For questions about or related to trigonometric series.

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0answers
13 views
2
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1answer
42 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 + ...
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0answers
18 views

Solving ctg(x)=x/b

B is small parameter. I have no problems finding first solution(both: b->0 and b->infinity). My solutions on photos. I got stuck trying to find solution when x->infinity. As I think, solution for x ...
2
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3answers
72 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 ...
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3answers
27 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 ...
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2answers
53 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
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1answer
124 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 ...
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2answers
119 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
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2answers
74 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 ...
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1answer
36 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 ...
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3answers
333 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 ...
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2answers
65 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) + ...
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1answer
43 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 ...
0
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1answer
48 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?
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0answers
61 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 $T$. ...
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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 ...
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1answer
121 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 $x\in [0,\pi]$, $$\left|\sum _{n=1} ^{N} (-1)^n \sin (nx)\right| < M\text{?}$$
0
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1answer
27 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 $\sum_{i=1}^{m}(cos(2 \theta_{i})) = \sum_{i=1}^{m}(cos(\frac{4\pi(i-1)}{m})) =0$ ...
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2answers
263 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 $\alpha > 1$, ...
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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: ...
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2answers
91 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 ...
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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
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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 $$\int_{0}^{2\pi}f(e^{i\theta})\overline {f(e^{i\theta})} d\theta =2\pi\sum_{k=0}^{n} c_k$$ ...
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1answer
22 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.
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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
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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
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2answers
235 views

Evaluation of a trigonometric partial sum

I just wanted to evaluate $$ \sum_{k=0}^n \cos k\theta $$ and I know that it should give $$ \cos\left(\frac{n\theta}{2}\right)\frac{\sin\left(\frac{(n+1)\theta}{2}\right)}{\sin(\theta / 2)} $$ ...
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2answers
1k views

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

Please, simplify these trigonometry term: $$ \frac{\sin 2x + \sin 4x + \sin 6x}{\cos 2x + \cos 4x + \cos 6x}=? $$
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3answers
31 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 ...
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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 ...
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1answer
48 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 ...
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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?
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3answers
66 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
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7answers
297 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, ...
2
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2answers
64 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
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1answer
24 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?
2
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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 = ...
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3answers
201 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, $$ a_{k} = \frac{1}{T}\int_{0}^{T} x(t)e^{-jkw_{0}t}dt $$ ...
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1answer
73 views

Convergence on sum of cos [closed]

How to find the range of x on this sum to converge? $$\sum_{n=1}^∞{{\cos nx}\over{n}}.$$
33
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2answers
412 views

Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$

Please help me to find a closed form for the infinite product $$\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$$ where $\tanh(z)=\frac{e^z-e^{-z}}{e^z+e^{-z}}$ is the hyperbolic tangent.
2
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1answer
57 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 ...
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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} \\ ...
10
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1answer
195 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 ...
2
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2answers
178 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 ...
0
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1answer
39 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
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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
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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.
1
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0answers
70 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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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
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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. ...