# Questions tagged [trigonometric-functions]

The trigonometric-functions tag has no usage guidance.

20
questions

4
votes

1
answer

281
views

### 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 ...

0
votes

1
answer

156
views

### 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 ...

0
votes

1
answer

96
views

### 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{...

8
votes

1
answer

467
views

### 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....

0
votes

0
answers

23
views

### 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 ...

1
vote

0
answers

116
views

### $\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 ...

0
votes

1
answer

100
views

### 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 ...

8
votes

1
answer

522
views

### 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/...

4
votes

1
answer

382
views

### 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 ...

22
votes

5
answers

2k
views

### 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 \...

4
votes

1
answer

145
views

### 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{...

6
votes

1
answer

368
views

### 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{\...

8
votes

1
answer

978
views

### 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 ...

1
vote

1
answer

164
views

### 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 (...

3
votes

4
answers

353
views

### 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 ...

0
votes

0
answers

114
views

### 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 ...

4
votes

0
answers

181
views

### 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} =
\...

9
votes

1
answer

706
views

### 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{...

3
votes

0
answers

235
views

### 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,\...

6
votes

2
answers

485
views

### 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 ...