1
vote
3answers
59 views

The value of $\int_0^{2\pi}\cos^{2n}(x)$ and its limit as $n\to\infty$

Calculate $I_{n}=\int\limits_{0}^{2\pi} \cos^{2n}(x)\,{\rm d}x$ and show that $\lim_{n\rightarrow \infty} I_{n}=0$ Should I separate $\cos^{2n}$ or I should try express it in Fourier series?
6
votes
2answers
57 views

Infinite Sum of Sines With Increasing Period

A while ago, I was thinking about the Weierstrass function, which is a sum of sines with increasing frequencies in such a way that the curve is a fractal. However, I wondered what would happen if one ...
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?
0
votes
1answer
35 views

Can every smooth sine function be given a smooth argument?

Here's a conjecture that I believe to be true, but I couldn't find a proof: Let $\alpha: \mathbb{R}\longrightarrow\mathbb{R}$ be a function such that $\sin\alpha$ is smooth. Then there is a smooth ...
8
votes
6answers
151 views

Prove that $ \int_0^{\pi} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x =\int_0^{\pi} \frac{(\sin x)^2}{1 + \cos x \sin x} \,\mathrm{d}x $

In a related question the following integral was evaluated $$ \int_0^{\pi} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x =\int_0^{\pi} \frac{\mathrm{d}x/2}{1 + \cos x \sin x} ...
0
votes
2answers
60 views

Using induction to prove a formula for $\sin x+\sin 3x+\dots+\sin (2n-1)x$

I'm working from the text "Intro To Real Analysis" by William Trench. Here is what I have thus far. I will prove using Mathematical Induction that $\sin x+\sin 3x+...+\sin (2n-1)x=\frac{1-\cos ...
0
votes
1answer
33 views

Accumulation points of trigonometric sequences

I am interested if the following sequences have accumulation points: $$x_{n} = \sin(2+\frac{1}{n})$$ and $$x_n = \tan (n)$$ Specifically for the first sequence is $1$ and $-1$ accumulation points? ...
11
votes
5answers
247 views

Proof: $\cos^p (\theta) \le \cos(p\theta)$

I came across this problem when I was at a book store inside of a book made to prepare Berkeley graduates to pass a mandatory exam. I wanted to buy the book, but, alas, I didn't have the money (forty ...
5
votes
5answers
184 views

Evaluate $\int_0^{\frac{\pi}{2}} \ln(1+\cos x)\, dx$

Find the value of $\displaystyle \int_0^{\frac{\pi}{2}} \ln(1+\cos x) $ I tried to put $1+ \cos x = 2 \cos^2 \frac{x}{2} $, but I am unable to proceed further. I think the following integral can be ...
3
votes
0answers
85 views

Squeeze Theorem: Finding the limit of a trig function

I'm stuck on finding the limit of a complex fraction/trig function. Could someone please assist, or point out where I'm going wrong? Determine $$\lim\limits_{x \to 0} ...
2
votes
0answers
104 views

Evaluating $\int_0^x \lvert \cos t \rvert dt$

in my mathbook there is given a solution to $$\int_0^x \lvert \cos t \rvert \, dt $$ but without any hints or tips. $$\int_0^x \lvert \cos t \rvert \, dt = \sin\left(x - \pi \left\lfloor \frac x ...
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
1answer
60 views

Prove $f$ isn't continuous at $\frac{1}{\pi}$

Let $f(x)=\left\lfloor {\sin {1 \over x}} \right\rfloor$ (meaning floor of $\sin x$). I need to prove that $f(x)$ isn't continuous at $x=\frac{1}{\pi}$. Proof: For a nehiborhood of $\frac{1}{\pi}$: ...
2
votes
1answer
115 views

How prove that $\max(|f(1)|,|f(2)|,|f(3)|,|f(4)|)\geq \frac{1}{2}$ if $f(x) = \cos(Ax)+\cos(Bx)$?

Let $ A, B$ be real numbers and $ f(x) =\cos(Ax) + \cos(Bx)$. How prove that $ \max(|f(1)|,|f(2)|,|f(3)|,|f(4)|)\geq \frac{1}{2}$?
2
votes
6answers
237 views

Algebraic proof of $\tan x>x$

I'm looking for a non-calculus proof of the statement that $\tan x>x$ on $(0,\pi/2)$, meaning "not using derivatives or integrals." (The calculus proof: if $f(x)=\tan x-x$ then $f'(x)=\sec^2 ...
3
votes
2answers
78 views

Duo Fresnel-like integrals $(??)$

I really wonder how I can prove the following integrals. $$\int_0^\infty \sin ax^2\cos 2bx\, dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}}\left(\cos \frac{b^2}{a}-\sin\frac{b^2}{a}\right)$$ and ...
8
votes
2answers
228 views

Integral $\int_{0}^{\pi/2} \arctan \left(2\tan^2 x\right) \mathrm{d}x$

The following integral may seem easy to evaluate ... $$ \int_{0}^{\Large\frac{\pi}{2}} \arctan \left(2 \tan^2 x\right) \mathrm{d}x = \pi \arctan \left( \frac{1}{2} \right). $$ Could you prove ...
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
46 views

Problem related to Mean Value Theorem

I found out a question that I can't figure out a way to solve it. Plz can anyone help me. Question is, Prove that $\exists\,C\in(0,\pi/4)\,\mathrm{s.t.}\,\tan(\pi/4+C)=3/C$ I know this should be ...
0
votes
0answers
62 views

Is $\sin (\mathbb N)$ dense in $[-1,1]$? [duplicate]

Let $\mathbb N$ be the set of positive integers, then is it true that $\sin (\mathbb N)$ is dense in $[-1,1]$ i.e. is it true that for every $x,y \in [-1,1]$ with $x<y$ , $\exists m \in \mathbb N$ ...
0
votes
2answers
16 views

A cycloid that goes through the beginning and through a general point

Parametric equations of the general cycloid through the beginning $(0,0)$ are $$x(t)=\frac{2t-\sin2t}{2d}$$ $$y(t)=\frac{1-\cos 2t}{2d}$$ How can we determine $d$ such that the cycloid goes through ...
1
vote
1answer
37 views

Evaluate position of first secondary maximum of $\frac{\sin N (x/2)}{\sin (x/2)}$

The function $$f(x) = \displaystyle \left | \frac{\sin \left( N \displaystyle \frac{x}{2} \right)}{\sin \left( \displaystyle \frac{x}{2} \right)} \right |$$ when evaluated for $x > 0$, has its ...
1
vote
2answers
113 views

Proving $\displaystyle \frac{\sin^3x}{x}\lt 0.69$ for any $x\gt 0$

Question : How can we prove strictly that the following inequality holds for any $x\gt0$?$$\frac{\sin^3x}{x}\lt 0.69$$ This seems difficult though it doesn't look so. Can anyone help?
3
votes
1answer
55 views

Proof for $\forall x$, $\exists y$ s.t $1/2 \le |x-y|\le 1$ and $ |\cos \pi x - \cos \pi y|\ge 1$

I am trying to understand a certain proof for the indifferentiability of the Weierstrass function, that strongly uses the following lemma: For all $x \in \mathbb{R}$ there exists $y \in \mathbb{R}$ ...
1
vote
2answers
37 views

Substitution of an implicit variable

I wasn't sure how to title this question: I want to manipulate the integral $$I(a,b) = \int_0^{\frac{\pi}{2}} \frac{d \phi}{\sqrt{a^2\cos^2 \phi + b^2 \sin^2 \phi}}$$ with this subsitution: $$\sin ...
1
vote
1answer
50 views

Handy way to find the $x$ value where $\sin x \cos \left( \frac{\pi}{2} \sin x \right)$ is maximum?

Like in the title, is there a handy way to compute the $x$ values for which the function $$f(x) = \sin x \cos \left( \frac{\pi}{2} \sin x \right)$$ reaches its maxima? The derivative is $$f'(x) = ...
1
vote
2answers
61 views

Trigonometry Inequality Involving Powers of Sin, Cos.

Prove that for $0 < x < \frac{\pi}4$, $$ \sin x^ {\sin x} < \cos x ^{\cos x}. $$ I don't have any nice ideas. I was thinking about taking the natural log and looking at the taylor series of ...
3
votes
1answer
85 views

Proving $\cos x < 1 - \frac{x^2}{2} +\frac{x^4}{24}$

I wish to prove the following inequality for $x\ne 0$: $$\cos x < 1 - \frac{x^2}{2} +\frac{x^4}{24}$$ Using the fact that I already prove: $$\cos x > 1 - \frac{x^2}{2}$$ My try: $\cos x = 1 - ...
2
votes
1answer
59 views

Taylor expansion and trigonometric functions

I've seen a proof including this claim: $$x-\frac{x^3}{6} \le \sin x$$ Now, for my understanding, the series $x-\frac{x^3}{6} +\frac{x^5}{120} -... $ is converging to $\sin x$ in an alternating way. ...
1
vote
2answers
25 views

Prove $a_n$ is within a specific range for all $n\in\mathbb{N}$

Let $a_n = \cos (a_{n-1})$ and $a_1 = {\pi\over 4}$. How to prove the range of $a_n$ is within the closed interval: $\left[ {{1 \over {\sqrt 2 }},{\pi \over 4}} \right]$? I thought about ...
0
votes
2answers
52 views

$\cos(x)$ domain and range

I'd like to refer the following answer: http://math.stackexchange.com/a/628992/130682 @robjohn claims that: ...
1
vote
1answer
40 views

Proof: $\sin(\frac{x-x_0}{2})\sin(\frac{x-x_1}{2}) = \frac{1}{2}\cos \frac{x_1-x_0}{2} - \frac{1}{2}\cos(x-\frac{x_1+x_0}{2})$

I looked up a proof for trigonometric interpolation, and it works with this equation: \begin{align} \sin(\frac{x-x_0}{2})\sin(\frac{x-x_1}{2}) = \frac{1}{2}\cos \frac{x_1-x_0}{2} - ...
5
votes
2answers
122 views

Show that $\sin(x)\sin(y)\sin(z) \leq \frac{1}{8}$

Let $(x,y,z) \in (\mathbb{R}^+)^3$ such that $x + y + z \leq \frac{\pi}{2}$. Show that $$\sin(x)\sin(y)\sin(z) \leq \frac{1}{8}$$ I have a solution using convexity of $\sin$ but I am looking ...
1
vote
2answers
72 views

Prove $|\sin(n)|$ or $|\sin(n + 1)|$ > $\sin(\frac{1}{2})$

Let $n$ be a positive integer. Prove that either $|\sin(n)|$ or $|\sin(n + 1)|$ > $\sin(\frac{1}{2})$. It makes sense to me if I look at the graph of $|\sin x|$. The interval from $k\pi$ to ...
1
vote
1answer
76 views

The function $e^{iz}$ maps $[0,2\pi)$ bijectively to $\{z\in\Bbb C:|z|=1\}$

Does anyone know a good proof or reference for this fact? I'm trying to write an elementary formal proof of this from just the series for $e^z=\sum_{n=0}^\infty z^n/n!$ (I don't have calculus ...
15
votes
4answers
641 views

Is there an elementary proof for Euler's product for Sine?

I've been looking at proofs of this equation: $$\displaystyle \frac {\sin \pi x} {\pi x} = \displaystyle \prod_{k \mathop = 1}^\infty \left({1 - \frac{x^2}{k^2} }\right)$$ All the proofs seem to ...
1
vote
1answer
185 views

Is this a good way to prove that $3^x+4^x =5^x $ has $x=2$ as the only real solution?

Divide both sides of the equation $3^x+4^x=5^x$ by $5^x$. $$ \Rightarrow \frac { 3^x }{ 5^x } +\frac { 4^x }{ 5^x } =\frac { 5^x }{ 5^x }$$ $$\tag1 \Rightarrow \left( \frac 3 5 \right)^x + \left( ...
2
votes
1answer
56 views

Weierstrass function

I got stuck on this exercise from Prof. Tao's real analysis notes. Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be the function $$f:= \sum_{n=1}^\infty 4^{-n} \sin(8^n\pi x)$$ Show that for every 8-dyadic ...
2
votes
6answers
115 views

explicit expression sought

Consider the equation $$ \cos^2\phi + \alpha\sin\phi\cos\phi-\beta=0\;, $$ where $\alpha,\beta\in\mathbb{R}$. I need to find an explicit expression for $\phi$. I have tried completing the square, but ...
1
vote
2answers
33 views

Show this function can be defined as the limit function

Let f: $ \mathbb{R} \rightarrow \mathbb{R} $ be defined by f(x) = 1 for x $\in \mathbb{Q} $, f(x) = 0 otherwise. We can see f is not regulated. Show that f may be obtained as a limit function: f(x) = ...
8
votes
2answers
192 views
+50

Calculation of $\int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2 \cos^2 x}}dx$

Calculate $$ \int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2 \cos^2 x}}dx$$ $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2\cos^2 ...
-2
votes
3answers
298 views

Limit of infinite loops of sin x as n tends to infinity [duplicate]

Show that $$lim_{n\to\infty} \text {sin sin ... sin x} = 0 $$ for all x. Note that the n here refers to the number of sin in the expression above.
0
votes
1answer
39 views

Limiting Function: Composition of Sine Curves

Define $f_1(x) = \sin(x)$. For each $k \geq 1$, define $f_{k+1}(x) = (f_k \circ f_1)(x)$. Are the properties of the limiting function (if it exists) well-known? That is, as $n$ tends to infinity, ...
0
votes
1answer
84 views

Differentiability of trigonometric piecewise functions

So I have a function of a real variable $x$: $f(x) = \left\{\begin{array}{lr} x \int_0^{tanx} \dfrac{t^2}{\sqrt{1+t^3}}dt & if \: x \ge 0\\ sin^2(x) & if \: x \lt 0 ...
5
votes
2answers
123 views

Concerning the sequence $\Big(\dfrac {\tan n}{n}\Big) $

Is the sequence $\Big(\dfrac {\tan n}{n}\Big) $ convergent ? If not convergent , is it properly divergent i.e. tends to either $+\infty$ or $-\infty$ ? ( Owing to $\tan (n+1)= \dfrac {\tan n + \tan ...
18
votes
2answers
891 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
2answers
226 views

How to prove $(\space|\sin n|\space)$ does not converge?

How do we prove that $(\space|\sin n|\space)$ is not convergent ? There is a beautiful proof of the non-convergence of $(\sin n)$ by considering the identities $\sin (n+1)=\cos1\sin n+ \sin 1 \cos n ...
-4
votes
2answers
146 views

Sufficient and essential condition for polynomials $P$ and $Q$ to satisfy $P(\sin x)= Q(\cos x)$ [closed]

The famous identity $\sin^2 x+\cos^2x =1$ can be written as follows: The polynomials $P(x)=x^2$ and $Q(x)=1-x^2$ satisfy $$P(\sin x)= Q(\cos x),\quad \text{for all }x\in\mathbb R$$ What are ...
2
votes
1answer
243 views

How do we find the limit of periodic function as x approaches to infinity $\left(\cos x^{\frac{1}{\sin x}}\right)$

Stuck with a periodic function limit problem: $$\lim_{ x\to \infty}\cos x^{\frac{1}{\sin x}}$$ That's how I tried to solve it: $$\lim_{ x\to \infty}\cos x^{\frac{1}{\sin x}}=\exp \left(\lim_{x \to ...
-2
votes
1answer
68 views

Is the series convergent? If convergent what will be the limit? [closed]

Is the series $$\sum\tan^{-1} \left(\frac1{2k^2}\right)$$ convergent? If convergent what will be the limit?