9
votes
2answers
148 views

$\arctan$ of a square root as a rational multiple of $\pi$

I know that if $x$ is a rational multiple of $\pi$, then $\tan(x)$ is algebraic. Is there a fairly simple way to express $x$ as $\pi\frac{m}{n}$, if $\tan(x)$ is given as a square root of a rational? ...
1
vote
2answers
399 views

How do i prove that $\frac{1}{\pi} \arccos(1/3)$ is irrational?

How do i prove that $\frac{1}{\pi} \arccos(1/3)$ is irrational?
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?
6
votes
3answers
1k views

Prove that the Tangent of 75 degrees equals 2 plus the square-root of 3

My (very simple) question to a friend was how do I prove the following using basic trig principles: $\tan75^\circ = 2 + \sqrt{3}$ He gave this proof (via a text message!) $1. \tan75^\circ$ $2. = ...
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 $x$ ...
37
votes
6answers
3k views

$\sin 1^\circ$ is irrational but how do I prove it in a slick way? And $\tan(1^\circ)$ is …

In the book 101 problems in Trigonometry, Prof. Titu Andreescu and Prof. Feng asks for the proof the fact that $\cos 1^\circ$ is irrational and he proves it. The proof proceeds by contradiction and ...
0
votes
1answer
239 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 ...
7
votes
4answers
2k views

When is $\sin(x)$ rational?

Obviously, there are some points (like $\pi,30$) but I am unsure if there are more. How can it be proved that there are no more points, or what those points will be? EDIT: I largely meant to ask ...
14
votes
2answers
2k views

ArcTan(2) a rational multiple of $\pi$?

Consider a $2 \times 1$ rectangle split by a diagonal. Then the two angles at a corner are ArcTan(2) and ArcTan(1/2), which are about $63.4^\circ$ and $26.6^\circ$. Of course the sum of these angles ...
4
votes
1answer
370 views

rational angles with sines expressible with radicals

An angle x is rational when measured in degrees. sin(x) is can be written using radicals. What are the conditions on x? If nested square roots are allowed? What I know so far: If sin(x) can be ...
3
votes
1answer
236 views

How can I determine a number is irrational?

I have a hypothesis about regular polygons, but in order to prove or disprove it I need a way to determine whether an expression is rational. Once I boil down my expression the only part that could be ...
6
votes
1answer
735 views

Sine values being rational

Can $$\sin r\pi $$ be rational if $r$ is irrational? Either a direct or existence proof is fine.