I know sine represents a ratio in a triangle, but when you're doing things like $\sin(x)$, when you plug that into a calculator, the output is a completely separate number, without it knowing anything else about the triangle. That would mean that you don't need the actual triangle to calculate $\sin(x)$ otherwise you'd need to plug in the measurements to your calculator when solving for $\sin(x)$, right? My general question is what equation is sin? It obviously represents some complicated function/equation/whatever that the calculator does to actually find what $\sin(x)$ equals, right? What is the equation used? How could I go about plugging in variables to possibly calculate $\sin(x)$ by hand without using any actual triangles/measurements? If you have an answer, please explain like I'm an idiot who barely knows anything, I'm just some autistic freshman in highschool who's WAY too into mathematical stuff. I probably used incorrect math jargon in this question but still, I'm really curious
In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotangent of a single angle without using a calculator?  Sometimes I don't feel right when I can't do things out myself and let a machine do it when I can't.  Or, if you could redirect me to a place that explains how to do it, please do so.  My dad said there isn't, but I just had to make sure.  Thanks.
How to remove an element from a list by index How do I remove an element from a list by index in Python?  I found the list.remove method, but say I want to remove the last element, how do I do this? It seems like the default remove searches the list, but I don't want any search to be performed. What is the reasoning that leads one to postulate this second form for the relativistic particle action? The action for the free relativistic particle with worldline $\gamma : I\subset \mathbb{R}\to M$ is  $$S[\gamma]=-m\int d\lambda\sqrt{-\dot{\gamma}^a(\lambda)\dot{\gamma}_a(\lambda)}\tag{1} $$  Now, one may postulate a second action  $$S'[\gamma,\eta]=\frac{1}{2}\int d\lambda \bigg(\eta(\lambda)^{-1}\dot{\gamma}^a(\lambda)\dot{\gamma}_a(\lambda)-\eta(\lambda)m^2\bigg).\tag{2}$$  These are classically equivalent actions.  My question is: usually what we have is (1) and we have a problem both with the square root and with the massless limit. Given this, how could we think about postulating (2)? In other words, how can we reach (2)?  Usually some people answer this by saying: "it doesn't matter, actions are postulated, you postulate it, compute the equations, prove it works and its over".  Now I beg to differ. I want to know how could someone reason exactly in order to know what to postulate.   I am perfectly comfortable with computing the equations of motion. I want to know is how given (1) we would have the idea to postulate (2).  Is it some special case of some general procedure that deals with constraints? Is the ideal $I = \{f\mid f (0) = 0\}$ in the ring $C [0, 1]$ of all continuous real valued functions on $[0, 1]$ a maximal ideal? Is the ideal $I = \{f \mid f (0) = 0\}$ in the ring $C [0, 1]$ of all continuous real valued functions on the interval $[0, 1]$ a maximal ideal? script custom previews in a menu I want to know how to create a menu with previews for an add-on. Until now I know that this is related with the bpy.utils.preview. I'm making an add-on that appends objects from a .blend file, I know how to append objects but what I want to do is to create a menu with previews of the objects /(similar to the matcaps menu - see below), select them and then press an &quot;import button&quot; to append the object. Here's some documentation about the module:    Also, as this is an add-on I don't know if I should have a def register() and a def unregister() inside this .py file as well as in the __init__.py file. Capture stdout and stderr into different variables In what occasion could the word “precious” be taken disapprovingly, or sarcastically? I understand that the word “precious” in the following quote of Maureen Dowd’s article “Heart of Darkness” (NYT, March 20) literally means “2. valuable or important and not to be wasted” as defined in OALD:     Congressman Jones read an e-mail from a former boss of General   Allen’s, giving the congressman this unvarnished assessment:   “Attempting to find a true military and political answer to the   problems in Afghanistan would take decades. Would drain our nation of   precious resources, with the most precious being our sons and daughters. Simply put, the United States cannot solve the Afghan   problem, no matter how brave and determined our troops are.   But I didn’t know “precious” has another meaning as defined in OALD: 5. (disapproving, people and their behavior) very formal, exaggerated and not natural, and Readers English Dictionary: 2.(colloquial) terrible, almost worthless, until I was told by my respected forum mate a few days ago.   Being encouraged by his suggestion for me to “ask another question if I’m not sure of what he means by describing his input as “precious” is actually rather “precious” phrasing, I would like to ask:   What is an example of “precious” being used as “disapproving” or sarcastic way (as neither OALD nor CALD provides examples)? Is there an easy way to discern the instance “precious” being used as “disapproving” from the case being used in the sense of genuine “great value” as used in the above quote other than judging from the context? How do I know when to use "the" versus "a" versus "∅" as an article on a noun? With proper nouns, we don't use the except for river names, newspaper names, etc.  I want to know why we use the with White House. I mean, under which rule can we categorize it? What could other similar examples for that rule be? Is there a way to arrange windows within the same application in the taskbar of Windows 7?  Right now we can freely adjust each application's position in the taskbar, but within the app itself the windows can't be reordered, for example multiple windows of firefox or chrome. Permalinks: custom post type -> custom taxonomy -> post I am having trouble working with WordPress rewrite rules, and could use some help.  I have a custom post type called _shows_.  All shows have a single custom taxonomy category _show-category_. A _show_ will never have more than one _show-category_.  I would like my urls to route in this manner:  www.mysite.com/shows/  =&gt;  archive-shows.php  www.mysite.com/shows/%category%/ =&gt; taxonomy-show-category.php  www.mysite.com/shows/%category%/%postname%/ =&gt; single-shows.php   So as a real world example, lets say we have a _show-category_ "Foo" and a _show_ post titled "Bar" that has "Foo" as it's _show-category_. I would expect my WordPress app to look like this:  www.mysite.com/shows/foo/ =&gt; shows all posts under the foo category www.mysite.com/shows/foo/bar =&gt; shows the indivual post   I try to avoid plugins when possible, but am open to any solution. How to Improve Mathematical Thinking and General Problem Solving Skills? I'm a sophomore in university and seriously feel that I'm bad at solving mathematical and algorithmic problems (be it discrete math, calculus or just puzzles). I noticed that I'm only good at solving questions that are similar to the ones that have been taught to us.  Here's how I generally approach it:   What is the problem? What do I need to do here? Does it look like I've encountered this before?  Can I think of a smaller problem to solve instead?   If the answer is no to all the above then I sort of blank out. I stare at it and force my brain to run through a wide variety of stuff, almost like a brute force attempt of solving it. Obviously that leads me to nowhere everytime. I simply can't think "outside the box."   What can I do to improve my situation? deleting windows system folder I want to use an old drive as a secondary hard drive without reformatting. But this old hard drive contains my old Vista installation which takes up gigs of space.  I tried changing ownership and it works to some extent but I can not delete anything in Windows/System32 folder.  Is there an easy way to change permissions so I can delete everything?    The solution is to use Takeown as Wil suggested and use cacls to take full control of the files. Since cacls doesn't seem to be recursive, I fired up cygwin and ran this command to recursively change all the files:  for d in $(find directory-of-interest -type d);do   cacls $d'/*' /e /G username:F; done   where directory-of-interest is in my case /cygdrive/g/Windows and username is my username.  I had to manually run it on directories that had spaces in its name. But after all this, I can now delete the directory. Warning: it seems to me that cacls can potentially royally screw things up if improperly used. After turning census block data into a raster, how do I redistribute the density to reflect land cover types I plan on rasterizing (terminology?) census block data relating to density to a regular 30 m by 30 m grid for use with NLCD data.  I then hope to distribute the density of the original census block among each result set of grid cells within the original polygon according to the land cover type dictated by the NLCD dataset.  For example, let's imagine a string of city blocks with a park, all found within one census block.  If within this census block the average density is 1000 people/km^2, we ultimately would want the resulting grid cell containing the park to have a lower population density than the grid cells containing the surrounding apartments.  This information may be captured by the NLCD dataset, with the park being classified as '21' (open but developed land) versus 24 for apartment buildings which could be classified as '24' (high density development).  Is there a method, or set of methods, within ArcMap that would allow me to:   Redistribute the density based on the NLCD value Perform this redistribution only within the original census block polygon   I don't know if this is possible, but I figured I would ask.  Edit  Here's what I have been trying:  I converted the census block data to a raster, snapping it to the NLCD land cover data and ensuring the output grid cells were the same size.  Each cell contains the same info as the original census block (housing umbers, population, etc). I have been trying to figure out a way to edit the new grid so that if the NLCD grid cell contains a land cover type that doesn't support a business or housing (park, forest, farm, etc.) that it's housing and population values area set to 0.  My next move was going to be figuring out how to subtract the non-residential/non-business grid cells area from the area of the original census block, and force a recalculation of the density.  Converted the altered census block layer to grid cells again, with each having the new density value, and once again set the cells' housing, population, and density values that don't support a business or housing (park, forest, farm, etc.) to 0.  Kind of circular, and I'm not sure of all the steps (raster calculator for editing individual grid cells based on another raster's grid cell value?), but it seems to logically work out if I can get the parts to work.  On the other hand, @dof1985's answer seems to simplify the first part of my workflow (by directing me to the correct toolset), but I will still have to nullify the grid cells where parks, etc., do exits after the density has been recalculated for the census data. Stopping all automatic updates Windows 10 We've upgraded some machines to Windows 10 and realized there were some updates which updated as required. However, I realized there was no option available to stop the download similar to that on Windows 7 and 8.1. The only way I could stop the download was to stop the Windows Update service.  My question is does anyone know of a way to stop auto updates or is stopping the service is the only solution? Is the distribution of an Ito diffusion at time t absolutely continuous wrt Lebesgue measure? Official way to ask jQuery wait for all images to load before executing something In jQuery when you do this:  $(function() {    alert("DOM is loaded, but images not necessarily all loaded"); });   It waits for the DOM to load and executes your code. If all the images are not loaded then it still executes the code. This is obviously what we want if we're initializing any DOM stuff such as showing or hiding elements or attaching events.  Let's say though that I want some animation and I don't want it running until all the images are loaded. Is there an official way in jQuery to do this?  The best way I have is to use &lt;body onload="finished()"&gt;, but I don't really want to do that unless I have to.  Note: There is a  in Internet Explorer which actually does wait for all images to load before executing code inside $function() { }. So if you're using that platform you'll get the behavior I'm looking for instead of the correct behavior described above. How can I get sql string, generated by Linq2SQL ORM? What is the maximum convergent $x$ in the power tower $x^{x^{x^{x\cdots}}}$? In the power tower $x^{x^{x^{x\cdots}}}$ where there is an infinite stack of $x$'s, what is the maximum convergent number? I know the answer by playing with the form $x^y=y$ and using Mathematica, but I don't know how to solve this by hand. In PHP, the HEREDOC string declarations are really useful for outputting a block of html.  You can have it parse in variables just by prefixing them with $, but for more complicated syntax (like $var[2][3]), you have to put your expression inside {} braces.  In PHP 5, it is possible to actually make function calls within {} braces inside a HEREDOC string, but you have to go through a bit of work.  The function name itself has to be stored in a variable, and you have to call it like it is a dynamically-named function.  For example:  $fn = 'testfunction'; function testfunction() { return 'ok'; } $string = &lt;&lt;&lt; heredoc plain text and now a function: {$fn()} heredoc;   As you can see, this is a bit more messy than just:  $string = &lt;&lt;&lt; heredoc plain text and now a function: {testfunction()} heredoc;   There are other ways besides the first code example, such as breaking out of the HEREDOC to call the function, or reversing the issue and doing something like:  ?&gt; &lt;!-- directly output html and only breaking into php for the function --&gt; plain text and now a function: &lt;?PHP print testfunction(); ?&gt;   The latter has the disadvantage that the output is directly put into the output stream (unless I'm using output buffering), which might not be what I want.  So, the essence of my question is: is there a more elegant way to approach this?  Edit based on responses: It certainly does seem like some kind of template engine would make my life much easier, but it would require me basically invert my usual PHP style.  Not that that's a bad thing, but it explains my inertia..  I'm up for figuring out ways to make life easier though, so I'm looking into templates now. How do I fix a NoSuchMethodError? How can I keep multiple live/bootable ISO images on a single USB drive?