Given $x$, $\sin(x) \in \mathbb{Q}$, where $x$ is in degrees, we want to find all $x$ in the range $(0,90)$.  One trivial solution is $x=30$.
I am using extensively trigonometric functions when an angle is given in degrees. Some of these functions like sine or cosine have rational values, for example, the well   known example is that $\cos(\theta) =0.6 $ and $\sin(\theta) =0.8 $.       However besides the case of multiplicity of $90^\circ$ it seems there are no rational numbers $\theta$  with simultaneously rational  values of  sine and cosine.      Is it possible somehow to prove that for rational values of an angle given in degrees there are  no values simultaneously rational of sine and cosine functions, beside obvious case of multiplicities of $90^\circ$?
I am looking for QGIS function to extract (or detect) edge from a raster, which was referred in here,  as answer by @MappaGnosis.  The name could have changed since then, or it may have been integrated into other tool... I cannot find it in my QGIS Processing Toolbox (2.14.10 and 2.18.2).  There are similar tools such as GRASS "i.zc Zero-crossing" and SAGA "Gaussian filter", but I am hoping to find this QGIS geoalgorithm. identity element of $(D_n ,\gcd)$ Let $x*y=\gcd(x,y)$ on $D_n=\{x\in \mathbb N:x |n\}$. I have to prove that $(D_n,*)$ is a commutative monoid and I have a problem when finding the identity element.  $x*e=x$  So $gcd(x,e)=x \to e=xk,k\in \mathbb N^*$ and $e \leq n$  A good $e$ is $0$ but it's not in $D_n$ A (not necessarily unitary) $R$ module $M$ splits into $B\oplus C$ where $B$ is unitary and $RC=\{0\}$. I have attempted a proof of this, but upon looking at my notes, I feel it might be incorrect: it is noticeably simpler than the one in my notes.  Proposition: any permutation in $S_n$ can be written as a product of disjoint cycles.  Proof (attempt): $s\in S_n$. Let $\langle s\rangle$ act on $X=\{1,\cdots,n\}$. Define $x\sim x'\iff x'\in\text{Orb}(x)$ for $x,x'\in X:$ this is clearly an equivalent relation, hence the orbits partition $X$. Each orbit corresponds a cycle in $s$, so we are done.  Does this hold, or have I missed something vital? Is there any way to favorite/bookmark an answer? Lots of questions are very generic, &quot;share an experience&quot; kinds of questions. In these cases, I am quite often interested in specific answers that may not have gotten that many upvotes. These questions also tend to elicit lots of responses, so going back to find the answers I'm interested in somewhere in the middle of a sea of responses is a pain. Is there a way to favorite or bookmark these answers? Why didn't Spock die in The Search for Spock? Data Not Backed Up, Partition Type: FFFFFFFF-FFFF-FFFF-FFFF-FFFFFFFFFFFF I was trying to install Linux on my MacBook for class when I bricked my computer, which now in an unbootable state.   I made my partition too small, so I was going to delete and create a larger one. However, Disk Utility did not allow me to delete the small partition. I followed the instructions from   However, not only was I still unable to delete it, but upon restarting I realized my primary partition (located on the same disk) changed into FFFFFFFF-FFFF-FFFF-FFFF-FFFFFFFFFFFF, and I had no options to boot.  (I am sure I performed the command on the correct disk and partition)  I saw some similar threads, but my main bootable drive is the one that got corrupt.   I don't want to lose any data. I'm not the most terminal savvy, so ANY help is appreciated, and I will kindly provide any additional information!  (Disk 0, partition 2 is my main partition, and that only one I care about)  Below are the results of  $ diskutil list and $ gpt -r show disk0.        UPDATE: :  I have followed @klanomath 's instructions. At the last step I received the following error:  Checking prerequisites Checking the partition list Checking for an EFI system partition Checking the EFI system partition's size Checking the EFI system partition's file system Problems were found with the partition map which might prevent booting Error: -69766: The partition map needs to be repaired because there's a problem with the EFI system partition's file system Underlying error: 8: POSIX reports: Exec format error    FINAL UPDATE: SUCCESS!  I just wanted thank @klanomath for helping me through this entire process. He spent 11 hours TeamViewing with me until not only my files were accessible, but until I could access my computer as could before this whole debacle (and this was after 5 hours of email correspondence).   To any future reader with this problem: PLEASE don't give up, it can be fixed! Hide some fields in the NewForm.aspx for a custom list Assume a p x n data matrix  (e.g. p independent samples of n variables). The n variables are correlated according to a correlation matrix $\rho$. Now if you consider the data matrix as some state t (t=time)  and based on this state I would like to calculate the next timestep (t+1). Therefore the correlation matrix would be allowed to have values &lt; 1 in the diagonal, which means that the diagonal elements would indicate the influence from each variable $x_{p,i}[t]$ on $x_{p,i}[t+1]$. So I would be interested to find a function f like this: $x_{p,i}[t+1] = f(x_{p,i}[t],\rho)$ so that the p x n data matrix  consisting of the $x_{p,i}[t+1]$ replicates the $\rho_{ij}$ (except the diagonal element). f should be something like this: $x_{i,j}[t+1] = \sum_{(k=1\ldots n)}\rho_{i,k} \cdot x_{kj}[t] + e_{i,j}.$ This means specifically, I not only want to calculate a data matrix, that replicates the correlation matrix as such, but also to let each observation be depending on its own state and linear dependent on the previous state of all the intercorrelated variables of its own state. For ease one can assume that the p variables are distributed normal and $e_{ij}$ are independent and distributed with mean$(e_{i,j}) = 0 $. Then for a numerical simulation it would be interesting  to find a formula to calculate the variance/standard deviation of each of the j variables (sd($e_{[.,j]} = ?).$ Thank you for any help! I ask a lot of question on SO, so this recommendation/idea comes from that experience. I'm not really sure how this would fit in with the current structure of SE.   Oftentimes after asking a question, maybe after an hour or so, I end up finding the answer I needed somewhere. And it comes time for me to answer the most difficult question of all: do I delete this question, or do I keep it in case someone answers and others may benefit from it. The problem is, if I don't delete it and no one answers, my acceptance rate goes down. So usually I'll just delete it.  However, sometimes I fear that someone is typing a really really long answer to my question since itäs only been an hour, and so I imagine the nightmare that person experiences when they find out their essay will be gone to waste. Overall, this never-to-be-seen-again answer could have benefitted many people had the question remained.  Since SE has gone "live-happy" lately, where answers and comments refresh live, I thought it might be interesting if the original asker of the question (or anyone else for that matter) could see how many people are currently responding to the question (just count, not necessary to display their names). This way, if I know someone is replying, I won't delete the question, and others can benefit from it. Observe the following results for searching the text "dictionary" on Google:    Notice that there is a custom search box at the bottom of the listing for dictionary.reference.com. Is there any documentation floating around that explains the criterion by which Google decides to display such a search box? #1115 - Unknown character set: 'utf8mb4' When I Import Mysql from local server to web ser it is showng me error 1115 - Unknown character set: 'utf8mb4' in Creating wordpress wp_comment table CREATE TABLE IF NOT EXISTS  `wp_commentmeta` ( // // ) ENGINE = INNODB DEFAULT CHARSET = utf8mb4 AUTO_INCREMENT =1;  #1115 - Unknown character set: 'utf8mb4'  My mysql version is Version information: 4.0.4, latest stable version: 4.4.10 (local version)  Version information: 4.0.10.7, latest stable version: 4.4.10( server version)  Please help me to solve out the error Getting CalloutException when testing Batchable class with Callouts and HttpCalloutMock How do I "parent" objects? I'm looking for a way to have two separate objects be connected, so that when I move the first object the second one will move also. The term parent is used, but how do I parent a object? My non-native English speaking friend just asked me: "Data is..." or "Data are..."?  I said both but that's because I've been desensitized from reading/writing both (especially from writing code and adding quick comments).  My question: Is it acceptable to utilize either for a university paper? Or is one safer than the other (when confronted with stickler professors)?    Related questions: Are there sets $A$ and $B$ such that $A \in B$ and $B \in A$? I think that the question is clear from the title:     Are there sets $A$ and $B$ such that $A \in B$ and $B \in A$?   My feeling is that the definition of any two such sets will be circular (we can't define $B$ till we define $A$ and we can't define $A$ till we define $B$), and hence will be excluded by some axiom.  On the other hand, I can't see an inherent contradiction, though I wouldn't be surprised if some form of Russell's paradox pops up, pushed behind one more set of braces. Show reference label when it is undefined When a reference is undefined, LaTeX shows simply [??] which doesn't give much information. I would like it to show the label instead. This is useful when more collaborators work on the same preprint and the references are yet to be done.  I'm using neither biblatex nor natbib. I do use bibtex but it is irrelevant for this question.  MWE:  \documentclass{article}  \begin{document}  This: \cite{undef} should look like: [{\bfseries ?? undef}]  \end{document} 14 months ago, an  supplement was published that discussed how the ranger described in the Player's Handbook was under powered. These rules reimagined the ranger's abilities, and suggested that an official version would be released later.     Finally, we come to implementation. If this iteration of the ranger, or a future revision of it, grades high enough, our plan is to present it as a revised ranger in a future D&amp;D sourcebook.   Does that Sourcebook exist yet? If not, have the devs given an update or told us their intentions? How to shop for mortgage rates ? I have already been prequalified with 3.8% (but I am sure I can do better :)). Now, I am about to make an offer on a house. So, once the seller accepts my offer, how do I rate shop ?    Do I need to do the rate shopping before making an offer, or after the seller accepts my offer ?  When the seller accepts my offer, then I will go to a banker and submit a full application, so will it be legally binding for me to go with the approved mortgage application ?  Or, once the seller accepts my offer, may I simultaneously apply for a mortgage with 3-4 bankers, and then choose the lowest rate ?     I asked my realtor, but she recommends to go with just one banker (her friend), and not to do any rate shopping. Is it the right way ?   (I live in a state of Arkansas, so I am thinking of going with the local bankers) I'm trying to find an alternative to the standard tombstone qed symbol for more informal papers. I initially thought of an empty square with Pub written inside, but what would actually be better would be a small beer mug symbol. (much like the one found in this set of icons )  Unfortunately there doesn't seem to such a symbol in the comprehensive latex symbol list (although many funny symbols can be found there). Has anyone thought about this? Is there any quick-and-dirty solution?