Questions tagged [trigonometric-functions]
For questions about trigonometric functions, i.e. sin, cos, tan, and their relatives and generalizations.
38 questions
14 votes
1 answer
987 views
How to prove $\sup_{n,k} \frac{1}{n}\sum_{j=0}^{n-1}\sin\left(\frac{2\pi k}{2^n-1}2^j\right) = \frac{\sqrt{15}}{8}$?
I am trying to prove the following conjectured identity: $$ \sup_{n\ge 1, \, 0\le k < 2^n-1} \underbrace{\frac{1}{n}\sum_{j=0}^{n-1} \sin\left( \frac{2\pi k}{2^n-1} 2^j\right)}_{S_n(k)} = \frac{\...
- 193
0 votes
0 answers
143 views
Product of combinatorial Laplacian eigenvalues is an integer
From Kirchhoff matrix tree theorem we know that the number of spanning trees in a graph is equal to the product of the combinatorial Laplacian eigenvalues (removing eigenvalue 0) divided by the number ...
- 549
8 votes
0 answers
187 views
Lower bounds on positive $\sin$-polynomials
I am aware that the structure $(\mathbb{R},+,\cdot,<,\sin)$ is extremely wild. Indeed, since the natural numbers are definable in such structure, one can define, via first-order formulas without ...
- 457
0 votes
2 answers
125 views
Calculating the Coefficients of a Sinusoid
Question: what is known about calculating the coefficients $a,\phi,\theta,d$ of $f(x)= a\sin(\phi x+\theta)+d$, resp. of $g(x)= a\sinh(\phi x+\theta)+d$ $\phantom{}$ that interpolate $\lbrace(x_0,...
- 13.9k
3 votes
0 answers
158 views
Vanishing linear combination of cubed arctangents
We can verify that, for example, $$15\arctan^2(1)-10\arctan^2(2)-2\arctan^2(3)+3\arctan^2(7)=0.$$ There are also other vanishing linear combinations (with non-zero integer coefficients) of squared ...
- 6,053
28 votes
3 answers
2k views
A Conjecture Involving Odd Primes Ending in $1$
Context and Motivation Consider the function: $$ f(n)= (a^n + a^{-n})(b^n + b^{-n}) ,$$ where, \begin{align*} a & =\tan{9^\circ}=\tan{\pi/20}=1+\sqrt{5}-\sqrt{5+2\sqrt{5}}\\ b & =\tan{27^\circ}...
- 559
3 votes
1 answer
410 views
Where to find or how to establish a general formula for the improper integral $\int_{0}^{\infty}\frac{\sin t}{t}(\ln t)^k\operatorname{d\!} t$?
When I tried to establish the Maclaurin power series expansion of the reciprocal $\frac{1}{\Gamma(z)}$ of the Euler gamma function $\Gamma(z)$, I came across the improper integral $$ I_k=\int_{0}^{\...
- 1,199
23 votes
2 answers
3k views
Boundedness of sum of sin(sin(n))
Playing with desmos I have accidentally noticed that the sequence of partial sums $$\left\{ \sum_{n=1}^{N}\sin(\sin(n)) : N\geq 1 \right\}$$ is bounded. However, I did not succeed in proving this ...
1 vote
3 answers
700 views
Huygens' trigonometric inequality
Prove that $$1-(4/3(\sin^3 \theta/2))/(\theta-\sin\theta)<(1-\cos\theta/2)(3/5-(3/1400)\pi^2/n^2)$$ holds for $0\le\theta\le\pi/2.$ Here $n$ is an integer greater than or equal to two. This an ...
- 371
1 vote
0 answers
109 views
Counting the number of local minima of a function that is the sum of square roots of cosines
Suppose you are given a set of functions $f_1, \ldots, f_n$. Every function is defined as follows $$f_i(x) = \sqrt{1+C^2_i-2C_i\cos (x-D_i)}$$ where $0<C_i<1$ and $0\leq D_i<2\pi$ are real-...
- 21
4 votes
0 answers
242 views
Generalization of the triangle inequality to complex exponents: What is $P\left(\left| x^{a+bi} + y^{a+bi} \right| \ge \left|z^{a+bi}\right|\right)$?
Let $x \le y \le z$ be the length of the sides of a triangle whose vertices are uniformly random on the circumference of a circle. In this question, it was proved that if $a \ge 1$, then the ...
- 5,608
12 votes
1 answer
1k views
Find the area of the region enclosed by $\sin^p x+\sin^p y=\sin^p(x+y)$, the $x$-axis and the $y$-axis (comes from a probability question)
Consider the graph of $\sin^p x+\sin^p y=\sin^p(x+y)$, where $x$ and $y$ are acute, and $p>1$. Here are examples with, from left to right, $p=1.05,\space 1.25,\space 2,\space 4,\space 100$. Find ...
- 5,039
7 votes
0 answers
245 views
Can an ellipse roll down a tilted sine curve without jumping?
Background Assume that we have a solid ellipse with uniform density, and that it rolls along a curve. In the following MO question, I asked along what curve an ellipse rolls down fastest. It was ...
- 5,009
4 votes
0 answers
172 views
Curiosity about "conditional trig identities"
Perhaps this should be cross-posted on Math Stackexchange, but it came up in the context of some research mathematics (quaternion orders, etc.) In this context, I have three angles $\alpha, \beta, \...
- 13.6k
7 votes
1 answer
448 views
Exponential trigonometric integral
I want to compute the normalization constant of some probability density on SO(3). After some simplification, I arrive at the following double integral: $$ \tag{1}\label{eq:1} \int_0^{2 \pi} \int_0^{\...
1 vote
0 answers
60 views
Exponential-like function equivalent for the Dixonian Elliptics
Is there some exponential-like function that acts as partner for the intricate Dixonian Elliptics, in a similar way that the Exponential function acts as a partner for the trigonometric functions ?
1 vote
0 answers
70 views
Finding the radical expressions of trig functions [closed]
I am trying to find the exact radicals of the sine and cosine of (m/n)*pi. I have been using sympy to do this and made this pretty good script: ...
9 votes
2 answers
4k views
Solving 'impossible' integrals with a new (?) trick
The following identities have been suggested based on formulas in a previous question of mine. If complex $\theta_1=\cos^{-1}(p)$ and $\theta_2=\sec^{-1}(p)$, where $p\in(-1, 0) \cup (1, \infty)$, ...
3 votes
1 answer
345 views
Integral of nested trigonometric function $\frac{\cos\left(p\cos\left(x\right)\right)}{p\cos\left(x\right) + c}$
while still working on my problem $\int J_{0}\left(\alpha x\right)J_{0}\left(\beta x\right)\cos(kx)\,\mathrm{d}x$ I came across the following definite integral \begin{equation} \int_{0}^{\pi}\frac{\...
- 173
5 votes
1 answer
437 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 ...
- 3,472
0 votes
1 answer
206 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
152 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{1}...
- 5,039
8 votes
1 answer
617 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....
user506548
1 vote
0 answers
152 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 ...
- 799
0 votes
1 answer
148 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 ...
- 103
8 votes
1 answer
589 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/...
- 13.9k
4 votes
1 answer
551 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 ...
- 1,558
24 votes
5 answers
3k 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 \...
- 732
5 votes
1 answer
196 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{...
- 4,002
6 votes
1 answer
405 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) $$ f(x)= \frac{\alpha \; \sin (\beta \; x)}x+4 \cos (\beta\; x) ,\qquad\qquad\qquad\qquad\qquad\qquad\\ g(x)=\frac{\...
user329770
11 votes
2 answers
1k views
A generalization of the law of tangents
The law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. Let $a$, $b$, and $c$ be the lengths of the three ...
1 vote
1 answer
269 views
How to eliminate angle in a Glissette equation of carried point of a line sliding along two lines not at right angles
Glissettes are the curves traced out by a point carried by a curve, which is made to slide between given points or curves. My problem specifically include a line which slides between two fixed lines (...
- 11
3 votes
4 answers
441 views
Prove that $(v^Tx)^2−(u^Tx)^2\leq \sqrt{1−(u^Tv)^2}$ for any unit vectors $u, v, x$
I believe I found a complicated proof by bounding the spectral norm $||uu^T-vv^T||^2_2:=\max_{||x||=1}|(u^Tx)^2-(v^Tx)^2|$. Using the fact that $dist(x,y):=\sin|x-y|$ is a distance function over unit ...
- 143
0 votes
0 answers
190 views
Addition formulas for q-analogs of trigonometric functions
Sine and Cosine functions possess notable formulas for addition of angles $$ \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). $$ One can ...
- 158
5 votes
0 answers
225 views
An addition theorem for three functions similar to $\sin,\cos$ and $\sinh,\cosh$ and one / some questions?
Define the functions $t_k(x) = \sum_{n=0}^{\infty}{\frac{x^{3n+k}}{(3n+k)!}}$ for $k=0,1,2$. The functions then satisfy: $ \begin{pmatrix}\exp(x) \\ \exp(\omega x) \\ \exp(\omega^2 x)\end{pmatrix} = \...
- 3,347
9 votes
1 answer
803 views
Three questions about three functions similar to $\sin,\cos$
In The Basel problem revisited? a question about the function, similar to sinc, $f(x)$ was asked: $$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \prod_{n=1}^\infty \left ( 1+ \frac{...
- 3,347
3 votes
0 answers
309 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
553 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 ...
- 335