This question involves finding the exact values of $x$ such that $0 \leqslant x \leqslant 2\pi$.  So far I have subtracted everything to the left side of the equation and then used the pythagorean identities to change $2\cos^2x\ \ \text{to}\ \ 2-2\sin^2x$, leaving   $$2-2\sin^2x-\sin x-1=0\,.$$ However, I need to further factor this, and I am not sure about what to do next.  Could someone please help me?
$2\cos^2x=1+\sin x$.  Determine the exact values of $x$ such that $0 \leq x \leq 2\pi$.    I am experiencing problems with factoring this question.  First I started by getting everything on to the left side so: $2\cos^2x-\sin x-1=0$.  Then seeing as there are two trig variable here ($\cos$ and $\sin$), I used the pythagorean identity and replaced $2\cos^2x$ with $1-2\sin^2x$ to make it into a one variable question: $1-1-2\sin^2x-\sin x-1=0$.   By collecting like terms it then becomes $-\sin^2x-\sin x=0$.  Now you could factor these with $0$ and $-1$, but then it creates uneven brackets so this musn't work.  I don't think the quadratic formula is necessary, but I am wondering if this could be treated as a difference of squares question.    If you replace the $\sin x$ and make the question into $-x^2-x=0$ it is a little easier to look at, however I am not sure if this is the right approach.  If someone could please clarify my answer, that would be appreciated!
Using package nomencl Which is correct: I'll be moving next month or I'll be shifting next month? For changing one's home from one place to another, I've heard people in western part of the world using the sentence:      I'll be moving next month.   In India, even in the English news channels, I've heard many people using the phrase:     I'll be shifting next month.   Which one is the correct usage? Or are both of them correct and the difference in usage is regional? This is from page 3 of  ().  Copying the relevant segment:     Stirling’s approximation tells us $\sqrt{2\pi n} (n/e)^n \leq n! \leq e^{1/12n} \sqrt{2\pi n} (n/e)^n$. In particular we can use this to say that   $$ {n \choose k} \leq \left(\frac{en}{ k}\right)^k$$   I tried the tactic of combining bounds from $n!$, $k!$ and $(n-k)!$ and it didn't work. How does this bound follow from stirling's approximation? Simple UV Mesh Unwrapping Scale issue I'm having problems using the UV mapping in blender 2.67. 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The mesh is only three faces so should only need this single seam. The image shows the mesh from infront and behind/underneath so it should make more sense.    Double Edit: One more picture for clarity on the seam issue. The mesh is unfolding correctly, it is just the scale of the unfolded map that is incorrect (in relation to one another) Why are the characters on the keyboard all capitalized? Why are all the letters (A, B etc) and all the writings (Home, Page Up etc) on the keyboard, capitalized?    Especially, the letters, because a new user could rightly expect uppercase letters to be entered upon pressing those keys to find the contrary. How to clear Chocolatey cache in the free version? Installation-file-cleaning is currently only available in  () and .  The package installation files can easily take gigabytes of storage space and to my knowledge they only increase in size as the user updates packages and installs new software, with only your hard drive capacity as the upper limit.   I don't know if Chocolatey stores cache in one central location — something is saved to ~\AppData\Local\Temp but the majority is located in the lib* folders in ~\ProgramData\chocolatey. Also the log files in the logs folder total to over 100 MB on my system. Is there any way to fully clear Chocolatey cache without manually deleting directories? How to apply a regex on a column fetched from DB I am using IBM DB2.  So, I am retrieving a column named SSN from a table. I have to apply the following regex on that column:  ([1-57-8][0-9]{2} |0([1-9][0-9] |[0-9][1-9]) |6([0-57-9][0-9] |[0-9][0-57-9])) ([1-9][0-9]|[0-9][1-9])([1-9]\d{3}|\d[1-9]\d{2}|\d{2}[1-9]\d|\d{3}[1-9])   Someone suggested to me that I have to write a function/procedure in the query to validate that the SSN picked from the db follows this regex else it should not be stored in the result set of the query.  I don't know how to write a function or a procedure in SQL, can you please help me out here? AlwaysOn AG: Current Host Server - Owner Node - Primary Instance I'm building up a SQL Server AlwaysOn AG and I'm on the step "failover", manual and automatic:  I noticed that in the Windows Server Failover Cluster a "Current Host Server" exists. My knowledge says that this is the "master" of the cluster that the cluster hosts. But now I noticed that the "Owner Node", which hosts the Availability Group(-role) a other node could be. (The owner node is also the primary instance of the AG, right?)  Now for me there exist two questions for the normal operating of an AlwaysOn AG:  Q1: Should the "Current Host Server" the same as the Owner Node of the AlwaysOn AG?  Q2: Imagine if I want to maintain a host. Shall I first failover the AG (via SSMS or with ALTER AVAILABILITY GROUP...) to the second node/another node and at second a failover of the WSFC-Current Host?  Maybe someone could help me with this questions and give me some advice.  Thanks Using (or not) the word "to" in a sentence ending with "be able" Which of these is grammatically correct?   As educators, we are always looking for ways to reach all the learners in our classroom as effectively as we are able.   As educators, we are always looking for ways to reach all the learners in our classroom as effectively as we are able to. Death in SAO before Kayaba announcement I was wondering if in the novel we knew about some deaths occurring before Kayaba announcement. I remember that in the anime, Kirito plays a bit with Cline before realizing that the log off option has disappeared. Then all the people are summoned to the city and learn about the fact that if they die in the game, they also die in real life.  I don't remember any players dying before the announcement in the anime, except for those who are not waking up and their relative pulling off their NerveGear thus causing death, but maybe in the light novel, some guys died without even knowing it just while being defeated by a mob (which seems awful for me).  Do we know about some players dying before that while thinking they are still in a normal game? Sometimes, I write the name of an author in a sentence before the citation. For this reason, I'd like to suppress it in the citation. How can I do that?  Is the only solution to create a new macro, e.g. \citewoauth, similar to \cite but without author names, or am I missing something? [The MWE (non-working actually) below is with \parencite but the general idea is similar.]    \begin{filecontents}{\jobname.bib}   @article{doe_2011,     title = {A title},     author = {John Doe},     journal = {Journal},     number = {99},     pages = {483--488},     date = {2015},   } \end{filecontents}  \documentclass[12pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage[french]{babel} \usepackage[T1]{fontenc} \usepackage{csquotes} \usepackage[style=verbose-trad2,language=french,backend=biber]{biblatex}  \addbibresource{\jobname.bib}  \begin{document} Some text\footnote{According to John Doe, it is \enquote{a complete nonsense} \parencite[485]{doe_2011}}  \printbibliography \end{document} How to search for upcoming conferences that will be held in a specific subject Why are these pistons pushing? I've lately become hooked into building computers in Minecraft and today I decided to build a giant hard drive. I've built HDDs before, but not at the scale I'm currently attempting, and I anticipated problems. As I was wiring some pistons, they activated and pushed forward oddly. As far as I can tell, there's nothing powering them.    I thought at first that it might be the redstone torches (top left of above pic) but they aren't powered.   Here's a top view of the same pistons. I also thought the line of redstone above  them might be the problem (middle of above pic), but when I destroy it, the pistons stay extended.  Is there a reason why my pistons are extending? If so, is there a way to get them to power (when I want them to) without disturbing the pistons above? Reference Text that develops Linear Algebra with Knowledge of Abstract Algebra Background: Due to some unfortunate sequencing, I have developed my abstract algebra skills before most of my linear algebra skills. I've worked through Topics in Algebra by Herstein and generally liked his approach to vector spaces and modules. Besides a very elementary course in linear algebra (where most of the time went towards matrix multiplication), I have not developed any other linear algebra skills.  But it seems that it is now necessary for me to do so. Most of the topics that I am looking at now require some background of linear algebra and I still lack understanding of ideas like: bilinear forms, invariant subspaces, eigenvalues, requirements for diagonalization of a matrix and so forth.  This brings me to my question (provided there are any)     Question: What are some good texts that develop the theory of linear algebra from a perspective of general algebra?   Are there any texts that develop the key (elementary) ideas of linear algebra in an abstract setting? Thank you for the help! What is the minimal set of parts for a circut with this AVR microcontroller? My Arduino Uno has a ATMEGA328P-PU microcontroller, and bunch of other stuff on the board. I'd like to program the chip without the Arduino software, and only the minimum of other components.  I want to eventually create things without the cost of Arduino, and I want to learn about the other parts on the board and add them back as as needed.  I'm comfortable with C and gcc, so I can probably figure out the software part. But what, if anything, besides these two products below, do I need to have in the breadboard with the microcontroller? shade plane with a color function color(x,y) Way to play shared game that is in use in steam? My brother just got me a new computer but he is always on so I can't play his steam shared games. Is there a way to play his games while he is playing a game too? Dynamic SELECT TOP @var In SQL Server How can I have a dynamic variable setting the amount of rows to return in SQL Server? Below is not valid syntax in SQL Server 2005+:  DECLARE @count int SET @count = 20  SELECT TOP @count * FROM SomeTable ConTeXt: Flush right formula numbers in cases environment I would like to have the formula numbers of the cases environment flush right like regular formulas (not at a fixed distant from the cases like it is by default with the default setting numberdistance=2.5\emwidth). How can I do that?  \starttext  \startplaceformula     \startformula         \startcases             \NC x = 1 \NR[+]             \NC y = 1 \NR[+]         \stopcases     \stopformula \stopplaceformula  \stoptext     What I want (but with the left bracket):  \starttext  \startplaceformula     \startformula         \startalignment[n=1]             \NC x = 1 \NR[+]             \NC y = 1 \NR[+]         \stopalignment     \stopformula \stopplaceformula  \stoptext I am not an expert of random forest but I clearly understand that the key issue with random forest is the (random) tree generation. Can you explain me how the trees are generated? (i.e. What is the used distribution for tree generation?)  Thanks in advance !