if any? I am ...
6
votes
2answers
53 views

What is the Taylor series of $\frac{1}{\sin(z)}$ about $z_0 = 1$?

This was a exam question so I know it cannot take too long to write out the proof. Only I cannot see an answer. I would imagine you write $\sin(z) = \sin(1+(z-1)) = \sin(1)\cos(z-1) + ...
1
vote
1answer
42 views

Why is the Taylor expansion of $\cos$ decreasing?

Why is the Taylor expansion of $\cos$ decreasing ? $\cos(t)=1-\frac{t^2}{2!}+\frac{t^4}{4!}-\frac{t^6}{6!}+...$ such that one can estimate $\cos(t)<1-\frac{t^2}{2!}+\frac{t^4}{4!}$ I ...
1
vote
0answers
49 views

L1 norm of a trigonometric polynomial

For a real $x$, $f(x) = \sum_{k=-T}^{T}e^{ikx}$ is the well known Dirichlet kernel. It is also known that $\|f\|_{L_1}=\int|f(x)|dx \le C_1\log T + C_2$ for some $C_1,C_2$ independent of $T$. ...
0
votes
1answer
47 views

Why do different trig functions sum differently?

Why does the $\sum_{n=1}^{\infty} \sin (\frac 1 {n^2})$ converge but the $\sum_{n=1}^{\infty} \cos (\frac 1 {n^2})$ diverge?
0
votes
0answers
7 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 ...
0
votes
1answer
21 views

Series of Certain Cosines

Let $m \in \mathbb{N}: m > 2$, and define $\theta_{i} = \frac{2\pi*(i-1)}{m} \forall i \leq m$. How can I show that $\sum_{i=1}^{m}(cos(2 \theta_{i})) = \sum_{i=1}^{m}(cos(\frac{4\pi(i-1)}{m})) =0$ ...
1
vote
1answer
55 views

Does the expansion of $\sin x$ contradict the normal formula $\sin x = \frac{\text{Perpendicular}}{\text{Hypotenuse}}$?

Lets say I have a right angled triangle with sides $3, 4$ and $5$ units. They form a perfect Pythagorean triplet. One of the angles in the triangle, say $\alpha$ must have the following condition: ...
1
vote
2answers
90 views

Trigonometric series problem

I have the following problem from my Fourier analysis book I would need some guidance with. I have tried it, but apparently I made some mistakes...here is my problem: We have: $$\sin \theta ...
1
vote
2answers
70 views

Positivness of the sum of $\frac{\sin(2k-1)x}{2k-1}$.

For $n\in \mathbb{N}$, $x\in (0,\pi)$. Prove that : $$f_n(x)=\sum_{k=1}^n \frac{\sin [(2k-1)x]}{2k-1} \geq 0.$$ I've tried to do it by differentiation : I Calculate $f_n'(x)$ (sum of ...
0
votes
2answers
50 views

What is the pupose of using minute and seconds with degree?

I want to know that converting from degree to radian and radian to degree is just understandable. But what is the purpose of expressing degree in term of minutes and seconds.. I know that there are 60 ...
0
votes
1answer
24 views

Integrating Trigonometric Series

Let $f(x)=c_0+c_1e^{i\theta}+c_2e^{2i\theta}+...+c_ne^{ni\theta}$ where $c_k\in \mathbb R$. We need to show $$\int_{0}^{2\pi}f(e^{i\theta})\overline {f(e^{i\theta})} d\theta =2\pi\sum_{k=0}^{n} c_k$$ ...
1
vote
1answer
20 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.
0
votes
2answers
62 views

Evaluating $\int_0^{\frac{\pi}2}\frac{\sin 2x}{\sqrt{x}}\,dx$

$$\int_0^{\frac{\pi}2}\frac{\sin 2x}{\sqrt{x}}\,dx$$ How to solve this trigonometric integral? I can't find any solutions. Some books suggest to use Fresnel integral. I would be grateful if you could ...
0
votes
0answers
50 views

is there any other method to solve this limit [duplicate]

If someone prove this $$ \lim_{x \to 0}\frac{\sin{x}}{x} =1$$ other than using L'Hospital or series expansion, it will be really appreciable
1
vote
3answers
30 views

Series representation of $\sin(nu)$ when $n$ is an odd integer?

So, out of boredom and curiosity, today I came up with a series representation for $\sin(nu)$ when $n$ is an even integer: $$\sin(nu) = \sum_{k=1}^\frac n2 ...
1
vote
3answers
38 views

How to prove that $\Sigma_{n=1}^{N-1}{\sin^2\frac{p\pi{}n}{N}} = \frac{N}{2}$

Here $p$ is a known integer constant ($p > 0$). I know that this is true for a fact (checked numerically in Matlab and it holds), but I'm just not able to prove it. I noticed a similar problem ...
0
votes
1answer
47 views

converting a signal back and forth with trigonometric functions

I'm trying to convert a signal back and forth using trigonometric functions. In the example below: 1) start off with a cos signal 2) convert the signal to a secant signal 3) would like to convert ...
0
votes
1answer
112 views

Proof that the sum $\sum _{n=1} ^{\infty} (-1)^n \sin (nx)$ is bounded

How can I prove that there is some constant $M>0$, such that for all $N\in\mathbb{N}$ and $x\in [0,\pi]$, $$\left|\sum _{n=1} ^{N} (-1)^n \sin (nx)\right| < M\text{?}$$
1
vote
1answer
65 views

Convergence of Trigonometric Dirichlet series

Can it be proved that the following series converges for some integer value of $s$? $$\sum_{n=1}^\infty\frac{1}{n^s|\sin(n)|}$$ If so what value(s) of $s$ would it converge for?
2
votes
3answers
63 views

Closed form for a trigonometric partial sum

I know that: $$\sum_{k=1}^n\arctan(2k^2)=\frac{\pi n}{2}-\frac{1}{2}\arctan(\frac{2n(n+1)}{2n+1})$$ Can a similar closed form expressions be given for $\sum_{k=1}^n \arctan(k^2)$? I was able to ...
3
votes
7answers
292 views

Prove $ \cos\left(\frac{2\pi}{5}\right) + \cos\left(\frac{4\pi}{5}\right) + \cos\left(\frac{6\pi}{5}\right)+ \cos\left(\frac{8\pi}{5}\right) = -1 $

$$ \cos\left(\frac{2\pi}{5}\right) + \cos\left(\frac{4\pi}{5}\right) + \cos\left(\frac{6\pi}{5}\right)+ \cos\left(\frac{8\pi}{5}\right) = -1 $$ Wolframalpha shows that it is a correct identity, ...
3
votes
2answers
253 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 $\alpha > 1$, ...
2
votes
2answers
62 views

Complex number trigonometry problem

Use $cos (n\theta)$ = $\frac{z^n +z^{-n}}{2}$ to express $\cos \theta + \cos 3\theta + \cos5\theta + ... + \cos(2n-1)\theta$ as a geometric series in terms of z. Hence find this sum in terms of ...
0
votes
1answer
23 views

Use Osborn's Rule to suggest an identity involving cothx^2 and cosechx^2 then to solve

i know 1+cothx^2=cschx^2 but how to suggest the identity then use to solve part b?
0
votes
3answers
179 views

Fourier Series coefficients/Trigonometric functions

I need some help about finding the Fourier Series coefficient of the given signal; $$ x(t) = \sin(10\pi t + \frac {\pi}{6} ) $$ I know that, $$ a_{k} = \frac{1}{T}\int_{0}^{T} x(t)e^{-jkw_{0}t}dt $$ ...
2
votes
1answer
89 views

new ArcTan serie 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 = ...
-2
votes
1answer
72 views

Convergence on sum of cos [closed]

How to find the range of x on this sum to converge? $$\sum_{n=1}^∞{{\cos nx}\over{n}}.$$
2
votes
1answer
55 views

$\prod_{k=1}^{n-1}\cos\left(\theta+\frac{k\pi}{n}\right)$

I know that the first one of the following identities holds, yet I don't know the identity of the general case as shown in the title and the bottom of the page. Is there anyone who knows the closed ...
0
votes
0answers
35 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} \\ ...
2
votes
2answers
175 views

$\sum_{k=0}^{n-1}\sin(k\frac{\pi}{n}+\theta)$

I'm trying to find the closed form of the above formula. This link shows the solution of tan version. Sum of tangent functions where arguments are in specific arithmetic series Though I'm trying to ...
10
votes
1answer
194 views

Limit of maximum of $f_{n}(x)=\dfrac{1}{n}(\sin{x}+\sin{(2x)}+\sin{(3x)}+\cdots+\sin{(nx)})$

let $$f_{n}(x)=\dfrac{1}{n}(\sin{x}+\sin{(2x)}+\sin{(3x)}+\cdots+\sin{(nx)}),x\in R,n\in N$$ let $$a_{n}=\max_{x\in R}{(f_{n}(x))}$$ Find this limit $$\lim_{n\to\infty}a_{n}$$ My try: since ...
0
votes
1answer
38 views

How can I compute this infinite series with cosines?

$\sum\limits_{h=1}^\infty (\frac{1}{2})^h\cos(\lambda h)$ where $\lambda$ is a real number. According to WolframAlpha the answer is $\frac{1-2\cos(\lambda)}{4\cos(\lambda)-5}$ Thanks!
0
votes
1answer
45 views

Trigonometry and complex numbers

Suppose $z_0=e^{i\theta_0}$ a complexe number as $\theta_0\in ]-\pi,\pi[ \setminus\{0\}$. For $n\in \mathbb{N}$, we pose $z_{n+1}=\dfrac{|z_n|+z_n}{2}$ and $z_n=r_ne^{i\theta_n}$ with ...
3
votes
0answers
63 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.
0
votes
3answers
317 views

How to find the sum of a geometric progression involving cos using complex numbers?

Use $ 2\cos{n\theta} = z^n + z^{-n} $ to express $\cos\theta + \cos3\theta + \cos5\theta + ... + \cos(2n-1)\theta $ as a geometric progression in terms of $z$. Hence find the sum of this progression ...
1
vote
0answers
69 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. ...
0
votes
2answers
1k views

How to simplify these trigonometry terms? (involving sin, cos, tan, cot)

Please, simplify these trigonometry term: $$ \frac{\sin 2x + \sin 4x + \sin 6x}{\cos 2x + \cos 4x + \cos 6x}=? $$
1
vote
2answers
69 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?
0
votes
1answer
140 views

There exists a countable set of mutually orthogonal trigonometric functions which form a basis for $L^2(T)$. Proof?

Evidently, this fact (for real or complex valued functions) is usually taken "for-granted" in derivations of Fourier series/transform, taking $\{e^{inx}|n\in\mathbf Z\}$ as the set of basis vectors. ...
3
votes
2answers
211 views

Showing $|\sum_{k=1}^n\frac{\sin(kx)}k|<2\sqrt{\pi}$

For any real $x$ and positive integer $n$, is it true that: $$\left|\sum_{k=1}^n\frac{\sin(kx)}k\right|<2\sqrt{\pi}\quad ?$$ Please justify.
4
votes
2answers
350 views

Compute $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$

Compute the series $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$ Hint: the answer is in fact 0
4
votes
3answers
102 views

Expansion of $\sin x$

I wanted to know, how can I derive: $$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}+\cdots$$
3
votes
1answer
92 views

Solve $x\sqrt{10} = \prod\limits_{k = 1}^{90} \sin(k), x\in \mathbb Q$.

Can someone help me with this question? I've found a solution but it's not a very nice one. I used 6 times the relation $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$. There's got to be a better way. ...
4
votes
1answer
120 views

Finite Series - reciprocals of sines

Find the sum of the finite series $$\sum _{k=1}^{k=89} \frac{1}{\sin(k^{\circ})\sin((k+1)^{\circ})}$$ This problem was asked in a test in my school. The answer seems to be ...
2
votes
2answers
72 views

Need help on part b of this trigonometric question.

The title of the question is: Using graphical technique determine the single wave resulting from a combination of two waves of the same frequency and then verify the result using trigonometrical ...
1
vote
2answers
52 views

Verifying the trigonometric identity $\cos{x} - \frac{\cos{x}}{1 - \tan{x}} = \frac{\sin{x} \cos{x}}{\sin{x} - \cos{x}}$

I have the following trigonometric identity $$\cos{x} - \frac{\cos{x}}{1 - \tan{x}} = \frac{\sin{x} \cos{x}}{\sin{x} - \cos{x}}$$ I've been trying to verify it for almost 20 minutes but coming up ...
1
vote
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 ...
3
votes
2answers
59 views

integrate product of trig functions

I need to find the Fourier cosine series for $\cos(3x)\sin^2(x)$, But I don't even know where to start to determine $$\int _0^{\pi }\cos(3x)\sin^2(x)\cos(k x)dx$$