Calculate cos 43 with an error of less than 7/20,000
Compute Taylor error of $10^{-5}$ for $\cos(\pi/20)$
Prove Equivalence Relation in G Hei, guys!  I'm having some trouble with the next problem:  Let $A$ and $B$ be subgroups of $G$. Show that $\sim$ is an equivalence relation when it is defined as follows: $g\sim g'\Leftrightarrow g' = agb$ for some $a$ from $A$, $b$ from $B$.  Should I prove this is valid for any a and b? It says for some elements a and b. But I might as well take the unit element.   Could you give me a clue on how to solve it?  Cheers! Why do I get TypeError: can't multiply sequence by non-int of type 'float'? I am typing to get a sale amount (by input) to be multiplied by a defined sales tax (0.08) and then have it print the total amount (sales tax times sale amount).  I run into this error. Anyone know what could be wrong or have any suggestions?   salesAmount = raw_input (["Insert sale amount here \n"]) ['Insert sale amount here \n']20.99 &gt;&gt;&gt; salesTax = 0.08 &gt;&gt;&gt; totalAmount = salesAmount * salesTax  Traceback (most recent call last):   File "&lt;pyshell#57&gt;", line 1, in &lt;module&gt;     totalAmount = salesAmount * salesTax TypeError: can't multiply sequence by non-int of type 'float' Disable OS X startup sound How can I permanently disable the startup sound (or chime) in Yosemite?  (I am using a late-2014 13" macbook pro)  I already tried several options found here and there :    installed StartNinja.app => did not work installed Auto-Mute.app  => did not work tried to install the old StartupSound prefPane => did not work tried the command "sudo nvram SystemAudioVolume=%80" => volume does not change (I also tried other values than 80, such as 00, 01, or nothing)   NOTE: I know that it is possible to mute the next startup sound by turning the volume down, but this is not a permanent solution. Which QGIS installation for Windows comes with r.in.lidar? Why is AES considered to be secure? The security of RSA is based on the integer factorization problem, which is a very well defined and understood mathematical problem. This problem must be solved in order to fundamentally break RSA.   What about AES (and others based on the same idea)? Why is it difficult to break? Is there any mathematical principle that ensures its security? I do as follows  $(1+x)^n= \binom{n}{0}+\binom{n}{1}x+\binom{n}{2}x^2+...+\binom{n}{n}x^n$   differentiating in both sides  $n(1+x)^{n-1}= \binom{n}{1}+2\binom{n}{2}x+3\binom{n}{3}x^2...+n\binom{n}{n}x^{n-1}$  then, for $x=1$  $n \cdot2^{n-1} = \sum_{k=1}^{n}k\binom{n}{k}$  From here, I don't know how to proceed, I have tried to multiply both sides for $(1+x)$ to make $n \cdot2^{n}$ but then the summatory changes, I don't know how to prove the equality. Very specific paragraph shape understanding "mount" as a concept in the OS I want to understand what mounting is. It is used in different contexts and situations and I can't find resources which:   Describe the mount concept Explain the actions taken by the computer/OS/utility when a mount is performed How and in which situations mount is used Which features in the Linux mount command are of frequent use and some examples   ( I hear mount applied to diverse entities directories, flash drives, network card, etc ) Shells have a useful feature where pressing the keyboard UP arrow replaces the contents of the current command line with the previous command that was entered. (And if you do it again, the command before that, and so on.)  I would like a similar feature, where pressing (some other button) replaces the contents of the current command line with the last line of output from the previous command (and if you do it again, the previous line of output , and so on.)  Is there any shell that provides a feature like this?  The motivation is that you often want to run a command, and then run another command with part of the previous commands's output as input. (No, I don't want to do anything involving piping the output of one command into the next. I'm looking for an interactive experience, where you run the first command, see its output, and then get a line of that output into your next command, without having to reach for the mouse to do a clumsy copy-and-paste.) Can I use I2C over 2m cables? As most of you know, in the Microsoft Store, there are . This means that on Windows I can emulate Ubuntu and have the Ubuntu command line directly on Windows.  The question is very easy: what can’t I do from the Ubuntu command line emulated in Windows that I can do on a proper Linux-based Ubuntu? Is it useful to download this Ubuntu application, install it, and work just with it instead of the real OS? Is it possible to install all the development libraries? Can I write (or not) device drivers? In other words: what are the limits?  What features of Ubuntu Linux-based are missing from Ubuntu-on-Windows?  New answers are welcome: I know that every software is always changing by improving characteristics and features! Is it possible to insert arguments into functions when they're used like Function@ or //Function? After reading a couple of answers and comments on some SQL questions here, and also hearing that a friend of mine works at a place which has a policy which bans them, I'm wondering if there's anything wrong with using backticks around field names in MySQL.  That is:  SELECT `id`, `name`, `anotherfield` ... -- vs -- SELECT id, name, anotherfield ... Water Wars: Time Travel TV Show? When to use static classes in C# Here's what :   static class CompanyInfo {     public static string GetCompanyName() { return "CompanyName"; }     public static string GetCompanyAddress() { return "CompanyAddress"; }     //... }       Use a static class as a unit of   organization for methods not   associated with particular objects.   Also, a static class can make your   implementation simpler and faster   because you do not have to create an   object in order to call its methods.   It is useful to organize the methods   inside the class in a meaningful way,   such as the methods of the Math class   in the System namespace.   To me, that example doesn't seem to cover very many possible usage scenarios for static classes. In the past I've used static classes for stateless suites of related functions, but that's about it. So, under what circumstances should (and shouldn't) a class be declared static? How useful is that rule and how many times we can use that in one game?  Under what conditions can the en passant rule be used?  How often does it come up and is it a good idea to use it? How to change folders color? How do i change the folders colors? i am using this icon theme   and there are several folders. like black, gray, purple, orange(which is the default), yellow, green, red, blue, brown and lightblue  How can i use any of them? for example have a blue folder called "family" in images and another right beside that have a purple folder called "others".  Basically why have he created all those folders with different colors? Cleveref with counters with same name I am trying to use the  package.  I have the following counters defined (hopefully this is the right way to do it!)  \newtheorem{thm}{Theorem}[section] \newtheorem{lemma}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary}   This works as I would like (i.e. Theorem 1.1, Corollary 1.2, Lemma 1.3, etc.)  My problem is \cref always returns Theorem 1.2, Theorem 1.3. I   Perhaps this is not right? (In my preamble)  \crefname{thm}{theorem}{theorem's} \crefname{lemma}{lemma}{lemma's} One can contract I have to I've when have is a helping verb, e.g.     I've got an octopus in my pants.   Is contracting the main verb technically incorrect or merely antiquated? My father loves to say,     I've a month-old smoked shoulder I can cook for dinner,   and he sounds like a crazy old man when he does. I would like to know if I can shut him down by informing him that he is incorrect. :) This next year, I am helping an effort to clean several SQL Server environments.    We have about 10,000 stored procedures and estimate that only about 1000 of them are used on a regular basis, and another 200 or so are used on a rare occasion, meaning we have a lot of work to do.    Since we have multiple departments and teams that can access these databases and procedures, we are not always the ones calling the procedures, meaning that we must determine what procedures are being called.  On top of that, we want to determine this over a few months, not in a few days (which eliminates some possibilities).  One approach to this is to use the SQL Server Profiler and track what procedures are being called and compare them to the list of what procedures we have, while marking whether the procedures are used or not.  From then, we could move the procedures to a different schema in case a department comes screaming.  Is using the Profiler the most effective approach here?  And/Or have any of you done something similar and found another way/better way to do this?