When we define trigonometric ratios using a right triangle then it denotes the ratio of its side but what does a trigonometric function of angle greater than 90° denotes in a unit circle ?
The trigonometric ratios of an angle greater than $90^\circ$ are equal to the supplementary angle's ratios.  I'm just clarifying this, but the ratios don't actually exist for angles greater than $90^\circ$ right? (since by definition these ratios are of a right angle triangle). Is it just mathematical custom to assume the ratios of an angle greater than $90^\circ$ to be equivalent to its supplementary angle's ratios? For convenience?
If I solve someones problem instead of answering their question, should I post a comment instead? White regions in DensityPlot — is this a bug with ColorFunctionScaling? Electron shells in atoms: What causes them to exist as they do? I have seen similar posts, but I haven't seen what seems to be a clear and direct answer.  Why do only a certain number of electrons occupy each shell? Why are the shells arranged in certain distances from the nucleus? Why don't electrons just collapse into the nucleus or fly away?  It seems there are lots of equations and theories that describe HOW electrons behave (pauli exclusion principle), predictions about WHERE they may be located (Schrödinger equation, uncertainty principle), etc. But hard to find the WHY and/or causality behind these descriptive properties.  What is it about the nucleus and the electrons that causes them to attract/repel in the form of these shells at regular intervals and numbers of electrons per shell?  Please be patient with me, new to this forum and just an amateur fan of physics. Does there exist a set L of pairwise disjoint lines, each line in an axis direction, in $\mathbb{R}^3$ such that their union is equal equal to $\{ x : x \text{ has at least one irrational coordinate}\}$? I am on Ubuntu and I did this command:  $ uname -a Linux slabrams-desktop 2.6.32-29-generic #58-Ubuntu SMP Fri Feb 11 19:00:09 UTC 2011 i686 GNU/Linux   Does it mean I am on 32bit or 64 bit processor?  The reason I am trying to figure this out is that I was getting errors which looked like this:  cannot execute binary file   and from Googling, I thought it was a processor issue.  Any ideas? Yield return from a try/catch block numbers of pages don't appear on pdf printed I am encountering two definitions of the special orthogonal lie algebra, and I would like to know if they are equivalent, and if there are advantages to working with one over the other.  If we begin with an $n$-dimensional vector space $V$ over a field $k$ and a chosen basis, we can define a bilinear form on $V$ by a matrix $S\in M_n(k)$, ie, let $\langle v,w\rangle=v^tSw$ for all $v,w\in V$.  Now $g\in GL_n(k)$ preserves the form ($\langle g(v),g(w)\rangle=\langle v,w\rangle$) if and only if $g^tSg=S$, so all such $g$ form a linear algebraic group $G$.  The tangent space at the identity of $G$ will be contained in that of $GL_n(k)$, so $T_eG\subset T_eGL_n(k)=M_n(k)$, and in fact, $T_eG=\{B\in M_n(k)\mid B^tS+SB=0\}$.  $T_eG$ becomes a lie algebra, $Lie(G)$, if we define the bracket to be the commutator of two matrices.  Now, if $S=I_n$, it follows that $G=O_n(k)$ is the orthogonal group of matrices satisfying $g^tg=I_n$, and $Lie(G)=\mathfrak{so}_n$ is the lie algebra of antisymmetric matrices.  In Humphrey's Introduction to Lie Algebras and Representation Theory, he defines $\mathfrak{so}_n$ to be all matrices $B$ satisyfing $B^tS+SB=0$, where $$ S=\begin{pmatrix} 1&amp;0&amp;0\\ 0&amp;O&amp;I_l\\ 0&amp;I_l&amp;O \end{pmatrix} \hspace{.5in}\text{or}\hspace{.5in} S=\begin{pmatrix} O&amp;I_l\\ I_l&amp;O \end{pmatrix} $$ depending on the parity of $n$.  The matrices obtained in this way are not antisymmetric, nor is the group $G$ preserving the form defined by $S$ the orthogonal group $O_n(k)$.  Are the two groups obtained from considering different $S$ isomorphic?  Are the two lie algebras isomorphic?  If so, why would one prefer one form to the other? Does separation of variables in PDEs give a general solution? When a partial differential equation is solved using the separation of variables method, is the produced solution the most general one that satisfies the equation or have we lost some forms of the solution because of the assumption that it is in the form of separated variables? How to install with Wubi when it says "Could not retrieve the required disk image files"? I tried to install Ubuntu 12.10 with the Wubi installer while running Windows, because my PC does not accept my BootCD as a bootable medium.  But Wubi showed me this error:  Ubuntu-Installationsprogramm (X) Ein Fehler ist aufgetreten: Could not retrieve the required disk image files Weiter Informationen finden Sie in der Protokolldatei: c:\users\username\appdata\local\temp\wubi-12.10-rev273.log   What can I do, because I want to get rid of my Windows 7 and use the way better Ubuntu 12.10?  Thanks for your huge answer, now my next problem after trying to install wubi with the .iso is, that the same error shows up again and i am not shure which part of the errorreport you need.  Can you tell me? Because i'm not that good at linux. How do I get past the "do not enter" traffic sign that stops me from installing "Sierra" instead of "High Sierra"?  I booted into rescue mode and deleted the original "High Sierra" installation, but booting from the USB stick with the "Sierra" installer on it still gives me the "no". Which piece of equipment is holding me back? I am primarily an amateur photographer but I do take small jobs from time to time.  I'm looking to upgrade my equipment but I'm not sure if I should be spending money on my body, glass, or both.  I have two common issues that I'd like to improve.   I shoot a lot at ice rinks and find that even in servo mode I get a lot of shots that are slightly out of focus.  I try to keep the AF target on face masks but many shots will come out focused an inch or two off (ears, hands, etc).  I'm shooting F/1.8-F/2.8 to get action stopping shutter speeds so you can really tell when the focus is even a hair off.  I don't know if this is a limitation of my camera, lens, or the human operator. Most of my paid work is indoors shooting candids where a flash is inappropriate.  I feel like I need to use a higher F-number to get more subject in focus but I usually can't do it without upping to ISO 3200 unless I'm using a flash.  By the time I clean up the high ISO noise in Camera Raw I end up with an image lacking detail.  So far this hasn't been a big problem because the photos are typically used in magazine prints only an inch or two wide but it's still driving me nuts.   Here is the equipment currently in my bag:   Canon Rebel T2i Canon EF-S 18-55mm f/3.5-5.6 II (no IS)   Collecting dust, came with my old Rebel XTi.  Canon EF-S 18-135mm f/3.5-5.6 IS   Purchased with my T2i.  This is my workhorse lens but I'm disappointed with the soft images it produces at higher focal lengths.  Stopping down makes the images acceptable but that's not always an option.  My eyes want more contrast and color but considering how inexpensive this lens is I can't really complain.  Canon EF 70-300mm f/4-5.6 IS USM   This was purchased with an older camera and doesn't get used much anymore.  The images are too soft beyond 200mm and the AF seems sluggish.  Canon EF 50mm f/1.4 USM   I am extremely happy with the images I get from this lens.  I use it indoors all the time and also paired with extension tubes for some macro work.  Some of my best ice hockey images have been from this lens even though I didn't have sports in mind for this lens.    My initial plan was to upgrade to a 60D or 7D body but after doing some reading it occurred to me that I might invest in better glass first.  I'm unsure of what will help improve focusing, image quality, high ISO shooting, etc.  Maybe it's a new camera, new glass, photography lessons, or a little of everything?  I'm looking to keep my next round of purchases under $3,000 and am willing to trade in some of my used equipment to offset costs.  Would anyone care to make a recommendation?  I'm looking at the EOS 60D and 7D bodies.  Not sure if the 7D is overkill for my needs.  I'm also looking at the EF 24-70mm f/2.8L USM lens to be my new workhorse but I'm concerned that it may not be wide enough on an APS-C body and the lack of IS might be an issue.  There's also the EF 70-200mm f/2.8L IS II USM but I'm thinking it wouldn't be wise to spend all my budget on that lens right now (or maybe not?). Initially, my question was: is "focussed" or "focused" the correct past tense of "focus", but since this applies to a lot of words, I would like to generalize and ask: is there supposed to be a rule when to double the consonant? What, precisely, is a confidence interval? Can a fundamentally deceptive character work? This question is in the spirit of another . I would like to explore a style of role-playing in DnD (specifically 3.5e) that may be slightly different from how characters usually work. The question aims to answer if and how such a character can work and be played for everyone's amusement. I am still relatively new to the game, so please forgive me if this question is naive. Probably people have tried before and discarded the idea.  Character concept The character is supposed to be deceptive, on both the character and the player level. He will thrive to achieve some higher goal that is different to the party's, but not in an open way. He will play along and work for the party's interest, but wait for his moment (ideally towards the end of the campaign) to strike. As such an evil religious fanatic would be ideal, who works towards achieving the goals of his deity by deceiving his party members, gaining their trust and acquiring resources together with them, but when given the choice would take the powerful magical item and bugger off with it. Motivation and Concerns The main motivation is to bring some diversity to the usual scheme. People tend to play funny characters and there is usually (at least in our games) happy peace in the party with the only dispute being about who gets the loot. I feel a character like the one suggested would make things more interesting since people can't take their party members for granted anymore. The concern is of course that it might not be very fun for the party to have their plans destroyed by one member of the party playing an &quot;extra cool&quot; character. I think done well, it could still be fun for the other players though, which is why the rest of the question will be about how to do it. How? First of all the DM should probably know, maybe he can even build it into his campaign. The exact story details will then depend on the exact setting, but I am sure together with the DM something could be worked out. I am mainly interested in if it can work and how it could be fun. $ \lim_{x\to o} \frac{(1+x)^{\frac1x}-e+\frac{ex}{2}}{ex^2} $ $$ \lim_{x\to o} \frac{(1+x)^{\frac1x}-e+\frac{ex}{2}}{ex^2} $$   (can this be duplicate? I think not) I tried it using many methods  $1.$ Solve this conventionally taking $1^\infty$ form in no luck  $2.$ Did this, expand $ {(1+x)^{\frac1x}}$ using binomial theorem got $\frac13$ then grouped coefficients of $x^0$ and it cancelled with $e$ then took coefficient of $x$ cancelled with $\frac{ex}{2}$ and so on   very messy right ? at last I got $\frac13$ but that's not the expected answer!  I must have went wrong somewhere can anyone help me with this. what is the difference between "yet" and "still"?   When we can use "still"? and when we can use "yet"?   Are they synonyms or not?  "The plan could yet succeed." Can I use "still" in that sentence instead of "yet"?  "I still don't understand." Can I use "yet" in that sentence too? Doesn't the end of Looper create a time paradox? I guess this is more a question about the nature of time travel more than anything else.  At the end of the movie,      Joe shoots himself in a field and changes the course of history.   Does this create a paradox as it changes the history that led up to that point?      If his future self had never existed he'd never have gone on the mission he did and thus he never would have ended up in the field.   Once he did what he did, he could no longer have done it, and so theoretically, time and space should have just folded in on itself and popped out of existence.  Is there something I'm missing that would explain how the movie could have ended where it did without the events at the beginning to lead it there? Use the definition of derivative to prove $\lim_{x\to 0}\ln(x+1)/x =1$ How would I use the definition of derivative to prove  $$\lim_{x\to 0} \frac{\ln(1+x)}{x} = 1$$  I got to $$\frac{\frac{\ln(1+x+h)}{(x+h)} - \frac{\ln(1+x)}{x}}{h}$$ but have no idea where to go from here.   On another site I found someones answer where they stated the following: $$ \lim_{x\to 0} \frac{\ln(1+x) - \ln(1+0)}{x-0}  = [\ln(1+x)]'\rvert_x = 0 $$ but I am unsure why the $x$ in the $x-0$ is removed. Can someone please explain? How to set Command Block(s) to test for a particular time and output a message?