By letting $$ \theta = a \arcsin(x) \, , $$ $$ x = \sin\left(\frac{\theta}{a}\right)$$ If $\theta / a$ is the angle in a right triangle then the hypotenuse of length $1$, and the opposite side $x$. I want to find $\sin(\theta)$ as $ \theta = a \arcsin(x)$. For $a = 2$ and $a = 4$ this is quite simple. When $a = 2$: $$\sin(\theta) = 2 \sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)$$ As $\theta/2$ is the angle in our triangle we can simply substitute this in terms of $x$ So $$ \sin(\theta) = \sin(2 \arcsin(x)) = 2x\sqrt{1 -x^2}$$ repeating this process for $a = 4$ $$ \sin(\theta) = 2 \sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right) = 4 \sin\left(\frac{\theta}{4}\right)\cos\left(\frac{\theta}{4}\right)\left( 2 \cos^2\left(\frac{\theta}{4}\right) -a\right)$$ In other terms $$\sin(4 \arcsin(x)) =4x\sqrt{1- x^2} (1-2x^2) $$ How can I attempt to generalise this at least for even values of $a$, thanks.
I'm wondering if there's a simple way to rewrite this in terms of $k$ and $x$, especially as a polynomial. It seems to me to crop up every so often, especially for $k=2$, when I integrate with trig substitution. But $k=2$ is not so bad, because I can use the double angle formula; it's the prospect of higher values of $k$ that motivates this question.   I think the law of sines may help? Or maybe even De Moivre's theorem, to find the length of the hypotenuse as the length of the angle changes, if we think of the right triangle drawn from $\arcsin(x)$ with side1 = $x$, hypotenuse = $1$, and side2 = $\sqrt{1 - x^2}$ as a complex number, though I'm not sure how that might work.
I am using Ubuntu 11.10 and my screensaver keeps coming on while I'm watching a movie.  I don't remember this happening in 11.04 and can't remember how to stop it? How to set security options for a PDF using pdflatex How do I uninstall the command line tools for Xcode? We have a bunch of Mac at work that no longer require Xcode or command line tools so we'd like to remove them so the compilers aren't hanging around and we don't have to keep updating them when updates arrive. , but now that Xcode is an app store app you are left with the tools if you ever install them. Can we clean this other than wiping the OS for macs running 10.10 and newer? Change Folder Permissions The title word filter is one of the worst ideas ever implemented on SO There are numerous (, , ) questions concerning this problem. I realize this is a borderline duplicate, but I'll re-iterate the futility of this particular filter and why it should be outright removed from Stack Overflow.  Do you know how effective this is in preventing users who just don't care from posting questions with bad titles? 100% ineffective. What do these users do?   Replace it with a synonym: "issue," "trouble," "predicament," "botheration" or what have you. Or worse, they simply work around the filter by deliberately misspelling the word or inserting space or punctuation:    () ?    and the coup de grâce:      I mean, even Jeff . This adds zero value to the title. In fact, it looks worse. Why, oh why are people  to scrap a bad feature? All this does is:   Prevent people who have a  to use the word "problem" in the title from posting questions with an appropriate title. Break the search feature because of workarounds or the need to use less descriptive titles than the error itself   This is an extraordinarily useless (and harmful) feature. Please, for the love of Cthulhu,  remove, eliminate, abolish, eradicate, extirpate this "feature" from Stack Overflow.    Bonus:     So I can't mention the Halting Problem in the title? – Keith Thompson Nov 1 at 19:05      @KeithThompson Maybe it's a "Halting Issue". Or "Need help with Halting". Or the "Halting thing in Computability Theory". – slhck Nov 1 at 19:14 I'm having problems understanding the following situation. Suppose two 1-tonne cars are going with the same orientations but opposite senses, each 50 km/h with respect to the road. Then the total energy is   $$\begin{eqnarray}E=E_1+E_2&amp;=&amp;\frac{1\mathrm t\times(50\mathrm{km}/\mathrm h)^2}2+\frac{1\mathrm t\times(50\mathrm{km}/\mathrm h)^2}2\\&amp;=&amp;1\mathrm t\times(50\mathrm{km}/\mathrm h)^2\\&amp;=&amp;2500\frac{\mathrm t\times\mathrm{km}^2}{\mathrm h^2}.\end{eqnarray}$$  Now if we look at it from the point of view of one of the cars, then the total energy is   $$\begin{eqnarray}E=E_1+E_2&amp;=&amp;\frac{1\mathrm t\times(0\mathrm{km}/\mathrm h)^2}2+\frac{1\mathrm t\times(100\mathrm{km}/\mathrm h)^2}2\\&amp;=&amp;\frac{1\mathrm t\times(100\mathrm{km}/\mathrm h)^2}2\\&amp;=&amp;5000\frac{\mathrm t\times\mathrm{km}^2}{\mathrm h^2}.\end{eqnarray}.$$  I know that kinetic energy is supposed to change when I change the frame of reference. But I understand that then there must be some other kind of energy to make up for it so that the energy in the system stays unchanged. But I don't see any other kind of energy here. I only see two total energies of the same system that seem to be different. Could you explain this to me?  Please note that while I don't understand any physics, I do understand college level mathematics, so if necessary please use it. (I doubt anything more than high school maths should be needed here, but I want to say this just in case.) Which day does "next Tuesday" refer to? At what point does next Tuesday mean      the next Tuesday that will come to pass     and no longer      the Tuesday after the Tuesday that will come to pass?   And, when does the meaning switch back? How could I detect (preferably with Python) duplicate MP3 files that can be encoded with different bitrates (but they are the same song) and ID3 tags that can be incorrect?  I know I can do an  checksum of the files content but that won't work for different bitrates. And I don't know if ID3 tags have influence in generating the MD5 checksum. Should I re-encode MP3 files that have a different bitrate and then I can do the checksum? What do you recommend? What software can I use to write on PDFs? I'm used to OS X where the system PDF viewer will let you draw and write text on top of any PDF document. You can then flatten it and save as a new PDF. It saves paper and time so you can fill in any form without printing it and scanning it back to the computer.  It also lets you insert a scanned image of your signature.  Is there software for Ubuntu that would do that? I have a WinForms app that I am trying to make full screen (somewhat like what VS does in full screen mode).  Currently I am setting FormBorderStyle to None and WindowState to Maximized which gives me a little more space, but it doesn't cover over the taskbar if it is visible.  What do I need to do to use that space as well?  For bonus points, is there something I can do to make my MenuStrip autohide to give up that space as well? FaceTime not working on iPhone 6 Plus purchased from Dubai I bought my iPhone 6 Plus in Dubai. I am using the phone in Sri Lanka and I cannot use the FaceTime. How can I use FaceTime while in Sri Lanka? The -k option (or --convert-link) will convert links in your web pages to relative after the download finishes, such as the man page says:     After the download is complete,   convert the links in the document to   make them suitable for local viewing.   This affects not only the visible   hyperlinks, but any part of the   document that links to external   content, such as embedded images,   links to style sheets, hyperlinks to   non-HTML content, etc.   So, if I didn't specify -k, can I run wget again after the download and fix that, and if so, what would be the proper command? My guess is wget -c [previous options used] [url] and run it in the same working directory as the file were downloaded to. What are Java command line options to set to allow JVM to be remotely debugged? I'm looking for the title of a short story by Ursula Le Guin about a scientist (?) who is given a drug that slows the timescale of his perception enormously, so that he can perceive that an alien (?) forest is actually behaving as a sentient being, reacting to current events, but at an enormously slow (from a normal human point of view) timescale. It might be in , but I don't remember the title (and it doesn't jump out at me from the list of titles) and don't have access to that collection right now. It is not &quot;The Word for World is Forest&quot;, nor &quot;Direction of the Road&quot; (from ), nor &quot;Vaster than Empires and More Slow&quot;, the latter two of which look at similar plant perception/time-scale issues. Update: it turns out that the description was right, but I had most of the details wrong: not Le Guin, not alien, not a scientist ... Best way to do multi-row insert in Oracle? I'm looking for a good way to perform multi-row inserts into an Oracle 9 database.  The following works in MySQL but doesn't seem to be supported in Oracle.  INSERT INTO TMP_DIM_EXCH_RT  (EXCH_WH_KEY,   EXCH_NAT_KEY,   EXCH_DATE, EXCH_RATE,   FROM_CURCY_CD,   TO_CURCY_CD,   EXCH_EFF_DATE,   EXCH_EFF_END_DATE,   EXCH_LAST_UPDATED_DATE)  VALUES     (1, 1, '28-AUG-2008', 109.49, 'USD', 'JPY', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (2, 1, '28-AUG-2008', .54, 'USD', 'GBP', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (3, 1, '28-AUG-2008', 1.05, 'USD', 'CAD', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (4, 1, '28-AUG-2008', .68, 'USD', 'EUR', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (5, 1, '28-AUG-2008', 1.16, 'USD', 'AUD', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (6, 1, '28-AUG-2008', 7.81, 'USD', 'HKD', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'); Listing, zebra effect and broken lines In , I've found one solution to produce a zebra effect for listing. Unfortunately, this only works if the lines are not broken. Is there a way to fix this?  Here is a code with an example showing the problem:  \documentclass[10pt,a4paper]{article}     \usepackage[utf8x]{inputenc}     \usepackage{ucs}     \usepackage[T1]{fontenc}     \usepackage{lmodern}     \usepackage[frenchb]{babel}     \usepackage[x11names, svgnames]{xcolor}     \usepackage{graphicx}     \usepackage{listings}     \usepackage{amsfonts}     \usepackage{amssymb}      \makeatletter         % Patch line number key to call line background macro         \lst@Key{numbers}{none}{%             \def\lst@PlaceNumber{\lst@linebgrd}%             \lstKV@SwitchCases{#1}%             {none&amp;\\%              left&amp;\def\lst@PlaceNumber{\llap{\normalfont                         \lst@numberstyle{\thelstnumber}\kern\lst@numbersep}\lst@linebgrd}\\%              right&amp;\def\lst@PlaceNumber{\rlap{\normalfont                         \kern\linewidth \kern\lst@numbersep                         \lst@numberstyle{\thelstnumber}}\lst@linebgrd}%             }{\PackageError{Listings}{Numbers #1 unknown}\@ehc}}          % New keys         \lst@Key{linebackgroundcolor}{}{%             \def\lst@linebgrdcolor{#1}%         }         \lst@Key{linebackgroundsep}{0pt}{%             \def\lst@linebgrdsep{#1}%         }         \lst@Key{linebackgroundwidth}{\linewidth}{%             \def\lst@linebgrdwidth{#1}%         }         \lst@Key{linebackgroundheight}{\ht\strutbox}{%             \def\lst@linebgrdheight{#1}%         }         \lst@Key{linebackgrounddepth}{\dp\strutbox}{%             \def\lst@linebgrddepth{#1}%         }         \lst@Key{linebackgroundcmd}{\color@block}{%             \def\lst@linebgrdcmd{#1}%         }           % Line Background macro         \newcommand{\lst@linebgrd}{%             \ifx\lst@linebgrdcolor\empty\else             \rlap{%                 \lst@basicstyle                 \color{-.}% By default use the opposite (`-`) of the current color (`.`) as background                 \lst@linebgrdcolor{%                 \kern-\dimexpr\lst@linebgrdsep\relax%                 \lst@linebgrdcmd{\lst@linebgrdwidth}{\lst@linebgrdheight}{\lst@linebgrddepth}%                 }%             }%             \fi         }     \makeatother      \lstset{ %    * Number for lines         numbers=left,                   % Where to put the line-numbers         numberstyle=\footnotesize,      % Size of the fonts used for the line-numbers         stepnumber=1,                   % Step between two numbers         numbersep=5pt,                  % How far the line-numbers are from the code % %    * Back returns for long lines         breaklines=true,                % Sets automatic line breaking         breakatwhitespace=false,        % Automatic breaks only happen at whitespace ?         breakindent=0pt,                % Space before the text of a break         postbreak=\mbox{                % Character used at the begining of a break             \rotatebox[y=0.9ex]{180}{\color{Red} $\Lsh$}%         },     }  \begin{document}  \begin{lstlisting}[language=Python,linebackgroundcolor={\ifodd\value{lstnumber}\color{gray}\fi}] for i in range(5): for i in range(5): for i in range(5): for i in range(5): for i in range(5): for i in range(5):     sum += i**3 print sum  print('Fin de la boucle...') \end{lstlisting}  \end{document} From what I can gather, there are three categories:   Never use GET and use POST Never use POST and use GET It doesn't matter which one you use.   Am I correct in assuming those three cases? If so, what are some examples from each case? I was wondering what is the purpose of piping the downloaded file by curl into sudo -E bash - in the following shell:   curl -sL https://deb.nodesource.com/setup_5.x | sudo -E bash - In Markov inequality proof, why is $\int_a^\infty xp(x) \, dx \ge \int_a^\infty ap(x) \, dx$ Markov inequality,   $$\Pr(X \ge a) \le \frac{E[x]}{a}$$  Proof  $$\begin{aligned}      E(X) &amp;= \int_0^\infty xp(x)\,dx = \int_0^a xp(x)\,dx + \int_a^\infty xp(x)\,dx \\       &amp;\ge \int_a^\infty xp(x)\,dx \ge \int_a^\infty ap(x) \, dx  \end{aligned}$$  Why is the last inequality true? Why is it ok to change $x$ to $a$, and we know it is less than or equal to? How did they figure it out? Thanks! Identify a movie in which aliens believed a sci-fi TV show to be true documentary I remember watching a sci-fi movie in the early 00s. It was full of comedy. Here are the details I recalled:   There was a space opera TV show in the movie. It seemed a true copy of Star Trek (they used to pilot a ship and there were captain, first officer, science officer etc on bridge). The name of the TV show was possibly the name of the movie. Once the main crew of the TV show were celebrating some achievement, they were approached by humans having weird behavior (they were actually aliens in disguise; in reality, they looked like octopus). The alien race was innocent. They couldn't comprehend the concept of art, lies etc. So, they believed that the TV show (they got the EM waves) was true historical documentary of mankind. So, they came looking for help from the brave captain of the show. From the TV show details, they had built a starship which really worked. The aliens needed the TV show crew to pilot the ship to defeat another alien race which terrorized them. The TV show crew didn't have any problems piloting the ship as the controls were familiar, only these controls were real.   Can you please identify the movie?