$sin(x)$ is the ratio between two sides of a triangle. One or more sides must be irrational for $sin(x)$ to be irrational. Why does a rational value of $x$ guarantee this?
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 $x$ is some product of a fraction of $\pi$.  Is $\sin(x) $ (or other trigonometric function) necessarily irrational if $ x $ is rational?  Edit: Excluding the trivial solution of 0.
Generalizing a sequence. Consider a sequence of the form:  $x[n]=-1,-1...,1,1,...,-1,-1...,1,1... \quad n&gt;0$  One can think of this as a square wave ($\pm1$) with a 50 % duty cycle (coming from EE).  For the simplest case of such a series i.e. $-1,+1,-1,+1,-1...$, a general equation is quite trivial ($x[n]=-1^n$).  Can we have e general equation for the other cases i.e. when there are multiple $-1$'s followed by $+1$'s ? Apparent Violation of Newton's $3^{\text{rd}}$ Law and the Conservation of Momentum (and Angular Momentum) For a Pair of Charged Particles Consider a system of the two identical positive point charges situated in free space (isolated from the influence of any other external fields) as shown in the attached diagram. Particle $1$ is at $(a,a,0)$ and particle $2$ is at $(0,a,0)$. Their velocities, at the considered instant of time, are as shown in the diagram ($\mathbf{v}_1$ along the $+x$ axis, $\mathbf{v}_2$ along the $+z$ axis).    Now, by applying the Biot-Savart law, we find that the magnetic field due to particle $2$ at the position of particle $1$ is along $+y$ axis, which means that the force acting on particle $1$ is along $+z$ axis according to Fleming’s left-hand rule. A similar analysis shows that there is no magnetic force on particle $2$ as the magnetic field of particle $1$ should vanish at positions (relative to particle $1$) located along its velocity vector.  Now, if we observe the net torque on the system of two particles about the $y$ axis then it is non-zero and is directed along the $-y$ axis. Here, there are no external forces or torques acting on the isolated two-particle system, and yet the net torque, as well as the net force on the particles, are nonzero. Why? Also, Newton's third law of motion seems to be broken in this scenario. Why?    Edit $1$  I have come to know from the responses that the electromagnetic field itself takes away some momentum and angular momentum about the considered axis. However, I think that if I consider only two charged particles as my system then the Abraham-Lorentz force can be assumed to be acting upon the system, and that is sufficient to make sure that we have considered the momentum being carried away by the electromagnetic field itself.   Even after considering the action of the Abraham-Lorentz force for the two-particle system, the scenario breaks both Newton's third law and the conservation of linear and angular momentum. This is because the Abraham Lorentz forces do not exactly counterbalance the force and the torques, on the two-particle system under consideration, due to the magnetic field.    Edit $2$  The previous edit was a result of confusion and misunderstanding on my part. The force associated with the momentum carried away by the electromagnetic field as a result of the electromagnetic interaction described in the question is the Lorentz force itself which simply doesn't obey the third law of motion. The Abraham-Lorentz force is a different story. It is associated with the momentum carried away by the radiation emitted by the accelerated charged particles. This is an additional force apart from the Lorentz force and corresponds to an additional carriage of momentum by the electromagnetic field. The momentum carried away by the electromagnetic field in correspondence with the Lorentz force doesn't correspond to radiation. For example: man(1), find(3), updatedb(2)?   What do the numbers in parentheses (Brit. "brackets") mean? Caps lock indicator not synced I have a laptop which I hook up to an external display when I'm at home. I also connected an external keyboard and mouse which makes it more comfortable for me.  I have noticed that pressing caps lock on the external keyboard does not light up the capslock indicator of the laptop keyboard (and vice versa).  When googling, I came across the setleds function, but when I type it into a console all I get is this:  KDGKBLED: Inappropriate ioctl for device Error reading current flags setting. Maybe you are not on the console?   Any idea if I can get this working? How to change boot priority? I have a laptop with Ubuntu installed as the only OS. I also have a USB stick with a Windows 7 installation on it. I want to install Windows from the USB stick on my laptop but when I try to restart the laptop to get access to BIOS, I can't seem to find the command for it, its like it skips everything and just says Ubuntu and then takes me to the login screen.  How do I change the boot priority to USB in Ubuntu? Strange GET requests in logs I'm getting the following GET requests in my apache logs:  109.230.251.14 - - [29/Mar/2011:16:28:18 +0100] "GET http://209.191.92.114/config/pwtoken_get?login=jackmcphee232&amp;src=ygodgw&amp;passwd=e59e2240415e6f1aba3da72b8f189f4e&amp;challenge=9TbU_9yfZhKmzlHtK9X4OkQlesTH&amp;md5=1 HTTP/1.0" 404 1226 "-" "-"   Any idea what it could be and how is that possibly a get request? That IP address seems to point to Yahoo! I'm very confused. How to find (and delete) duplicate files I have a largish music collection and there are some duplicates in there. Is there any way to find duplicate files. At a minimum by doing a hash and seeing if two files have the same hash.  Bonus points for also finding files with the same name apart from the extension - I think I have some songs with both mp3 and ogg format versions.  I'm happy using the command line if that is the easiest way. \section*[toc]{normal section title} not working for me for my thesis I want to have a section appear differently in the ToC than in the text.  After googling a bit, I found \section[short]{long} as a proposed answer, but this does not seem to work for me.  Here is a MnotWE:  \documentclass[12pt,a4paper,reqno]{amsbook} \usepackage{fontspec} \setmainfont[SlantedFont={Latin Modern Roman Slanted},          SmallCapsFont = {Latin Modern Roman Caps},          ]{Latin Modern Roman}    \usepackage[activate=true,nocompatibility},final,tracking=true,factor=1100,stretch=10,shrink=10]{microtype}  \usepackage[bindingoffset=1cm]{geometry} \usepackage{polyglossia} \setdefaultlanguage{english} \usepackage{csquotes}  \title{\textsc{Some Title}} \author{me}  \begin{document} \maketitle \frontmatter \tableofcontents \mainmatter  \part[toc part]{normal part} \section*[toc section]{normal section} some text  \end{document}   It just takes "toc part" as normal and as ToC heading for the \part command.  For the section I am interested in the starred version. Here the [toc section] part of the command seems to be completely ignored.  I use LuaLaTeX to build my document. Does anyone have a suggestion how to get a \section* appear with a different title in the ToC? LINQ's Distinct() on a particular property Probability of a 2 pair in poker What is wrong with my method of find the probability of a 2 pair?  For the first card we have 52 choices, one this is picked( call it A) we have 3 choices(different suite than A but this same value) for the second card, ( lets call this card B). Now we have AB, to we have to throw away 2 more cards so we don't have 3 of a kind or a four of a kind. Therefore one our third card we have 48 choices and on our fourth card we have 3 choices again similar to above.  On our last card we only have 44 choices left. So the probablity of getting a 2 pair should be (52*3*48*3*44)/(52 choose 5).  But this isn't correct.     Note: I edited this to "2 pair"; the original version was wrong. C++ vectors (instead of arrays) There are now errors when updating and I cannot install most software due to a corrupted  /etc/apt/sources.list   file. Is there a copy I can download to replace it?  The file would be for Ubuntu 12.04 (Final Beta) in the United States. Let $A=\begin{pmatrix}0&amp;5\\ 7&amp;4\end{pmatrix}$.      1) Write $A$ as a product of $4$ elementary matrices.      2) Write $A^{-1}$ as a product of $4$ elementary matrices.   My work. I have managed to find $A^{-1}$, which came out to be this:  \begin{pmatrix}-\frac{4}{35}&amp;\frac{1}{7}\\ \frac{1}{5}&amp;0\end{pmatrix}  However, I am struggling to figure out how I would split each of these matrices into $4$ elementary matrices. Any help? Construction of uncountably many non-isomorphic linear (total) orderings of natural numbers I would like to find a way to construct uncountably many non-isomorphic linear (total) orderings of natural numbers (as stated in the title).  I've already constructed two non-isomorphic total orderings. They are $$ 1 &lt; 2 &lt; 3 &lt;\ldots\\ 1 &gt; 2 &gt; 3 &gt;\ldots $$ where "$&lt;$" and "$&gt;$" are  strict total ordering relations.  Thanks in advance for your help. I have recently registered some domains on the basically brand new .directory gTLD. Because of operating costs to maintain a gTLD, it is assumed that many of the hundreds being rolled out this year and next will fail. I will admit I am not very knowledgeable on how the TLD process works.   What exactly does a new TLD failing mean? I realize that they lose ICANN accreditation, but does that just mean that they will be no longer be sold? If the .directory TLD were to go under, would my .directory domains be affected? I bought it from domain registrar Gandi if that helps. How to work around Eclipse Luna/GTK bug 339174 (change to not use oxygen) The bug is described here:  For me it fails with:  java: /build/buildd/oxygen-gtk3-1.0.2/src/animations/oxygencomboboxdata.cpp:87: void Oxygen::ComboBoxData::setButton(GtkWidget*): Assertion `!_button._widget' failed.   Problem is if I go to GTK+ Apperance in Control Module and change to anything other than oxygen Eclipse will look totally awful.  Regardless what theme I use (tried QtCurve and Raleigh) it has the same look with huge padding around all controls etc.  Is it possible to have a sensible look without oxygen GTK?  I have kubuntu-desktop installed over the stock LTS distro.   $dpkg -l libgtk* | grep -e '^i' | grep -e 'libgtk-*[0-9]'     ii  libgtk-3-0                             3.4.2-0ubuntu0.8                          GTK+ graphical user interface library     ii  libgtk-3-bin                           3.4.2-0ubuntu0.8                          programs for the GTK+ graphical user interface library     ii  libgtk-3-common                        3.4.2-0ubuntu0.8                          common files for the GTK+ graphical user interface library     ii  libgtk2.0-0                            2.24.10-0ubuntu6.1                        GTK+ graphical user interface library     ii  libgtk2.0-bin                          2.24.10-0ubuntu6.1                        programs for the GTK+ graphical user interface library     ii  libgtk2.0-common                       2.24.10-0ubuntu6.1                        common files for the GTK+ graphical user interface library Where are the evolutionary "inbetweeners"? The problem I've always had with evolution is the actual lack of variation between animals. More specifically, the lack of observable gradual change between species.  Take for example the hammerhead shark. It obviously has a strong connection with a normal shark, but where's the "half-hammerhead" shark? Where's the species that only has a SLIGHTLY wider head?  I really doubt that a single mutation on a normal shark resulted in a head shaped like that, it had to happen gradually. Right? As far as I understand, evolution works by random mutation in a child being somehow slightly beneficial to that animal so it get's to live longer and reproduce more, eventually resulting in it's genetic material in dominating the inferior ones before it. But if it was for whatever reason beneficial to have a slightly wider head, why don't all the sharks have slightly wider heads now? On the other hand, if it wasn't beneficial, how come the hammerhead still exists?  Furthermore, I feel like some characteristics of a certain animal can't really have gradual benefits. There's a lizard that shoots blood out of it's eyes. The original ancestor certainly didn't, and I'm sure that if it's child only bled slightly from it's eyes it wouldn't really help if much in life. So for it to actually become beneficial, it seems to me that there had to be a ton of lizards that had a disadvantage of bleeding slightly. In fact, there had to be so many that they evolved even further in the same direction and kept doing that until they bled so much it's an actual benefit (sorry for this turning so gross). So why did the inbetweeners survive long enough to evolve even more, and why did they THEN die out? Word for resembling the truth How do we refer to the act of resembling the truth, regardless if it is indeed the truth?  Some context: suppose you are watching a film and the story bears a striking resemblance to "real life". Perhaps you remember similar events from your own background or the story has a strong internal coherence. So you feel that the story could be true, even though you know it is fictional. Or, as the legend has it, when Michelangelo finished his "Pietà" and the sculpture seemed so "real" and "natural" to him, that he was amazed that the sculpture couldn't talk. One last example: you hear the promises of a politician in election campaign and he seems to be telling the truth, but of course you have no way to figure out, beforehand, if he is being honest or not.  The word "likelihood" comes to mind, but this word suggests some kind of numerical (even statistical) evaluation, which is akin to "probability". The word I am looking for describes the feeling you get, which leads to a subjective "binary" judgement (either it resembles the truth or not).  In Portuguese, such word is "verossimilhança" (from Latin verum, truth + similis, similar). Every word is marked as misspelled in TeXstudio Does a photon exert a gravitational pull?