2
votes
0answers
33 views

Alternative Proof for “Roots of Mertens Function-Farey Sequence-Cosines Relations”

You can write Merten's function as $$ M(n)= \sum_{a\in \mathcal{F}_n} e^{2\pi i a} , $$ where $\mathcal{F}_n$ is the Farey sequence of order $n$. The sum may be split into imaginary and real ...
1
vote
1answer
69 views

When does $\cos\frac{\pi}{m}=2\cos\frac{\pi}{r}\cos\frac{\pi}{n}$ with $m,n,r \in \mathbb{Z}$ hold?

Consider the equation below: $$\cos\dfrac{\pi}{m}=2\cos\dfrac{\pi}{r}\cos\dfrac{\pi}{n},$$ where $m,n$ and $r$ are non-zero integers. Equality holds when $m=2$ and $r=2$ (or $n=2$), and also when ...
21
votes
2answers
397 views

Is $\{\tan(x) : x\in \mathbb{Q}\}$ a group under addition?

A student asked me the following today : Is $S:= \{\tan(x) : x\in \mathbb{Q}\}$ a group under addition? I am quite perplexed by it. Clearly, the only non-trivial part is to check For any $x, ...
3
votes
0answers
63 views

Proof of a series containing normal as well as hyperbolic functions

Show that $$\sum _{n=0}^{\infty}\frac{\sinh(\pi n \sqrt 2)- \sin(\pi n \sqrt 2)}{n^3{\cosh(\pi n \sqrt 2})- \cos( \pi n \sqrt 2)}=\frac{ \pi^3}{18 \sqrt 2}$$ I have no hint as to how to even start.
4
votes
0answers
114 views

Prove That $1/\sin(i \pi/n)+1/\sin(j \pi/n) \ne 2$ When $n$ is Odd

The original problem is: Prove that $1/\sin(i \pi/n)+1/\sin(j \pi/n) \ne 2$ when $n$ is odd. I tried, and found that although everything looks similar, I actually know nothing about this kind of ...
2
votes
1answer
159 views

Triangle with integral side lengths

$ABC$ is a triangle with integral side lengths. Given that $\angle A=3\angle B$, find the minimum possible perimeter of $ABC$. I got this problem from an old book (which did not provide even a hint). ...
0
votes
2answers
274 views

Solutions to Trigonometric Problem Class

Is there a way to prove that the general solution of: $\sin^2 \pi x + \sin^2 \frac {ab\pi}{x} = 0$ is: $x = \pm 1,\pm a, \pm b, \pm ab$ and more specifically to derive the proof analytically, ...
9
votes
5answers
231 views

When does $a \cdot\sin(x) = \sin(a \cdot x)$?

I am examining the expression $a \cdot \sin(x) =\sin(a \cdot x)$ where $a$ is a rational constant. Is there a way to determine which values of $x$ would be valid? Does it only hold true for certain ...
3
votes
1answer
66 views

Prove that there are no integers $\csc {\frac{j\pi}{n}}-\csc{\frac{k\pi}{n}}=2$

Prove that there are no integers $j,k,n$ with odd $n$ satisfying $$\csc {\dfrac{j\pi}{n}}-\csc{\dfrac{k\pi}{n}}=2$$ This problem from $AMM,1999,10630$, but this solution is very ugly,and it's not ...
0
votes
3answers
121 views

Finding the number of integer solutions, why is this wrong?

The question is to find the number of solutions such that $(x, y)$ are integers: $(x-8)(x-10)=2^y$. Here's what I did: $u(u-2)=2^y$. From the quadratic formula, $u=1+\sqrt{1+2^y}$. This is where I ...
13
votes
2answers
312 views

How to do a very long division: continued fraction for tan

I want to compute $$\tan(r) = \cfrac{r}{1 - \cfrac{r^2}{3 - \cfrac{r^2}{5 - \cfrac{r^2}{7 - {}\ddots}}}}$$ by dividing the power series for sin and cos as it is said can be done in ...
1
vote
1answer
176 views

Generate a polynomial w/ integer coefficients whose roots are rational values of sine/cosine?

I'm a high school calculus/precalculus teacher, so forgive me if the question is a little basic. One of my (very gifted) students recently came up with a construction yielding a quartic, one of whose ...
10
votes
2answers
971 views

When is $\sin x$ an algebraic number and when is it non-algebraic?

Show that if $x$ is rational, then $\sin x$ is algebraic number when $x$ is in degrees and $\sin x$ is non algebraic when $x$ is in radians. Details: so we have $\sin(p/q)$ is algebraic when ...
9
votes
1answer
393 views

Why $\arccos(\frac{1}{3})$ is an irrational number?

I was reading the following question. It is a very nice question with a very nice answer! I would like to know why $\arccos(\frac{1}{3})$ is an irrational number.
50
votes
1answer
3k views

Trigonometric sums related to the Verlinde formula

Original question (see also the revised, possibly simpler, version below): Let $g > 1, r > 1$ be integers. Playing around with the Verlinde formula (see below), I came across the expression ...
5
votes
1answer
686 views

Sine values being rational

Can $$\sin r\pi $$ be rational if $r$ is irrational? Either a direct or existence proof is fine.
5
votes
3answers
206 views

When is $\frac1{n}\sin(nt)=\frac1{n+2}\sin((n+2)t)$?

I'm trying to classify (at least as fully as possible) the values of $t$ in $(0,\pi/2)$ for which the following equation has a solution for some natural number, $n$. ...
12
votes
1answer
452 views

Is $\sin(n^k) ≠ (\sin n)^k$ in general?

Is it true that $\sin(n^k) ≠ (\sin n)^k$ for any positive integers $n$ and integers $k ≠ 1$? What if $n > 0, k ≠ 1$ are rational?