Prove    $$\sin\frac\pi7 \cdot \sin\frac{2 \pi}{7}\cdot\sin\frac{3 \pi}{7} = \frac{\sqrt7}{8}$$   I tried using the product to sum identities but those got me nowhere. Any help would be great.
Evaluation of $$\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7} = $$   $\bf{My\; Try::}$ I have solved Using Direct formula::  $$\sin \frac{\pi}{n}\cdot \sin \frac{2\pi}{n}\cdot......\sin \frac{(n-1)\pi}{n} = \frac{n}{2^{n-1}}$$  Now Put $n=7\;,$ We get   $$\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7}\cdot \sin \frac{4\pi}{7}\cdot \sin \frac{5\pi}{7}\cdot \sin \frac{6\pi}{7}=\frac{7}{2^{7-1}}$$  So $$\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7} =\frac{\sqrt{7}}{8}$$  Now my question is how can we solve it without using Direct Formula, Help me  Thanks
Finding an uncountable chain of subsets the integers How to find an uncountable subset of $P(\mathbb{N})$ such that every two elements of it can be compared. In fact, give an uncountable subset of $P(\mathbb{N})$ such that has totality property. We mean by $P(\mathbb{N})$, the powerset of natural numbers set $\mathbb{N}$. This is an old problem that deserves a bugfix, but it's getting beyond a joke now and I can't seem to work around it.  I keep getting the dreaded failed To download extra data files error in relation to ttf-mscorefonts-installer:    Repeatedly asking it to try again doesn't help, additionally, repeatedly trying-  sudo rm -rf /var/lib/update-notifier/package-data-downloads/partial/* sudo apt-get --purge --reinstall install ttf-mscorefonts-installer   Isn't working either.  I just get at least one 404 error from one of the sourceforge links that stops it working-  Reading package lists... Done Building dependency tree        Reading state information... Done 0 to upgrade, 0 to newly install, 1 reinstalled, 0 to remove and 20 not to upgrade. Need to get 0 B/29.5 kB of archives. After this operation, 0 B of additional disk space will be used. Preconfiguring packages ... (Reading database ... 352049 files and directories currently installed.) Preparing to unpack .../ttf-mscorefonts-installer_3.4+nmu1ubuntu2_all.deb ... mscorefonts-eula license has already been accepted Unpacking ttf-mscorefonts-installer (3.4+nmu1ubuntu2) over (3.4+nmu1ubuntu2) ... Processing triggers for fontconfig (2.11.94-0ubuntu1.1) ... Processing triggers for update-notifier-common (3.168.2) ... ttf-mscorefonts-installer: processing... ttf-mscorefonts-installer: downloading http://downloads.sourceforge.net/corefonts/andale32.exe Get:1 http://downloads.sourceforge.net/corefonts/andale32.exe [198 kB] Fetched 198 kB in 0s (339 kB/s)                                                 W: Can't drop privileges for downloading as file '/var/lib/update-notifier/package-data-downloads/partial/andale32.exe' couldn't be accessed by user '_apt'. - pkgAcquire::Run (13: Permission denied) ttf-mscorefonts-installer: downloading http://downloads.sourceforge.net/corefonts/arial32.exe Get:1 http://downloads.sourceforge.net/corefonts/arial32.exe [554 kB] Fetched 554 kB in 0s (575 kB/s)                                                 W: Can't drop privileges for downloading as file '/var/lib/update-notifier/package-data-downloads/partial/arial32.exe' couldn't be accessed by user '_apt'. - pkgAcquire::Run (13: Permission denied) ttf-mscorefonts-installer: downloading http://downloads.sourceforge.net/corefonts/arialb32.exe Get:1 http://downloads.sourceforge.net/corefonts/arialb32.exe [168 kB] Fetched 168 kB in 0s (306 kB/s)                                                 W: Can't drop privileges for downloading as file '/var/lib/update-notifier/package-data-downloads/partial/arialb32.exe' couldn't be accessed by user '_apt'. - pkgAcquire::Run (13: Permission denied) ttf-mscorefonts-installer: downloading http://downloads.sourceforge.net/corefonts/comic32.exe Err:1 http://downloads.sourceforge.net/corefonts/comic32.exe   404  Not Found W: Can't drop privileges for downloading as file '/var/lib/update-notifier/package-data-downloads/partial/comic32.exe' couldn't be accessed by user '_apt'. - pkgAcquire::Run (13: Permission denied) E: Failed to fetch https://netix.dl.sourceforge.net/project/corefonts/the fonts/final/comic32.exe  404  Not Found  E: Download Failed Setting up ttf-mscorefonts-installer (3.4+nmu1ubuntu2) ...   The specific point at which it fails to download changed - it seems to be a sourceforge reliability problem, but this has now been going on weeks and results in annoying popups as above all the time.  How do we get this fixed and/or how can I work around it for now. I'm having a problem with author display for a statute - the author is 'City and County of San Francisco' but the output (using the apalike.bst style file provided by Elsevier) gives the author as 'City and San Francisco', missing the 'County' in the authorname.  Any suggestions? How do I check if a given string is a legal/valid file name under Windows? I want to include a batch file rename functionality in my application. A user can type a destination filename pattern and (after replacing some wildcards in the pattern) I need to check if it's going to be a legal filename under Windows. I've tried to use regular expression like [a-zA-Z0-9_]+ but it doesn't include many national-specific characters from various languages (e.g. umlauts and so on). What is the best way to do such a check? How do I suspend a whole process (like the Process Explorer does when I click Suspend) in C#.  I'm starting the Process with Process.Start, and on a certain event, I want to suspend the process to be able to do some investigation on a "snapshot" of it. I think my device got a malware. A lock screen with ads sometimes appear, and I must swipe an area at the bottom of the screen to open the phone. I am in Eastern Europe, some ads are in Slavic languages, and they are seemingly related to travel.    Image resized. Click to view the full image  I have seen ads for Lot Air (Polish carrier). I recently went to their site.  Considering my recent surfing activity and the nature of the ads, I guess the malware is connected to airlines active in Eastern Europe.  The lock screen also promotes the app "DU Quick Charger" by claiming it has recharged the phone extra fast during screen off. (!?) It also shows battery level. I have never installed DU Quick Charger. I doubt DU Quick Charger is installed on the phone.  The problem started after installing Ryanair app. The problem is maybe, but not surely, related to this. Uninstalling Ryanair and restarting the phone doesn't remove the occasional appearance of this lock screen. After a while the situation has changed for the worse, now this lock screen appears after EVERY screen off.  I have always used Avast. I am not rooted and only install via Google Play.  What is this? And how do I remove it? Are the decimal places in a CSS width respected? Was mathematics invented or discovered? What would it mean to say that mathematics was invented and how would this be different from saying mathematics was discovered?   Is this even a serious philosophical question or just a meaningless/tautological linguistic ambiguity? Arduino only works in root I've installed Arduino but as the title says, I can only upload with sudo. And yes my user is part of the dialout group so that's the weird thing.   Arduino did function before on my system. What could be wrong?  Output ls -l:  ls -l /dev/ttyUSB* crw-rw-rw- 1 root dialout 188, 0 apr  2 09:23 /dev/ttyUSB0 The bevel depth (the radius of the tube) is set to 9cm but it actually isn't when compared to the background grid. This is is confirmed by importing it into sketchup and scaling it to the proper units it turns out the radius is only around 2.286cm. This problem only shows up on certain projects. What's wrong and how do I fix it?  (sorry that I can't upload the file, it's a long story) How do I iterate over a range of numbers in Bash when the range is given by a variable?  I know I can do this (called "sequence expression" in the Bash ):   for i in {1..5}; do echo $i; done   Which gives:     1    2    3    4    5   Yet, how can I replace either of the range endpoints with a variable? This doesn't work:  END=5 for i in {1..$END}; do echo $i; done   Which prints:     {1..5} Does it make sense to combine PCA and LDA? Assume I have a dataset for a supervised statistical classification task, e.g., via a Bayes' classifier. This dataset consists of 20 features and I want to boil it down to 2 features via dimensionality reduction techniques such as Principal Component Analysis (PCA) and/or Linear Discriminant Analysis (LDA).  Both techniques are projecting the data onto a smaller feature subspace: with PCA, I would find the directions (components) that maximize the variance in the dataset (without considering the class labels), and with LDA I would have the components that maximize the between-class separation.  Now, I am wondering if, how, and why these techniques can be combined and if it makes sense.  For example:   transforming the dataset via PCA and projecting it onto a new 2D subspace transforming (the already PCA-transformed) dataset via LDA for max. in-class separation   or    skipping the PCA step and using the top 2 components from a LDA.   or any other combination that makes sense. Can't move/snap Nautilus window to occupy left or right half of the 1366x768 screen - is it just me? I'm running Ubuntu 18.04 LTS with GNOME Shell. Nautilus does not want to occupy 50% (left or right) of the screen sized at 1366x768.    Other applications (such as GNOME Terminal, Eye of GNOME, Gedit) are normally snapping to the left/right half of the screen.  But Nautilus can be resized to the width lower than half of screen (I can get 650 px, which is smaller than 1366/2 = 683 px).  The first usable screen resolution seems to be 1400x1050, then goes 1440x900. But large amount of current devices .  Is it just me?     Updates: 1. standard shortcut Super+← or Super+→ do not work too. 2. I reported  which has linked upstream . 3. It does not work on GNOME FlashBack session too. I was working on a userscript today, which had a goal of displaying chat pings immediately at the main site. This would allow me to respond more quicker-like to chat pings.  However, I was met by this fearsome warrior:     Can we please set some decent header things? I want to be able to do this really bad. Multiple object instances represented by two slightly shifted rectangles In a software diagram, I'd like to be able to represent multiple instances of an object by two rectangles with one of them slightly shifted. I recently learned about the pic style so I'm using it with two nodes to achieve what I want.  The following example does work, but I'd like to know if I can do the same without using the fill=white. My hack kind of assumes the entire picture background is white and that may not always hold. I'd like to be able to crop the unwanted picture part out.  \documentclass[crop,border=3pt]{standalone}  \usepackage{pgf,tikz} \usetikzlibrary{positioning}  \begin{document} \begin{tikzpicture}[     every node/.style={font=\sffamily\small},     block/.style={         rectangle, draw, align=center, minimum width=2cm,         minimum height=1.4cm, font=\sffamily\small     },     multinode/.pic={         \node [block] at (0.1,0.1) {};         \node [block, \tikzpictextoptions, fill=white] at (0,0) {\tikzpictext};% &lt;-- FILL WHITE     } ]     \node [block] (STG) at (2,5) {single\\instance};     \pic [pic text={multi\\instance}] at (5,5) {multinode}; \end{tikzpicture} \end{document}   The following images show first one version with the fill white hack, and then one version without it. How to preserve the same parskip in minipage how to prove that $\ln(1+x)< x$ I am trying to draw a cantor set in LaTeX and I am having issues doing this. I am trying multiple \put commands, but they just go to the right of one another. Does anyone know how to do this?  Any help would be much appreciated. How to draw density estimate on histogram by pgfplots? I know how to draw a density histogram using pgfplots from this answer: .  But how can I draw an estimate line on the histogram like this?    \documentclass{article} \usepackage{pgfplots} \begin{document} \pgfplotsset{     small,     every axis plot post/.style={         fill=orange!75,         draw=orange!50!black     },     trim axis left }  \begin{tikzpicture} \begin{axis}[ymin=0, title=\texttt{hist=density}] \addplot [hist=density] table [y index=0] {random.txt}; \end{axis} \end{tikzpicture} \end{document} How to escape quotes in shell? I'm having trouble with escaping characters in bash. I'd like to escape single and double quotes while running a command under a different user. For the purposes of this question let's say I want to echo the following on the screen:  'single quote phrase' "double quote phrase"   How can I escape all the special chars, if I also need to switch to a different user:  sudo su USER -c "echo \"'single quote phrase' \"double quote phrase\"\""   Of course, this doesn't produce the right result.