0
votes
0answers
24 views

Non-trivial solutions for $(x;\sin{x})\in\mathbb{Q}^2$? [duplicate]

Is it possible? For $\cos{x}$ were analogous solutions also okay. $$(x;\sin{x})\in\mathbb{Q}^2$$
3
votes
0answers
122 views

Find all $\theta$ such that sin$\theta$ and cos$\theta$ are both rational number.

Find all $\theta$ such that sin$\theta$ and cos$\theta$ are both rational number. I thought this question might have been asked by someone else, but I couldn't find any. Currently I'm studying ...
21
votes
2answers
433 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, ...
12
votes
5answers
1k views

Are there infinitely many rational outputs for sin(x) and cos(x)?

I know this may be a dumb question but I know that it is possible for $\sin(x)$ to take on rational values like $0$, $1$, and $\frac {1}{2}$ and so forth, but can it equal any other rational values? ...
1
vote
2answers
206 views

Can all possible angles on a rational triangle be represented as a rational multiplied by Pi?

The reason I ask: I was wondering if it was possible to find the angle of a rational triangle by only using the lengths of its sides and knowledge of $\pi$ (that is, no inverse trig functions). So, ...
3
votes
2answers
134 views

Define two rational numbers $\alpha$ and $x$ such that $\sin( { \alpha }) =x$

Of course for $x\neq 0 $ and $\alpha$ in radians. Can you define them?
12
votes
2answers
1k views

Is sin(x) necessarily irrational where x is rational?

My friend and I were discussing this and we couldn't figure out how to prove it one way or another. The only rational values I can figure out for $\sin(x)$ (or $\cos(x)$, etc...) come about when $x$ ...
6
votes
0answers
143 views

Rational multiples of $\pi/2$ whose sines are also rational

Let $f(x)=\sin(x\frac{\pi}{2})$. Let $R$ the set of $x$ such that : $0\le x\le 1$ $x \in \mathbb Q$ $f(x) \in \mathbb Q$ Hence, $0\in R$ as $f(0)=0$. $1\in R$ as f(1)=1. And $\frac{1}{3}\in R$ as ...
0
votes
1answer
238 views

When are $\theta$ and $\sin\theta^\circ$ both rational? [duplicate]

Possible Duplicate: Sine values being rational I'm guessing that if I look in Ivan Niven's elementary book on irrational numbers, I'll find the answer to this quickly, but I'm posting it ...