Find the value of $\cos 1^\circ + \cos 2^\circ + \cos 3^\circ +.....+\cos 180^\circ $  My Attempt: $$=\cos 1^\circ + \cos 2^\circ + \cos 3^\circ +....+\cos 180^\circ $$ $$= \cos 1^\circ + \cos 2^\circ + \cos 3^\circ +....-1$$  How do I complete it?
Possible Duplicate:          Prove $$\cos(\alpha) + \cos(\alpha + \beta) + \cos(\alpha + 2\beta) + \dots + \cos[\alpha + (n-1)\beta] = \frac{\cos(\alpha + \frac{n-1}{2}\beta) \cdot \sin\frac{n\beta}{2}}{\sin\frac{\beta}{2}} $$
As our users regularly upload ~6MB images to use on the website (and aren't too familiar with how to resize them first), WordPress stores the originals as well as resizing them to several different sizes.  I'd like a function or plugin that takes the uploaded image, resizes it down to something more manageable and then replaces the original.  I've seen some functions that delete the original but do not replace it, meaning it is impossible to regenerate the thumbnails at a later date. I need this to be replaced so the user can upload a large image and it is automatically resized down and stored for future resizing if needed. Let $\mathbb S = \varnothing$.      Then from the definition: $ \bigcap \mathbb S = \left\{{x: \forall X \in \mathbb S: x \in X}\right\}$      Consider any $x \in \mathbb U$.      Then as $\mathbb S = \varnothing$, it follows that: $\forall X \in \mathbb S: x \in X$ from the definition of vacuous truth.      It follows directly that: $\bigcap \mathbb S = \left\{{x: x \in \mathbb U}\right\}$      That is: $\bigcap \mathbb S = \mathbb U$. ()   Proofwiki uses the above "proof" to "prove" that intersection of the empty set is the whole universe.   My question is, is the use of vacuous truth really allowed in axiomatic set theory, like ZFC? I don't see how the use of vacuous truth is justified.   The next problem I can think of is that we cannot really "define" the elements of empty set (to my knowledge, there is no element in empty set) so how can we then prove as the above proof did? This seems to contradict the use of vacuous truth.  And of course, there is issue of using the whole universe as a set, and I don't think this is allowed.... (Maybe proof above is using a different axiomatic set theory, as I am using ZF-minded thoughts...) ViewPager and fragments — what's the right way to store fragment's state? Is Angular momentum conserved in Impure rolling How to fix pagination for custom loops? MediaElement not playing audio from stream WP7 Block cipher fixed points (plaintext equal to ciphertext) The isomorphism $\mathbb{Z}[i]/(a+bi) \cong \Bbb Z/(a^2+b^2)\Bbb Z$ is , when the integers $a$ and $b$ are coprime. But what happens when they are not coprime, say $(a,b)=d&gt;1$?      — For instance if $p$ is prime (which is not coprime with $0$) then $$\mathbb{Z}[i]/(p) \cong \mathbb{F}_p[X]/(X^2+1) \cong \begin{cases} \mathbb{F}_{p^2} &amp;\text{if } p \equiv 3 \pmod 4\\ \mathbb{F}_{p} \times \mathbb{F}_{p} &amp;\text{if } p \equiv 1 \pmod 4 \end{cases}$$ (because $-1$ is a square mod $p$ iff $(-1)^{(p-1)/2}=1$).  — More generally, if $n=p_1^{r_1} \cdots p_m^{r_m} \in \Bbb N$, then  each pair of integers $p_j^{r_j}$ are coprime, so that by CRT we get $$\mathbb{Z}[i]/(n) \cong \mathbb{Z}[i]/(p_1^{r_1}) \times \cdots \times \mathbb{Z}[i]/(p_m^{r_m})$$  I was not sure how to find the structure of $\mathbb{Z}[i]/(p^{r}) \cong (\Bbb Z/p^r \Bbb Z)[X] \,/\, (X^2+1)$ when $p$ is prime and $r&gt;1$.  — Even more generally, in order to determine the structure of $\mathbb{Z}[i]/(a+bi)$ with $a+bi=d(x+iy)$ and $(x,y)=1$, we could try to use the CRT, provided that $d$ is coprime with $x+iy$ in $\Bbb Z[i]$. But this is not always true: for $d=13$ and $x+iy=2+3i$, we can't find Gauss integers $u$ and $v$ such that $du + (x+iy)v=1$, because this would mean that $(2+3i)[(2-3i)u+v]=1$, i.e. $2+3i$ is a unit in $\Bbb Z[i]$ which is not because its norm is $13 \neq ±1$.  — I was not able to go further. I recall that my general question is to known what $\mathbb{Z}[i]/(a+bi)$ is isomorphic to, when $a$ and $b$ are integers which are not coprime (for instance $a=p^r,b=0$ or $d=(a,b) = a^2+b^2&gt;1$).  Thank you for your help! Do I need internet/3G to use GPS? I have a Samsung Galaxy S2 that has some problems with the GPS: If I use internet my navigation works but when I turn the 3G off, GPS does not work.  Do I need internet/3G to use GPS? Built-in unit strings recognized by Quantity? I can write  VAR=$VAR1 VAR=${VAR1} VAR="$VAR1" VAR="${VAR1}"   the end result to me all seems about the same. Why should I write one or the other? are any of these not portable/POSIX? I quite enjoy paying attention to how I design my documents and presentations. I usually spend hours thinking over and designing my slides for a workshop or presentation, so that they are aesthetically pleasing and as intuitive as possible. Likewise I recently started revising my CV I figured and I wanted to make it stand out a bit more. (Just to make it clear I don't mean making a clown of a document but just better use of colors, contrast and design elements.)  I have long wondered whether or not this is something that can backfire, since most documents in academic context are extremely plain, at least in my experience. It's very common to see the default Powerpoint slides (white bg, black arial text) or .   My question is as follows: is putting time and effort into design of academic documents something that can backfire? Will I risk being prejudged with first impressions such as "well he put much effort in the presentation his documents, perhaps because the content is sub-par"?     I realize that the question might be somewhat subjective from person to person but I encourage everyone to consider it in terms of this SE blog entry: . What is the rule for omitting/including the definite article in the following sentences:     I used to play piano.   I used to play the piano.   I would pick the first sentence, but I've heard people say the second sentence even when they are not referring to a specific piano. Which one is correct? Okay setting the scene. I have three tables, (Table1, Table2 and DataTable) and I want to insert into Table1 and Table2 using DataTable as source. So for every row in DataTable I want a row in Table1 and Table2, and Table2 needs to have the inserted id (PK) from Table1...  If I were to do this...  INSERT INTO Table1 SELECT A, B, C FROM MyTable INSERT INTO Table2 SELECT IDENTITY_INSERT(), D, E, F FROM MyTable   I'd get the ID of the last inserted record into Table1.  Is a CURSOR or WHILE loop the only ways to do this? I've connected a USB to Serial bridge dongle, and running dmesg | grep tty outputted the following:  [    0.000000] console [tty0] enabled [603199.380677] usb 2-2: cp210x converter now attached to ttyUSB0   So now I'm trying to write to it as per  by running the following:  cat hello.txt &gt; /dev/ttyUSB0   and   sudo cat hello.txt &gt; /dev/ttyUSB0   But both result in the following error:  bash: /dev/ttyUSB0: Permission denied   What am I doing wrong?  Thanks Can I acquire a Windows 8.1 licence by free upgrade from 7 to 10, then downgrading So, I have Windows 7 pro, and I'm looking to upgrade so that I can use some of the apps that are only available in Windows 8+. I considered just taking advantage of the free Windows 10 upgrade, and actually installed it at one point, but then I found out that there's forced automatic updates and decided that wasn't for me(I'm a bit paranoid after Android 4.4.4 killed Flash Player without telling me. :P). So, I did some research.   From what I read, the Windows 10 upgrade "consumes" your Windows 7 license into a Windows 10 license, and so even if you revert back after upgrading then you can still change your mind later and jump back to Windows 10. () I also read that Windows 10 has "downgrade" rights to Windows 8 and Windows 7, which to my understanding meant that if you have a Windows 10 key, you can also use it to install Windows 8 or 7 if you want. ()  My plan for the process is re-upgrade to 10, pull the license key, downgrade to 8 and use the license key, then upgrade to 8.1 via Windows Store, and end up on 8.1. So my questions are: Is this possible?  Will it wipe all my data in the process?(not a dealbreaker if it does, but I'd need to know that so I can back it up) Do I need to re-upgrade to 10 in order to downgrade to 8?  Any input is definitely appreciated. Thanks! How do I remove duplicate "Open With" context menu items in Finder.app? For whatever reason, the "Open With" context menu in Finder is always listing every app four times (exactly).     I've read about (and tried) rebuilding the Launch Services database to restore the context menu to its initial state, but nothing seems to work. I've also used Onyx.app to clear user and system caches, but I'm still left with four repetitions of each app under the "Open With" service. I have restarted the system a few times just to be sure it was not some temporary corruption of the list.  I'm running OS X 10.6.6 on a 2011 17" MacBook Pro with a fresh install of OS X (i.e. I didn't use Migration Assistant.app or a Time Machine backup). I did, however, sync all my preferences and other files using MobileMe like I always do, so I imagine it's feasible that a preference file somewhere is causing problems? I didn't notice whether this was happening before or after my MobileMe sync.  How can I fix this so one app shows? Free visual similarity image for a local hard drive search? I'm looking for a desktop search engine for finding images on my local hard drive with high visual similarity. I've only found paid for ones so far. It could be Linux or windows one, either would be helpful.  A Google desktop plug-in would be nice! Why is the set of commutators not a subgroup? My table is:     id home datetime player resource     1 10 04/03/2009 john 399   2 11 04/03/2009 juliet 244   5 12 04/03/2009 borat 555   3 10 03/03/2009 john 300   4 11 03/03/2009 juliet 200   6 12 03/03/2009 borat 500   7 13 24/12/2008 borat 600   8 13 01/01/2009 borat 700     I need to select each distinct home holding the maximum value of datetime. Result would be:     id home datetime player resource     1 10 04/03/2009 john 399   2 11 04/03/2009 juliet 244   5 12 04/03/2009 borat 555   8 13 01/01/2009 borat 700     I have tried: -- 1 ..by the MySQL manual:   SELECT DISTINCT   home,   id,   datetime AS dt,   player,   resource FROM topten t1 WHERE datetime = (SELECT   MAX(t2.datetime) FROM topten t2 GROUP BY home) GROUP BY datetime ORDER BY datetime DESC  Doesn't work. Result-set has 130 rows although database holds 187. Result includes some duplicates of home. -- 2 ..join  SELECT   s1.id,   s1.home,   s1.datetime,   s1.player,   s1.resource FROM topten s1 JOIN (SELECT   id,   MAX(datetime) AS dt FROM topten GROUP BY id) AS s2   ON s1.id = s2.id ORDER BY datetime   Nope. Gives all the records. -- 3 ..something exotic:   With various results.