I need help Proving this: &quot;For Every $n\in \mathbb{N-{0}}$ Prove the following: $$\sin(1)+\sin(2)+...+\sin(n)=\frac{\sin(\frac{n+1}{2})\sin(\frac{n}{2})}{\sin(\frac{1}{2})}"$$ I assume we have to use Induction On this but I failed to do so If anyone can provide a simple and clear proof I would be grateful!
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? For example here is the sum of $\cos$ series:  $$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \cos \biggl( \frac{ 2 a + (n-1)\cdot d}{2}\biggr)$$  There is a slight difference in case of $\sin$, which is: $$\sum_{k=0}^{n-1}\sin (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \sin\biggl( \frac{2  a + (n-1)\cdot d}{2}\biggr)$$   How do we prove the above two identities?
The minimum value of $x^8 – 8x^6 + 19x^4 – 12x^3 + 14x^2 – 8x + 9$ is The minimum value of $$f(x)=x^8 – 8x^6 + 19x^4 – 12x^3 + 14x^2 – 8x + 9$$ is  (a)-1  (b)9  (c)6  (d)1  Apart from trying to obtain $1$, which in this case is simple and $f(2)=1$ is there a standard method to approach such problems.  Please keep in mind that this is an objective question in one of the competitive exam and you get around 5 mins to solve it.  Also this is asked in elementary section, so only knowledge of basic calculus and polynomials is assumed. How to write a LaTeX package that bundles not only .sty and .cls files but also some logos in .pdf or .eps formats? I would like to make a LaTeX package with the following subdirs:   tex/latex/common (a common .tex file shared among beamer and letter) tex/latex/resources (lots of pdf and eps files with graphical resources) tex/latex/mybeamer (a custom beamer theme with 5 .sty files) tex/latex/myletter (a .cls file for a letter template)   I would like to package these files into a single LaTeX package that users could install in an easy way. I read about the .ins and .dtx files and I can see how to pack the .sty and .cls files but what about the extra graphics - the resources folder which contains several PDF and .eps files? Why is Leia okay with being tracked to Yavin? In A New Hope, after Han and Luke fight off the TIE Fighters while escaping from the Death Star, these exchanges happen:             INT. DEATH STAR - CONTROL ROOM             Darth Vader strides into the control room, where Tarkin is             watching the huge view screen. A sea of stars is before him.                                   TARKIN                      Are they away?                                   VADER                      They have just made the jump into                       hyperspace.                                   TARKIN                      You're sure the homing beacon is                       secure aboard their ship? I'm taking                       an awful risk, Vader. This had better                       work.             INT. MILLENNIUM FALCON - COCKPIT             Han, removes his gloves and smiling, is at the controls of             the ship. Chewie moves into the aft section to check the             damage. Leia is seated near Han.                                   HAN                      Not a bad bit of rescuing, huh? You                       know, sometimes I even amaze myself.                                   LEIA                      That doesn't sound too hard. Besides,                       they let us go. It's the only                       explanation for the ease of our                       escape.                                   HAN                      Easy... you call that easy?                                   LEIA                      They're tracking us!                                   HAN                      Not this ship, sister.             Frustrated, Leia shakes her head.                                   LEIA                      At least the information in Artoo is                       still intact.                                   HAN                      What's so important? What's he                       carrying?                                   LEIA                      The technical readouts of that battle                       station. I only hope that when the                       data is analyzed, a weakness can be                       found. It's not over yet!   Leia doesn't even know if there is a weakness, let alone whether they'll find one. If she (correctly) realizes that they're being tracked, why doesn't she suggest that the Millennium Falcon go somewhere besides Yavin 4 instead of just frustratedly shaking her head? I know the plans , but it seems unlikely that continuing to fly to the secret Rebel base in the ship she's certain is being tracked is the only option. They are already in hyperspace when this conversation happens, but they apparently have time to talk for a little while after, so presumably they'd have time to stop and reroute. I am looking for a correct proof of this statement: If $G$ is a group such that $G/Z(G)$ is cyclic, then $G$ is commutative.  Proof: $G/Z(G)$ is isomorphic to $\operatorname{Inn}(G)$ and is cyclic, and then for every $a$ and $b$ in $G$ the inner isomorphisms $\gamma_a$ and $\gamma_b$ satisfy $\gamma_a \gamma_b = \gamma_{ab} = \gamma_{ba} = \gamma_b \gamma_a$, and therefore for every $a,b \in G$, $ab = ba$.  Is that proof complete, or am I missing something? Thanks a lot for the help. Parser for C# Which parsers are available for parsing C# code?  I'm looking for a C# parser that can be used in C# and give me access to line and file informations about each artefact of the analysed code. How to use Boost in Visual Studio 2010 Has confinement been experimentally observed? So, confinement has obviously been shown by lattice gauge theory to be a predicted aspect of QCD. However, to what extent has it been observed in experimental physics? Find the volume of the tetrahedron with the vertices $P(1,1,1)$, $Q(1, 2, 3)$, $R(3, 1, 2)$, and $S(2, 3, 1)$. Determining a matrix from its characteristic polynomial Let $A\in\mathcal{M}_{n}(K)$, where $K$ is a field. Then, we can obtain the characteristic polynomial of $A$ by simply taking $p(\lambda)=\det(A-\lambda I_n)$, which give us something like $$p(\lambda) = (-1)^n\lambda^n + (-1)^{n-1}(\text{tr } A)\lambda^{n-1} +  \cdots + \det A$$ Now, how can we obtain the matrix $A$ knowing the characteristic polynomial? Uninformative error message when using AUCTeX How to find an end point of an arc given another end point, radius, and arc direction? How to install a QGIS plugin when offline? Due to various IT policies at my workplace, QGIS is installed on a machine that is not connected to the internet. I wish to install a couple of QGIS plugins on this system.  I have downloaded the required plugins from . How do I install them in QGIS? injective linear transformation basis. Let $T : V → W $ be an injective linear transformation, where $V, W$ are vector spaces of dimension $n$ over the field $K$. If $\{e_1,\ldots , e_n\}$ is a basis for $V$ , show that $\{T(e_1),\ldots , T(e_n)\}$ is a basis for $W$. Is there a method to split thousands of polylines by employing a tool like the "Editor" - "Split" method and using distance, in ArcGIS?      When I utilize the "split" - "distance" tool (fig1), it only splits one section leaving the remainder unsplit (fig2).    I have attempted the "COGO" tool - using "Proportion", and can utilize the "duplicate" button to assign the remainder of segments which seems like a viable option.  But I would like to find some way to automate this process.  Has anyone found a method to do this?  Or is Python (coding) the only way?  I am trying to split over 2000 lines into equal distance segments (fig 3) of ~600m. Each segment will have its angle measured, and will ultimately result in a rose diagram representing angular statistics.  The distance of each segment is important in my splitting regime (a long line will have a greater representation of a particular angle on my rose diagram than the neighboring shorter line segments).  I currently am using ArcGIS 10.2 with an advanced user license. Word for a person who continually tries to teach you when it's not necessary I've experienced this way too many times, and the best example I can give is a person who treats you like you have no idea how to do X and who keeps teaching you how to do it. I feel handicapped. Take for example how to play a card game. You know how to play the game because you've played it 5 times. However, there is a person always insists on explaining the game to you - like you are always new to the game - when you don't actually need any help and you just want to think for yourself. Another example situation is doing a repetitive task which you already have experience with, and yet the person continually tries to 'teach' you when you don't need help/already know how to do it. It almost makes you mad because you really don't need help. As an example which recently happened to me:  &quot;Hey, do you remember how to start the wood chipper?&quot; &quot;Yes I do&quot; &quot;Okay, well, let me show you one more time&quot;  Again, is there a term for someone who does this? I'm considering the ring $\mathbb{Z}[\sqrt{-n}]$, where $n\ge 3$ and square free. I want to see why it's not a UFD.   I defined a norm for the ring by $|a+b\sqrt{-n}|=a^2+nb^2$. Using this I was able to show that $2$, $\sqrt{-n}$ and $1+\sqrt{-n}$ are all irreducible. Is there someway to conclude that $\mathbb{Z}[\sqrt{-n}]$ is not a UFD based on this? Thanks. about://(null) popup? I encountered a popup similar to this one this morning while checking the weather report at  in Safari, though I have not been able to reproduce it.  The title is "about://(null)" and the text "Why you no play Candy Crush!!"  The popup occurred twice.  I tapped Cancel both times.  I suspect the source of this is rogue JavaScript in an ad network, judging from the fact that the screenshot below shows it in an ad-supported application, and when I saw it, I was on an ad-supported website.  My phone is not jailbroken.  I'm wondering if anyone is able to identify the source of the popup definitively.  Speculation is running rampant on Apple's support communities and elsewhere, from push notifications (delete Facebook and all your games!) to carrier notifications (which don't work on iPod Touches…), and my gut tells me little of it is helpful.  But I could also be very wrong.  (This is not my screenshot; it comes from .)    UPDATE 1: The Candy Crush people have this to say, but I'm not (yet) convinced there's actually something installed that is to blame:   UPDATE 2: Just reproduced it in the iOS Simulator while clicking around woodtv.com, but I'm not sure how to track it down further from here. Why "think bright" is used with adjective instead of adverb (think brightly)? I find "think bright" is much more common usage, but why? In my opinion, "Think" is a verb, so it is reasonable to use adverb to describe it.  Examples:      THINK BRIGHT!      The National University of Science and Technology MISIS   () babel shorthand "| doesn't work in macros The package babel with the ngerman option (and most other languages as well, but not english) provides the shorthand "|, which prevents a ligature at the given position (see p. 81 of the , e.g. Auf"|lage will result in the word Auflage without an fl-ligature (as is desired because, in German, you don't want ligatures across morpheme borders).  However, when placed in a macro, "| doesn't work anymore. Why is this and how can I circumvent it?  \documentclass{article}  %\usepackage[T1]{fontenc}% doesn't help \usepackage[ngerman]{babel}  \newcommand{\foo}{"|}  \begin{document}  ff  f"|f  f\foo f  \end{document} operators as strings I need to evaluate a mathmatical expression that is presented to me as a string in C#.  Example noddy but gets the point across that the string as the expression.  I need the evaluate to then populate an int.  There is no Eval() in C# like in others langugaes...  String myString = "3*4";   Edit:  I am on VS2008   Tried the Microsoft.JScript. = Its deprecated method (but still complies - warning)  However the Microsoft.JScript dll that I have doens work on      public object InvokeMember(string   name, BindingFlags invokeAttr, Binder   binder, object target, object[] args);   Complains that there is a missing ";"  go figure...  EDIT 2  Solution - was the codeDom one - it worked for as there are no security issue - only me ever going to be running the code.  Many thanks for the replies ...  And the link to the new Dragon Book  awesome   EDIT 3  Matt dataTable.Compute() also works - even better for the security conscious.  (parameter checking noted)