I am not able to grasp the logic behind how $\cos(\frac \pi2+h) = -\sin(h)$. I was able to find an explanation on Reddit but it is not clear. Can anybody elaborate in a better way?  Here is the link to the explanation -
How do I go about proving this? I know one method is:  $\eqalign{    \cos (90^\circ + \theta ) &amp;\equiv \cos90^\circ \cos\theta  - \sin90^\circ \sin\theta   \cr    &amp;  \equiv (0)(\cos\theta ) - (1)(\sin \theta )  \cr    &amp;  \equiv  - \sin \theta  \cr} $  I'd appreciate others, particularly ones that will allow me to visualize this identity on the unit circle.  Thanks alot.
How make f[{x,y}] evaluate as f[x,y]? Often confusion arises when the  spell is used, an  is worn, or the 's anti-magic ray is employed, whether a given character's ability will still function.   The spell does give you some guidelines on how to adjudicate the effects of the loss of magic but says very little about how to determine what is affected.  In short, how do I tell if an ability is magical? Considering phrases of the form "I left [the] X", what causes some words to need a "the" before them, while it sounds awkward with others?  Needs "the":   I left the office I left the bank I left the house I left the courthouse   Awkward with "the":   I left work I left social media I left New York   Either way works:   I left [the] school I left [the] church Here is a question from my number theory class.  Prove  $$(p-k)!(k-1)!\equiv (-1)^k \text{ mod p} $$ Help please! I can't use the terminal while gedit command is running I'm trying to identify a novel about an alien race coming to Earth. They are rabbit-like.    I thought it was Niven or Pournell but I cannot remember the title.  It was Quotzal or something like that. Apollonius circle, its radius and center I have a graph. This is my code.    \documentclass[10pt]{article}     \usepackage{pgf,tikz}     \usetikzlibrary{arrows,patterns}     \pagestyle{empty}      \definecolor{ttqqcc}{rgb}{0.2,0,0.8}     \definecolor{ffqqtt}{rgb}{1,0,0.2}     \definecolor{ttqqff}{rgb}{0.2,0,1}     \definecolor{uququq}{rgb}{0.25,0.25,0.25}      \begin{document}      \begin{tikzpicture}[line cap=round,line join=round,&gt;=triangle 45,x=1.0cm,y=1.0cm]      \draw[-&gt;] (-3.63,0) -- (3.67,0);     \foreach \x in {-3,-2,-1,1,2,3}       \draw[shift={(\x,0)}] (0pt,2pt) -- (0pt,-2pt)          node[below] {\footnotesize $\x$};     \draw[color=black] (3.48,0.07) node [anchor=south west] { $x$};      \draw[-&gt;] (0,-1.93) -- (0,5.57);     \foreach \y in {-1,1,2,3,4,5}       \draw[shift={(0,\y)}] (2pt,0pt) -- (-2pt,0pt)          node[left] {\footnotesize $\y$};     \draw[color=black] (0.06,5.19) node [anchor=west] { $y$};      \draw (0pt,-10pt) node[right] {\footnotesize $0$};      \clip(-3.63,-1.93) rectangle (3.67,5.57);     \draw[pattern color=ffqqtt,pattern=north east lines,fill opacity=0.1, smooth,samples=50,domain=0:1.0]      plot(\x,{\x^4-4*\x^2+3}) -- (1,0) -- (0,0) -- cycle;     \draw[pattern color=ttqqcc,pattern=crosshatch,fill opacity=0.1, smooth,samples=50,domain=1.0:1.7320508075688772]      plot(\x,{\x^4-4*\x^2+3}) -- (1.73,0) -- (1,0) -- cycle;      \begin{scope}[xscale=-1]     \draw[pattern color=ffqqtt,pattern=north east lines,fill opacity=0.1, smooth,samples=50,domain=0:1.0]        plot(\x,{\x^4-4*\x^2+3}) -- (1,0) -- (0,0) -- cycle;     \draw[pattern color=ttqqcc,pattern=crosshatch,fill opacity=0.1, smooth,samples=50,domain=1.0:1.7320508075688772]        plot(\x,{\x^4-4*\x^2+3}) -- (1.73,0) -- (1,0) -- cycle;     \end{scope}      \draw[smooth,samples=100,domain=-3.6342419080068153:3.665758091993186] plot(\x,{(\x)^4-4*(\x)^2+3});      \fill [color=uququq] (-1.73,0) circle (1.5pt);      \fill [color=uququq] (-1,0) circle (1.5pt);      \fill [color=uququq] (1,0) circle (1.5pt);      \fill [color=uququq] (1.73,0) circle (1.5pt);      \end{tikzpicture}     \end{document}   When I see the DVI output DVI, I have    And if I look at the PDF file, I have Where should the period be put when an entire sentence is quoted at the end of a sentence? As in the following:     A common proverb is: “A rolling stone gathers no moss.”        A common proverb is: “A rolling stone gathers no moss”.        A common proverb is: “A rolling stone gathers no moss.”.   If the proverb weren't a full sentence, I would immediately put a single period after the quote, at the very end.  But if it is a sentence, which period should be removed so that there is only one period? Two periods look absolutely dorky. I am reading Schwarz QFT and I reached the mass renormalization part. So he introduces, after renormalization, a physical mass, defined as the pole of the renormalized propagator, and a renormalized mass which can be the physical mass, but can also take other values, depending on the subtraction schemes used. Are these masses, other than the physical one, observable in any way experimentally, or are they just ways to do the math easier (using minimal subtraction instead of on-shell scheme, for example)?   Also, in the case of charge renormalization, the explanation was that due to the vacuum polarization, the closer you are to the charge, the more you see of it, so the charge of the particle increases with the momentum you are testing the particle with. However, I am not sure I understand, from a physical point of view, why do you need to renormalize the mass. Is this physical mass (defined as the pole of the renormalized propagator) the same no matter how close you get, or it also changes with energy? And if it changes, what is shielding it at big distances? Wrong place for hyperref link in scaled tikz picture I have a TikZ picture with a node containing a minipage with an itemize environment; each item is a hyperref to other sections of the document.   When scaling the image (for example with \begin{tikzpicture}[scale=.7, transform shape] I've a strange behavior: links in itemize are not in the right place: they present a somewhat vertical offset, as if they don't scale as the rest of image.   Here the MWE:  \documentclass[a4paper]{article} \usepackage{tikz} \usepackage[pdftex]{hyperref} \begin{document}  Here something interesting\label{text:interesting-one} \newpage  Here something interesting, too\label{text:interesting-two} \newpage  and here there are very very interesting stuff\label{text:interesting-three} \newpage  \newpage Now the problem: links rectangles have some offset, making links unusable in case of long itemize  \begin{figure}[htb] \begin{tikzpicture}[scale=.7, transform shape] \node at (0,0) (mynode) { \begin{minipage}{4cm} \begin{itemize} \item \hyperref[text:interesting-one]{First link} \item \hyperref[text:interesting-two]{Second link} \item \hyperref[text:interesting-three]{Third link} \item \hyperref[text:interesting-one]{First link again} \item \hyperref[text:interesting-two]{Second link again} \item \hyperref[text:interesting-three]{Third link again} \end{itemize} \end{minipage} }; \end{tikzpicture} \end{figure} \end{document}   And here a snapshot of the problem    Note: This is evident using the default settings of hyperref package, which draws the red rectangle around links, but it took me some time to found it when only the text is coloured, i.e. when using \usepackage[linkcolor=blue,colorlinks=true]{hyperref} If "clearly evident" is redundant, what word or phrase suggests being indisputable but not readily apparent? A  asks whether the modifier "clearly" changes the meaning of "evident". I would have said, prior to examining a few dictionaries, that evident did not necessarily mean readily apparent, but to my surprise it does.  This brings me to ask what word or expression would mean something can be taken as a true fact, but not through an easily constructed (not obvious or readily apparent) rationale?   For example, if Sherlock Holmes says something is true while everyone else is still sitting on their hands, then it is evident to Holmes, but not to the rest of the world. So, from everyone else's point of view, it is not readily apparent, and (by definition) apparently it is not evident, clear, or clearly evident at all, based on the dictionary definitions. But (let's stipulate) it is true and deducible from the observable evidence. What would we call this?  Update  An update is highlighted above. I can see from some of the answers and comments that I may have gone off my mark with the phrase "not through an easy-to-follow rationale”, and I have modified the question to correct this. I understand that there can be rationales, arguments, or proofs that, once explained, may or may not be easy to follow. What I intend is that the rationale to support the indisputable conclusion is not (initially) easy to recognize or construct (not readily apparent) in the first place. I apologize for the confusion. (The original can be seen in the change history.)  Update 2  Shortly after offering a bounty to this request, I recalled a single word that I now believe is a very good match to the parameters specified in this inquiry as it is stated above. That word has not yet been suggested (or approached) by any answer or comment (active or deleted) at the time of this edit. This request is changed to a single word request. (However, if you feel a single word fits well but not without some additional supporting words, please offer your thoughts in an answer.) While my curiosity for the term has already been met, the request and bounty remain as an open challenge. I will post my own answer once the bounty has expired and been awarded. Is SHA-256 secure as a CTR block cipher? Generate a 256-bit random nonce. XOR it with a 256-bit reusable symmetric key. This is x.  We represent numbers in simple binary instead of a counting function. 0 in dec = [256 zeros] in binary, 1 = [255 zeros]1, 23092348 = [241 zeros]1011000000101110001111100, etc.  For each nth block needed for the length of the plaintext, XOR x with n and hash it with SHA-256 to form b[n].  Let p[n] be the nth 256-bit block in the plaintext and c[n] be the nth block in the ciphertext. Then c[n] := b[n] XOR p[n].  Send the unencrypted nonce along with the ciphertext.  To decrypt c[n], XOR it with b[n] to retrieve p[n].  Is this secure? If so, then why do we need AES-256?  If this is not secure, then is there any mode of operation for which SHA-256 is secure as a block cipher? Specifically, it would need to be used in cipher feedback, output feedback, or any other mode that only uses the cipher in the encryption direction. I need a software, that emulates a printer, converts the printed documents to PDF/TIFF and send it via eMail to a specified address (for windows).  The virtual printer / PDF conversion is possible with a lot of PDF-converters. But I need a software, which is also able to directly mail me these PDFs. It's always the same mailaddress, and I have an SMTP server.  Any suggestions? Intersection and union of simply connected domain I have some basics concept that, in $\mathbb{C}$, a "simply connected domain $D$" is a region in $\mathbb{C}$ with no holes. I am not sure whether it has various formal definition , but the one used in my complex analysis class looks very complicated. I barely understand what it means :  $\textbf{Definition}$ Let $\gamma_0, \gamma_1$ be two continuous closed path in a domain $D \subseteq \mathbb{C}$ parameterized by $I = [0,1]$;that is, $\gamma_i(0) = \gamma_i(1)$ for all $i = 0, 1.$ We say $\gamma_0, \gamma_1$ are $\textbf{homotopic as closed paths on D}$ if there exists function $\delta : I \times I \rightarrow D$ such that   $1) \ \delta(t,0) = \gamma_0(t) \ \forall t \in I \ \ $    $2) \ \delta(t,1) = \gamma_1(t) \ \forall t \in I \ \ $     $3) \ \delta(0,u) = \delta(1,u) \ \forall u \in I.$  A continuous closed path is $\textbf{homotopic to a point}$ if it is homotopic to a constant path(as a closed path).  $\textbf{Definition}$ A region $D \subseteq \mathbb{C}$ is called $\textbf{simply conected}$ if every continuous closed path in $D$ is homotopic to a point in $D$.  $\textbf{Problem :}$ 1) Let $D_1, D_2$ be simply connected plane domains whose intersection is nonempty and connected. Prove that their intersection and union are both simply connected.  2) Let $P, Q$ be smooth functions on a domain $D \subseteq \mathbb{C}$, Find necessary and sufficient condition for the form $P dz + Q d\bar{z}$ to be closed. How can a piece of A4 paper be folded in exactly three equal parts? This is something that always annoys me when putting an A4 letter in a oblong envelope: one has to estimate where to put the creases when folding the letter. I normally start from the bottom and on eye estimate where to fold. Then I turn the letter over and fold bottom to top. Most of the time ending up with three different areas. There must be a way to do this exactly, without using any tools (ruler, etc.). Aligning equations within multline How to integrate $\int\sqrt{x^2+1}dx$ $$\int\sqrt{x^2+1}dx$$ I've been attempting this problem for days and I've made very little progress. I understand that I should substitute $x=\tan(x)$ and $dx=\sec^2(x)dx$ but beyond that I am very much lost. How to restore default group/user ownership of all files under /var? I accidentally changed /var owner/group to my username and then changed it back to root, but not all the /var folders' owners are root, so is there anyway to change back owner/group of files/folders to default state?  Or at least those files/folders that are created by packages? I would like to know how potential difference is created between two points?Is potential difference created by adding more electrons at one side?If that's the case how to add more electrons at one side?  I'm really looking to know what really constitutes potential difference?