How do i calculate $ \sum_{k=1}^{n} \sin(2k-1)x $, using $e^{i*x}=cos(x) + i*sin(x)$?
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?
sample make file for creating more than one exe files with different directory I agree that every page in a site should have a unique TITLE, and that every page should have one and only one H1 element, but should the text of each H1 be completely unique too?  Instinctively one would think so, but there are potentially many cases where (IMHO) mind this doesn't make sense and attempting to do so will impair the user experience &amp; presentation of the page - for example, where pages are part of a closely related cluster, e.g. a multi-step process/tool/wizard.  Surely the process name should go in the H1 (e.g. "Pregnancy Planner"), and the current step you're on should be marked up as a H2 (e.g. "Step 1 - Trying for a baby"), rather than shoehorning both into a single H1 together (e.g. "Pregnancy Planner - Step 1 - Trying for a baby")?  Not only does this make the H1 rather long and potentially impact the presentation of the page, I would argue that in combining both into one you also lose semantic goodness, as the page is surely semantically richer by keeping parent (process) and child (step) as separate, nested headings?  Furthermore, the current proposal I have read is that on the first page the H1 should be the name of the process and the first step an H2, but on subsequent pages the same text should not be a H1 at all (just text styled exactly the same as an H1) and the current step should become the H1 instead. For me this is introducing unnecessary complexity to the page authoring/production process, and inconsistency in the semantic structure of the pages within the stepped process.  So it all comes back to the basic assertion that each H1 must be completely unique, which I question. Can you convince me that this should be so and that the cost/complexity of implementing this (especially in situations such as that above) is worth it?  Your thoughts please... Note: the eventual outcome of this discussion involved dropping this particular off-topic reason, along with other changes. See:  &nbsp; &mdash;Shog     I was using the reason "demonstrate a minimal understanding" myself quite frequently, but the more I review, the more I have doubts about it and I think this reason can either be dropped entirely or at least rephrased, so I started to use "problems with code you've written must describe the specific problem" instead:     Questions asking for code must demonstrate a minimal understanding of the problem being solved.   So the poster has posted some description, maybe even a snippet of code and in his mind that already proves that he has that minimal understanding required. After all he described the problem and what he tried to solve. So telling him that he lacks that understanding is rather rude, and from the posters point of view doesn't even apply. What both version of the reason not emphasize is that a description of a problem is not various versions of     it doesn't work   but that we rather mean really specific details (see below). I can often see in the reopen review queue that the "improved" version just includes a more verbatim description, while what usually would be expected is some specific error message, a trace or a log, or whatever applies.  Then it continues with:     Include attempted solutions, why they didn't work, and the expected results. See also: Stack Overflow question checklist   comparing it to:     problems with code you've written must describe the specific problem — and include valid code to reproduce it — in the question itself. See SSCCE.org for guidance.   Which says essentially the same, only in more detail.   So the "minimum understanding" reason is IMO a more rude version of the reason telling the user that an SSCCE is missing.  The intent of giving this reason is to tell the user that he needs to give exact details to be able to help him, so IMO a checklist would be much better. Something like:     Your question was closed because it may not provide enough details. Things you can do to improve your question to get an answer include:         Format the code in a way that makes it readable and easily understandable.   Provide the smallest working sample of code demonstrating the problem (link to SSCCE)   Include your input and output.   If applicable provide logs (i.e. Android debugging log, stack trace, etc.).   If applicable provide the exact error messages (compiler, etc.).      And something like this should be best included already in the banner that appears when a question is closed instead of being hidden in some pages you often enough have to actively looking for.  It might be helpful to adjust this, based on the tags or close reason, as I realize that I'm more focused on the programming area, and the above list may not apply to other areas of SO.  Update  Another thing I forgot to mention (but wanted to) is this:     Questions asking for code   Is IMO essentially an invitation of the type of questions     I have a problem, can you provide the solution? (with an optional please)   while     Questions concerning problems with code you've written   puts at least some emphasis on the fact that the poster has to put some effort himself in trying to solve the problem by providing the code which he used to attempt the solution. Textbook recommendations for self-studying high school math? I'm a high school student who is trying to figure out a complete course of self-study for each year of high school. How can I self-learn grades of math without devoting too much time? This is a complex issue for me, as other students at my competitive high school have tutors and the like. Please recommend textbooks that have detailed explanations and progressive practice problems, for self-study for each area such as:  Algebra  Geometry  Trigonometry and Analytic Algebra  Pre-calculus  BC Calculus  Other people have skipped grades of math due to help from tutors and parents. Can I cover all of geometry and trigonometry in 8 months without going insane and be able to skip a grade? Writing string at the same position using Console.Write in C# 2.0 I have a console application project in C# 2.0 that needs to write something to the screen in a while loop. I do not want the screen to scroll because using Console.Write or Console.Writeline method will keep displaying text on console screen incremently and thus it starts scrolling.  I want to have the string written at the same position. How can i do this?  Thanks How can I show that if the second fundamental form of a surface is identically equal to zero, then the surface is a plane? This is my question:  Let P be a plane considered as a surface in 3-space. Show that its second fundamental form is zero. Conversely, show that if the second fundamental form of a surface is identically zero then the surface is a plane.  So far I have that the general equation of the plane is $$ax+by+cz+d=0.$$ Then $$M(u,v)=(u,v,\alpha u + \beta v + \gamma),$$ where $a,b,c,d,\alpha,\beta, \gamma$ are all constants and $c\not=0$. Then $M_u=(1,0,\alpha)$ and $M_v=(0,1,\beta)$. I know that $\vec{n} =\frac{M_u \times M_v}{|M_u \times M_v|}$ which gives $$ \vec{n}=\frac{(-\alpha,-\beta, 1)}{\sqrt{\alpha^2+\beta^2+1}}.$$ Hence, the unit norm is a constant. $$ M_{uu}=M_{uv}=M_{vv}=0$$ Hence $l=m=n=0$, so the second fundamental form is $$ ldu^2+mdudv+ndv^2=0.$$  Now I need to show the converse, but I don't know how. Please help :) I cannot install any package. It seems the command apt-get update must be run but it throws some errors:  $ sudo apt-get update Get:1 http://security.ubuntu.com/ubuntu xenial-security InRelease [102 kB]      Hit:2 http://ve.archive.ubuntu.com/ubuntu xenial InRelease                      Hit:3 http://ve.archive.ubuntu.com/ubuntu xenial-updates InRelease              Hit:4 http://ve.archive.ubuntu.com/ubuntu xenial-backports InRelease            Fetched 102 kB in 23s (4337 B/s)                                                *** Error in `appstreamcli': double free or corruption (fasttop): 0x000000000210f4b0 *** ======= Backtrace: ========= /lib/x86_64-linux-gnu/libc.so.6(+0x777e5)[0x7fac8d8317e5] [...] Aborted (core dumped) Reading package lists... Done E: Problem executing scripts APT::Update::Post-Invoke-Success   'if /usr/bin/test -w /var/cache/app-info -a -e /usr/bin/appstreamcli;   then appstreamcli refresh &gt; /dev/null; fi' E: Sub-process returned an error code Is it possible to extend an Indian business visa from India? I'm an Egyptian girl from Cairo. I'm planning a six-month trip to India in December for training purposes, but getting a six-month visa is so hard from my country.  If I can only get 3 months instead of 6 months, is it possible to extend the business visa in India? What does the notation like 8.6e-28 mean? What is the 'e' for? I have a problem with the interpretation of a test result in which the p-value is 8.6e-28. How should it be interpreted? What is the e for?  Specifically, I used White's Test for the homoscedasticity assumption of a linear regression:  White's general test statistic :  150.0535  Chi-sq( 9)  P-value =  8.6e-28 How do I identify SMD components? (or how do I identify any component) How do I identify the markings on an SMT component and match it up with a part number so I can be a good designer and actually look at a datasheet (and read the whole thing)? Or identify a part to replace an unknown part on a PCB? Best way to determine if two path reference to same file in C# In the upcoming Java7, there is a  to check if two file object are same file reference.  Are there similar API provided in the .NET framework?  I've search it over MSDN but nothing enlighten me.  I want it simple but I don't want to compare by filename which will cause problems with hard/symbolic links and different style of path. (e.g. \\?\C:\, C:\).   What I going to do is just prevent duplicated file being drag and dropped to my linklist. I know that $re^{i\theta} = x + iy$ for any complex number $x + iy$ by Euler's formula. How do you calculate relative and global phase? How can I deduce/identify an app from a notification icon? Clip tool gives ERROR 000732? At 9th level, the Armorer Artificer (from Tasha's Cauldron of Everything) gains the &quot;Armor Modifications&quot; feature :  You learn how to use your artificer infusions to specially modify your Arcane Armor. That armor now counts as separate items for the purposes of your Infuse Items feature: armor (the chest piece), boots, helmet, and the armor’s special weapon. Each of those items can bear one of your infusions, and the infusions transfer over if you change your armor’s model with the Armor Model feature. In addition, the maximum number of items you can infuse at once increases by 2, but those extra items must be part of your Arcane Armor.  This special weapon is called 'Thunder Gauntlets' (a simple melee weapon) for the 'Guardian' Armor Model, and 'Lightning Launcher' (a simple ranged weapon) for the 'Infiltrator' Model. But I am wondering which Artificer Infusions are appropriate for this special weapon ? Is it only the 'Enhanced Weapon' and 'Radiant Weapon' infusions, or are there others ? How can I download an older version of OS X/macOS? Sometimes it's hard to search for older versions of macOS &amp; OS X  on the App Store.  This is an attempt to provide the canonical QA for this issue, as per the Meta QA -  I expect it to be modified over time to hopefully become a complete information resource.  How can I get any older version of a Mac OS to be able to run as a VM, or to reinstall on my Mac? See all my questions across all Stack Exchange sites? Is it possible to earn more than 100 lollipops per second? I have pumped my lollipop rate to 100/s, but it took a long time to go from 99 to 100. Is it even possible to get higher or should I stop? How to convert a factor to integer\numeric without loss of information? Cross-validation for timeseries data with regression