Questions tagged [trigonometric-functions]

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Diophantine equation $\cos(2\pi x)\cos(2\pi y) = \cos(2\pi z)$

While working on finite order elements of $\operatorname{SO}_n$, I meet this question: Find all identities of the form $\cos(2\pi x)\cos(2\pi y) = \cos(2\pi z)$ with $x, y, z$ rational numbers. As ...
WhatsUp's user avatar
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For which $p>p^*$ does the inequality $\cos^2(−π/4+π/p)>1/2+π/p^2$​ hold?

I have the inequality $\cos^2(−\frac{\pi}{4}+\frac{\pi}{p})>\frac{1}{2}+\frac{π}{p^2}$​​ and have found that it holds for $p>p^*$, where $p^*$ is some positive number (around ~2.8). I'm looking ...
blockchain_dietmar's user avatar
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If $a_1=1$ and $a_n=\sec (a_{n-1})$ then what does the proportion of positive terms approach, as $n\to\infty$?

Consider the sequence $a_1=1$ and $a_n=\sec (a_{n-1})$ for $n>1$. What does the proportion of positive terms approach, as $n\to\infty$? At first I thought the limiting proportion might be $\frac{...
Dan's user avatar
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Integer solutions of $2\cos\left(\frac{p\pi}n\right)+2\cos\left(\frac{q\pi}n\right)+4\cos\left(\frac{p\pi}n\right)\cos\left(\frac{q\pi}n\right)=1$

I asked this question on MSE yesterday. Due to observations that I have made only after that, I feel that it may be a much harder problem than it looks like. So, I am posting the question here as well....
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Determine plot bounds of a given function to highlight interesting behavior?

Am working on plotting software. Am trying to find an algorithm that would automatically generate x and y bounds for the plot such that the interesting behavior is shown. There are some algorithms ...
Joshua Guevara's user avatar
1 vote
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$\sin(\frac{\pi}{p}) $ not expressible by positive radicals and $\sin(\frac{\pi}{q_i})$?

We have the following identities: $\sin(\frac{\pi}{1})=0$ $\sin(\frac{\pi}{2})=1$ $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$ $\sin(\frac{\pi}{4})=\sqrt{\frac{1}{2}}$ Lets start with a definition. Rules ...
mick's user avatar
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Trigonometry/spherical angles/minimum-least-squares [closed]

An issue from 3D tessellated geometry: Find the direction vector of a plane that minimizes the silhouette of a set of triangles. To say it another way, find the direction vector that is most ...
mattica's user avatar
  • 103
8 votes
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Trivial (?) product/series expansions for sine and cosine

In an old paper of Glaisher, I find the following formulas: $$\dfrac{\sin(\pi x)}{\pi x}=1-\dfrac{x^2}{1^2}-\dfrac{x^2(1^2-x^2)}{(1.2)^2}-\dfrac{x^2(1^2-x^2)(2^2-x^2)}{(1.2.3)^2}-\cdots$$ $$\cos(\pi x/...
Henri Cohen's user avatar
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Trigonometric Diophantine equation

Is there a general method to solve the equation $P(x_1,x_2,...,x_n)=0$ with $P$ is a polynomial in $n$ variables with integer coefficients and $x_k=\cos(q_k\pi)$ with $q_k$ is a rational number? This ...
Veronica Phan's user avatar
22 votes
5 answers

Axiomatic construction of trigonometric functions

I am able to construct functions $\sin,\cos\colon \mathbb R \to \mathbb R$ satisfying the following properties: $\sin^2 x + \cos^2 x = 1$, $\sin(x+y)=\sin x \cos y + \sin y\cos x$, $\cos(x+y)=\cos x \...
Emanuele Paolini's user avatar
4 votes
1 answer

Definite integral of power of sine ratio

I stumbled on the following rather appealing trigonometric definite integral, \begin{equation} \int_0^y \left(\frac{\sin x}{\sin (y-x)}\right)^a \mathrm{d}x = \pi \frac{\sin(ya)}{\sin(\pi a)} \end{...
Timothy Budd's user avatar
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6 votes
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Is there any hope to prove that $g(x)>-4$ if $f(x)<0$?

I have these two functions for $x>0$, $\beta>0$ and $\alpha$ (all reals) $$ f(x)= \frac{\alpha \; \sin (\beta \; x)}x+4 \cos (\beta\; x) ,\qquad\qquad\qquad\qquad\qquad\qquad\\ g(x)=\frac{\...
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A generalization of the law of tangents

The law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. Let $a$, $b$, and $c$ be the lengths of the three ...
Emmanuel José García's user avatar
1 vote
1 answer

How to eliminate angle in a Glissette equation of carried point of a line sliding along two lines not at right angles

Glissettes are the curves trances out by a point carried by a curve, which is made to slide between given points or curves. My problem specifically include a line which slides between two fixed lines (...
barakugav's user avatar
3 votes
4 answers

Prove that $(v^Tx)^2−(u^Tx)^2\leq \sqrt{1−(u^Tv)^2}$ for any unit vectors $u, v, x$

I believe I found a complicated proof by bounding the spectral norm $||uu^T-vv^T||^2_2:=\max_{||x||=1}|(u^Tx)^2-(v^Tx)^2|$. Using the fact that $dist(x,y):=\sin|x-y|$ is a distance function over unit ...
Dan Feldman's user avatar
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Addition formulas for q-analogs of trigonometric functions

Sine and Cosine functions possess notable formulas for addition of angles $$ \sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b) \qquad \text{or} \qquad \cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b). $$ One can ...
Matteo's user avatar
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An addition theorem for three functions similar to $\sin,\cos$ and $\sinh,\cosh$ and one / some questions?

Define the functions $t_k(x) = \sum_{n=0}^{\infty}{\frac{x^{3n+k}}{(3n+k)!}}$ for $k=0,1,2$. The functions then satisfy: $ \begin{pmatrix}\exp(x) \\ \exp(\omega x) \\ \exp(\omega^2 x)\end{pmatrix} = \...
mathoverflowUser's user avatar
9 votes
1 answer

Three questions about three functions similar to $\sin,\cos$

In The Basel problem revisited? a question about the function, similar to sinc, $f(x)$ was asked: $$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \prod_{n=1}^\infty \left ( 1+ \frac{...
mathoverflowUser's user avatar
3 votes
0 answers

Trigonometry and plane geometry

This will be a variation on the theme of this question, or maybe a rephrasing of it with a somewhat readjusted emphasis. In this posting I introduced the function \begin{align} & f_3(\theta_1,\...
Michael Hardy's user avatar
6 votes
2 answers

Need a reference for a trigonometric inequality

In my old high school notebook (20 years ago), the following inequality appears with its proof: $$1+\cos x + \frac{1}{2}\cos 2x + \cdots + \frac{1}{n}\cos nx \geq 0$$ for any real $x$ and positive ...
Vu Thanh Tung's user avatar