What trigonometric identity was used?
$\sum \cos$ when angles are in arithmetic progression
How did Suren Manvelyan create the macro pictures of eyes? I came across some really cool  today. How did he manage to get these photos? I couldn't find any EXIF information anywhere, and I don't have enough experience with macro to figure out how he did it.  It's obvious that the light has a lot to do with the result. How and what kind have been used? And what kind of camera equipment? Why doesn't Neo teleport at random? If Python does not have a ternary conditional operator, is it possible to simulate one using other language constructs? "invalid EFI file path" while booting Windows WPF DataGrid : CanContentScroll property causing odd behavior Mathematical Induction divisibility $8\mid 3^{2n}-1$ Does the new guidance on duplicate questions suggest closing a question as duplicate, if the original answers the OP's question? In other words, should we now look at the content of the answers when we are evaluating questions for possible duplication?  Is it sufficient to say "Your answer can be found in this question over here, therefore your question is a duplicate?"  The new guidance is here:    Among other things, it describes the new close banner:     This question already has an answer here:   Link to the Duplicate Question   ... which seems to suggest that, if some other question in the system answers the OP's question, then the OP's question can be closed as a duplicate on that basis (even if the question itself is not necessarily duplicated).  Example Here:    Note the comments below the question, where I dispute the assertion made there that it is a duplicate question.  The original question doesn't mention unit testing. I'm looking to find out if there's any easy way to calculate the number of ways to tile a $3 \times 2n$ rectangle with dominoes. I was able to do it with the two codependent recurrences  f(0) = g(0) = 1 f(n) = f(n-1) + 2g(n-1) g(n) = f(n) + g(n-1)   where $f(n)$ is the actual answer and $g(n)$ is a helper function that represents the number of ways to tile a $3 \times 2n$ rectangle with two extra squares on the end (the same as a $3 \times 2n+1$ rectangle missing one square).  By combining these and doing some algebra, I was able to reduce this to  f(n) = 4f(n-1) - f(n-2)   which shows up as sequence , confirming that this is the correct recurrence.  The number of ways to tile a $2 \times n$ rectangle is the Fibonacci numbers because every rectangle ends with either a verticle domino or two horizontal ones, which gives the exact recurrence that Fibonacci numbers do. My question is, is there a similar simple explanation for this recurrence for tiling a $3 \times 2n$ rectangle? This is a  about Monitoring Software. Also Related:   I need to monitor my servers; what do I need to consider when deciding on a monitoring solution? Why use a ReentrantLock if one can use synchronized(this)? I'm just wondering where these values are being set and what they default to? Mine is currently 18446744073692774399. I didn't set it anywhere that I can see.  $ cat /proc/sys/kernel/shmmax  18446744073692774399  $ sysctl kernel.shmmax kernel.shmmax = 18446744073692774399 I am looking for a word that would describe being obsessed with the details of a larger entity such that the "looker" neglects to see the whole or (perhaps more importantly) the purpose of the whole.  Basically, "not seeing the big picture".  What can [more] succinctly describe this?  Edit: I just found this post: . I think I want the exact opposite. Markov/Chebychev inequality If Z is a square integrable random variable such that $E[Z]&gt;0$, then prove that  $$P(Z\geq k) \geq \frac{(E(Z)-k)^2}{E(Z^2)} \forall k \in [0,E(Z)]. $$  This is a homework question and I don't even understand why the inequality is reversed comparing to Markov/Chebychev inequality. Should Cauchy-Schwartz inequality be used? If yes, then how to begin? Thanks a ton! What is the significance of momentum? I just want to get an idea what  is. I know the mathematical meaning that momentum is $mv$ where $v$ is velocity. But I don't know its significance. Like I know that Acceleration is how velocity is changing with respect to time, I wanted to get feel of what momentum really is? Casting two cantrips, both modified by metamagic? Can you cast one cantrip, modifying it with "Twinned Spell" (Fire Bolt) to target 2 creatures, and then cast another cantrip as a Bonus Action (Fire Bolt), using "Quickened Spell" to target one of the previous creatures or another creature? Fantasy book, YA, Girl with magical powers & tattoos I read this book series in middle school. I believe that it would be classified as young adult fiction/fantasy. A girl has magical powers, and with her growing powers comes runes/tattoos in her arms. This is shown on every book cover. I know that one cover is blue. This one shows the girl in a black tank top and black pants. One is a yellow cover, with her in a dress (I think).  The girl has a friend/mentor, who she develops feelings for. The girl is the key to stopping a war between multiple factions/people. Each side is trying to use her to their advantage. I think the series was written in the early 2000s. I found this question in a book.   How many terms of the Maclaurin expansion of $(1+x)^{82}$ are needed to guarantee finding a value of $1.1^{82}$ to an accuracy of $10^{-6}$?  This is how I tried to do it:  We want the Lagrange error to be less than $10^{-6}$:  $$R_{n}(0.1)&lt;10^{-6}$$  Now to find $R_{n}(0.1)$:  $$R_{n}(0.1)=\frac{f^{(n+1)}(0.1)\cdot0.1^{n+1}}{(n+1)!}$$  After writing the values of ${f^{(n+1)}}(x)$, I find a pattern and say that  $${f^{(n+1)}}(x)=\frac{82!}{(82-n)!}(1+x)^{82-n}$$  $${f^{(n+1)}}(0.1)=\frac{82!}{(82-n)!}(1.1)^{82-n}=\frac{82!(1.1)^{82-n}}{(82-n)!}$$  Now just substitute all of this back to $R_{n}(0.1)&lt;10^{-6}$:  $$\frac{\frac{82!(1.1)^{82-n}}{(82-n)!}\cdot0.1^{n+1}}{(n+1)!}&lt;10^{-6}$$  Great! Now I have an inequality that I can solve. I can't graph this so I just use trial and error (which seems like quite an intensive way to do it, so if anyone has a better way I'd love to know).  $$\frac{\frac{82!(1.1)^{82-n}}{(82-n)!}\cdot0.1^{n+1}}{(n+1)!}-10^{-6}&lt;0$$  This seems to work first for $n=28$. That means the highest power of $x$ that I need is 28, so the number of terms in the expansion needs to be $n+1=29$. But the book's answer is       30   What am I doing wrong here? I'm still quite new to the power series expansions so if there's anything unclear in my working, please let me know (also, as a side note, the book itself may be wrong, though that isn't very likely I'd assume). I have an upcoming trip where I will be flying to and from Copenhagen, through Toronto.  I notice that my departure and return gate in the Philadelphia airport is not in the international terminal.  Where will I clear customs? When performing apt-get update, I get the following error:   root@ADS3-Debian6:/home/aluno# apt-get update Atingido http://sft.if.usp.br squeeze Release.gpg Ign http://sft.if.usp.br/debian/ squeeze/contrib Translation-en Ign http://sft.if.usp.br/debian/ squeeze/contrib Translation-pt Ign http://sft.if.usp.br/debian/ squeeze/contrib Translation-pt_BR   (...)  Obter:10 http://security.debian.org squeeze/updates/non-free i386 Packages [14 B] Baixados 612 kB em 4s (125 kB/s)                     Lendo listas de pacotes... Pronto There is no public key available for the following key IDs: 8B48AD6246925553 The Boolean modifier is not working I'm trying to model an , but it's not going so well when it comes to the magazine of the weapon.   Next to the magazine is a mesh. I want to cut holes in the mag with that mesh. But the magazine just goes invisible or connects with the mesh when trying to execute the Boolean Modifier. I have tested to invert the normals and some other stuff, but it isn't working.