0
votes
3answers
37 views

convergente of the sum of sines of the terms of the alternating harmonic series

I want to know about the convergence or divergence of the following series: $$\sum \sin (a_n) $$ where $$a_n=\frac{(-1)^n}{n}$$ The tests that I tried were inconclusive. Is it possible to know? ...
0
votes
2answers
29 views

$x/|x|$ question about division

What is $\frac{x}{|x|}$ can it be simplified? Because look at this. $\frac{r\cosh(x)}{\sqrt{\cosh^2(x)}} = \frac{r\cosh(x)}{|\cosh(x)|}$ How do you do this?
2
votes
4answers
265 views

How to deduce the following trig relation?

How can I deduce: $$\sqrt{|x|}\sin(\frac{1}{x}) \le \sqrt{|x|}$$?? I know of the relation. $$\sin(u) \le u$$ $$u = \frac{1}{x}$$ $$\sin(1/x) \le \frac{1}{x}$$ But nothing related to $\sqrt{x}$ ...
1
vote
2answers
40 views

Misunderstanding about the definition of a limit (Spivak Calculus)

In Spivak's text, I quote: "In general, if $\epsilon > 0$ to ensure that $|x^2\sin(\frac{1}{x})| < \epsilon$ we need only require that $|x| < \epsilon$ and $x \ne 0$" This can easily be ...
1
vote
3answers
45 views

Show $|\sin(y)y - \sin(x)x| \leq C|y - x|$ for some $C > 0$

Show $|\sin(y)y - \sin(x)x| \leq C|y - x|$ for some $C > 0$. This is one of the steps in a bigger problem I'm trying to solve, and while it first appeared it would be entirely straightforward, I ...
2
votes
0answers
60 views

simultaneous trigonometric equations

Consider the pair of simultaneous equations $p_5\cos(2\omega\tau)+p_4\omega\sin(2\omega\tau) = p_1\omega^2-p_3-p_6\cos(3\omega\tau) $, $p_4\omega\cos(2\omega\tau)-p_5\sin(2\omega\tau) = ...
2
votes
2answers
78 views

Pick a smart function

Our teacher wants us to find a function $f$ on $(0,\pi)$ such that $$\sqrt{\sin(x)} f(x)^{\frac{1}{4}} =k_1 + \cos(x)$$ and $$\sqrt{\sin(x)} f(x)^{-\frac{1}{4}} = k_2 + \cos(x).$$ The two constants ...
0
votes
2answers
47 views

Period of $\frac{\sin(Ny)}{sin y}$ with $N$ odd?

The function $$f(y) = \displaystyle \frac{\sin(Ny)}{\sin y}$$ is periodic with period $2 \pi$ in general. But tracing the graphic of that function for $N$ odd it seems that for $0 \leq x < \pi$ ...
1
vote
1answer
48 views

$\lim\limits_{R \rightarrow \infty}{|\cos z|}$ in complex numbers

Let $z=Re^{i \alpha}$ $\alpha$ is given number and $\alpha \in (0, \frac{\pi}{2})$ Find (if exist) $\lim\limits_{R \rightarrow \infty}{|\cos z|}$ What happen if function $f(z)=|\cos z|$ when ...
2
votes
0answers
36 views

Finding the sum of a trigonometric series, fourier series

I need to compute that for $x \in [0, 2\pi]$ $$\sum_{n=1}^\infty\frac{\sin(nx)}{n^3} = \frac{1}{12}x(x-\pi)(x-2\pi)$$ by using the uniform convergence $$\sum_{n=1}^\infty\frac{\sin(nx)}{n} = ...
3
votes
1answer
52 views

Equivalence of the two cosine definitions

There are at least two ways to define the cosine function: You can define it with a right triangle in the unit circle and extend the definition to $\mathbb{R}$. (classic definition) The other ...
1
vote
3answers
51 views

Adding sin(x + a) + sin(x + b)

I'm trying to prove $$\forall a,b \in \mathbb{R} \exists c,d,e\in \mathbb{R}: f(x):=\sin(x+a) + \sin(x+b) = c \sin(x+d) + e$$ I attempted using $\sin(s) = \frac{e^{is} - e^{-is}}{2i}$ and ended up ...
2
votes
1answer
42 views

sum of an arctan series using mathematical induction

How to solve this problem using mathematical induction: $$\arctan (1) + \arctan \Big(\frac13\Big) + ... + \arctan \bigg(\frac{1}{n^2+n+1}\bigg)=\arctan (n+1)$$
0
votes
3answers
50 views

Find min and maxima

Find local min and maxima of $ \sin(x^3)$ on the interval $]-2,2[$. I take the derivative and get: $$3x^2 \cdot \cos (x^3)$$ I set this equal to zero and get $$x^3 = \cos^{-1}(0)$$ $$ \Rightarrow ...
0
votes
1answer
55 views

Infinite trigonometric series, find the constant C_n

Hi this is my first post :) I am not sure how to do part b. You get the infinite series of $\displaystyle c_n\cdot \sin(\frac{n\cdot \pi\cdot x}{L})$ from $n=1$ to infinity And this is equal to ...
2
votes
4answers
113 views

Solving the integral $\int_0^{\pi/2} \frac{\sin((2n+1)t)}{\sin t} \mathrm{d}t$

I really don't know how to solve this integral $$\int_0^{\pi/2} \frac{\sin((2n+1)t)}{\sin t} \mathrm{d}t$$ Should I use firstly a formula of $\sin(a+b)$?
18
votes
2answers
890 views

Tough contest problem

I found this problem in a collection of contest problems of a Russian competition in 1995 and wasn't able to solve it. Solve for real $x$: $$ \cos (\cos (\cos (\cos(x))))=\sin (\sin (\sin (\sin ...
1
vote
3answers
54 views

Show that $dx = \frac{2}{1 + u^2} du$ where $ u = {\tan(\frac{x}{2})} $

Hello everyone I have been trying to show that $dx = \frac{2}{1 + u^2} du$ where $ u = {\tan(\frac{x}{2})} $ but I keep ending up with something like this: $2d{\sin(\frac{x}{2})}\cos(\frac{x}{2}) $ ...
2
votes
2answers
88 views

Graph of Sin(x) along the line y=x

Well, I want the equation of $\sin(x)$ which has the line $y=x$ as it's axis. Basically I want the $\frac\pi4$ rotation of the curve y=$\sin(x)$. I already attempted differentiating the curve and ...
3
votes
4answers
81 views

Limit involving $(\cos x)^{1/x^4}$

I am having trouble calculating the following limit. $$\lim_{x \to 0}(\cos x)^{1/x^4}$$ In Problems in mathematical analysis by Demidovich there is a hint that in case of $1^{\infty}$ indeterminate ...
0
votes
1answer
46 views

Condition on a sequence to generate an entire function

That conditions are necessary and sufficient for the succession $\{ \beta_n\}$ for the infinite product $$\prod_{n=1}^{\infty} \frac{ \sin(\beta_n z) }{\beta_n z} $$ converges an entire function. ...
0
votes
2answers
104 views

Local extrema for the function $f(x,y)= x^2+y^2 e^{x^2} + x\sin x$?

I would like to find the stationary points if they exist and so I start by finding the partial derivatives for $x$ and $y$ and equal them to zero and from the second equation I know that $y=0$ but I ...
2
votes
0answers
81 views

conjecture regarding the cosine fixed point

context/motivation if the angle on a calculator is set to radians, then it is very easy to demonstrate that iteration of $cos x$ (for arbitrary initial x) converges - simply keep pressing the ...
-2
votes
2answers
65 views

Sinus equality proof [closed]

Show that the following equalities hold true for every $n$ from $\mathbb{N}$ and every $x$ from $\mathbb{R}$ $$\sin^{(n)}(x)=\sin(x+n\fracπ2)$$ How do I solve this?
0
votes
2answers
45 views

limit of $\cos(a_n)$, where $\{a_n\} \to 0$

Let $$\lim_{n \to \infty}a_n \to 0$$ How would you prove the following: $$\lim_{n \to \infty} \cos(a_n) = 1$$ I'm trying to do this using the definition of a cauchy sequence: $$ \exists N \in ...
0
votes
3answers
59 views

Calculate the limit:

I need to calculate: $$\lim_{x\rightarrow \frac{\pi }{4}}\frac{\sin2x-\cos^{2}2x-1}{\cos^{2}2x+2\cos^{2}x-1}$$ I replaced $2\cos^{2}x-1=\cos2x$ and $\cos^{2}2x=1-\sin^{2}2x$, so this limit equals ...
2
votes
3answers
92 views

Value of tan(pi/2)

I understand that this is a very stupid question but I'm not getting the answer. At x=pi/2,what s the value of tan(x)?Should it be -infinity or +infinity? Texts tell it to be +infinity.But why? ...
0
votes
1answer
75 views

What is the sine of arcsine of $x$? Problem with using trigonometric substitution in integral.

I'm having problems with this $\int \sqrt{1-x^2}\,dx$. Now the text book (Spivak's Calculus) says we can replace $x$ by $\sin u$ ($u = \arcsin x$). Now my question is how can we replace $u$ by ...
4
votes
2answers
403 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
0
votes
1answer
196 views

Show that there exists unique real solution

Show that there exists unique real numbers $a$ and $b$ satisfying $$3\sin a-2\cos b=6a-12,$$ $$\cos a + 3\sin b=6b+6.$$ Thank you!
0
votes
1answer
75 views

how to prove an identity related to $\int_0^\infty\sin(x^{1+a})dx$?

i have made some experiments in maple evaluating the integral $$\int_0^\infty\sin(x^{1+a})dx$$ and the computer give me the following result ...
1
vote
1answer
40 views

trigonometric inequality - how to prove it?

Let $ 0 < x < \frac {\pi}{2}$ How to prove it? $$2 \sin x \le x- \frac {\pi}{3} + \sqrt {3} $$
2
votes
1answer
88 views

minimum value of a trigonometric equation is given. the problem is when the minimum value attains

Suppose the minimum value of $\cos^{2}(\theta_{1}-\theta_{2})+\cos^{2}(\theta_{2}-\theta_{3})+\cos^{2}(\theta_{3}-\theta_{1})$ is $\frac{3}{4}$. Also the following equations are given ...
2
votes
2answers
115 views

$x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$

$x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$.
1
vote
4answers
162 views

$\lim_{x\rightarrow\infty}\sin(x)$?

In physics I came across these kind of equations when I am trying to find the asymptotic behaviour of some function. Can anyone explain if there is any sense in talking about $\sin(x)$ or $\cos(x)$ ...
2
votes
0answers
109 views

At large times, $\sin(\omega t)$ tends to zero?

While doing a calculation in quantum mechanics, I got a expression $\sin(\omega t)$, and my prof said if I consider the consider at large times, then i can assume that this goes to zero because at ...
5
votes
1answer
218 views

Expressing $\sin\pi/n$ in terms of radicals of integers

Are values of $$\sin \frac{\pi}{n}$$ where $n$ is a positive integer all expressible in terms of radicals of integers? If not, what is the first $n$ for which it is not?
0
votes
2answers
61 views

Trigonometric proof query

I am having trouble proving the following identity (where $m,n \in \mathbb{R}$ are arbitrary): $$\sin(mx)\sin(nx) = \frac{1}{2}[\cos(m -n )x - \cos(m + n)x] \quad (1)$$ By expanding the RHS, I can ...
4
votes
1answer
121 views

Polynomial expression of $\frac{\sin x}{x} $

Could you explain to me why $$\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi ^2}\right)\left(1-\frac{x^2}{(2 \pi) ^2}\right)\left(1-\frac{x^2}{(3 \pi )^2}\right)\cdots$$ I've read in this article ...
13
votes
4answers
604 views

Showing that $ |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$

For every nonnegative integer $n$ and every real number $ x$ prove the inequality: $$\sum_{k=0}^n|\cos(2^kx)|= |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$$
6
votes
0answers
222 views

Integral of a gaussian function of trigonometric functions

I need help with the analytical solution of this integral: ...
3
votes
2answers
872 views

Divergence of the sequence $\sin(n)$ [duplicate]

Possible Duplicate: Prove the divergence of the sequence $(\sin(n))_{n=1}^\infty$. How can I show that the sequence $$ a_n = \sin(n) $$ is divergent? I tried to show that $\sin(n+1) - ...
2
votes
2answers
119 views

Proving this trigonometric inequality?

Let $|\theta-\theta_0|\leqslant \frac{\pi}4$. How can I prove that $$2(1-\cos(\theta-\theta_0))\geqslant \frac{|\theta-\theta_0|^2}{2}?$$
0
votes
1answer
248 views

Why Euler's formula is true? [duplicate]

Possible Duplicate: How to prove Euler’s formula: $\exp(i t)=\cos(t)+i\sin(t)$? I need to know why Euler's formula is true? I mean why is the following true: $$ e^{ix} = \cos(x) + i\sin(x) ...
8
votes
3answers
321 views

$\pi$, Dedekind cuts, trigonometric functions, area of a circle

(I should say at the outset that this question is broad, and may need splitting up. Although I ask several questions, I present them as one because they are not independent of one another, and I am ...
0
votes
1answer
63 views

How to convert trignometric polynomial to standard form?

I notice there are actually two standard forms of trignometric polynomials: $ c_0+ \sum_{k=1}^n \sum_{\alpha +\beta =k} c_{\alpha ,\beta}\sin^\alpha(x) \cos^\beta(x)$ $ c_0+\sum_{k=1}^n\{a_k\sin(kx) ...
2
votes
2answers
426 views

Showing $f(x)=\sum_{n=1}^{\infty}{\sin\left(\frac{x}{n^2}\right)}$ is continuous.

Let $$f(x)=\sum_{n=1}^{\infty}{\sin\left(\frac{x}{n^2}\right)}.$$ a) Show that the series converges for $x\in [0,\pi/2]$. b) Show that $f$ is monotone and continuous on this interval. This is what ...
2
votes
0answers
110 views

Inequality with even powers of trigonometric functions

For $m>0$, $0 < n\leqslant m+1$ ($m,n\in \mathbb{Z} $) , and $0 < a < 1$ , prove that $$2^{n}\cdot \left( a^{n}\cos ^{2m}\dfrac {\pi a} {2}+\left( 1-a\right) ^{n}\sin ^{2m}\dfrac {\pi ...
2
votes
1answer
119 views

A pseudo Fejér-Jackson inequality problem

$x\in (0,\pi)$ ,Prove that: \begin{align} \sum_{k=1}^{n}\frac{\sin{kx}}{k}>x\left(1-\frac{x}{\pi}\right)^3 \end{align} the inequality holds for all integer $n$ I tried Fourier, or Dirichlet ...
0
votes
1answer
1k views

Showing that $\cos(x)$ is a contraction mapping on $[0,\pi]$

How do I show that $\cos(x)$ is a contraction mapping on $[0,\pi]$? I would normally use the mean value theorem and find $\max|-\sin(x)|$ on $(0,\pi)$ but I dont think this will work here. So I think ...