How does the unit circle work for trigonometric ratios of obtuse angles? I know that the x coordinate is cos(θ) and the y coordinate is sin(θ). But I understand these in context of only acute angles? I don't understand why the unit circle definition works for other than acute angles? Somebody please provide me some good intuition.
How does the unit circle work for trigonometric ratios of obtuse angles? I know that the x coordinate is $cos (\theta )$ and the y coordinate is $sin (\theta )$. But I understand these in context of only acute angles? I don't understand why the unit circle definition works for other than acute angles? Somebody please provide me some good intuition.
How can I invert a 'clip' selection within TikZ? I wish to draw an arc for a reflex angle on the outside of a triangle with coordinates (A), (B) and (C).  If I wanted to draw the arc inside the triangle I could use something like  \begin{scope} \path[clip] (A) -- (B) -- (C) -- cycle; \draw (B) circle (2mm);    % this is the little angle marker \end{scope}   Is there a straightforward way to exclude the part of the circle that's in the triangle, as in the pseudo-code below?  \begin{scope} \path[magicalinverseclipcommand] (A) -- (B) -- (C) -- cycle; \draw (B) circle (2mm);    % this is the little angle marker \end{scope}   EDIT: Although I included the code above as a motivating example of what an inverse clip would do, I would prefer a general solution that works systematically with any closed path, and not just in the particular example of a triangle with a circle clipped out of it. Looking at my question, I realise this isn't clear, and I will be happy with the best solution to my particular problem, but wonder if there is a less ad hoc solution than those listed so far. allow you to create code which defines the methods of classes that implement it. You cannot however add any code to those methods.    allow you to do the same thing, along with adding code to the method.   Now if you can achieve the same goal with abstract classes, why do we even need the concept of interfaces?   I've been told that it has to do with OO theory from C++ to Java, which is what PHP's OO stuff is based on. Is the concept useful in Java but not in PHP? Is it just a way to keep from having placeholders littered in the abstract class? Am I missing something? What is a tty, and how do I access a tty? I have the following enumeration:  public enum AuthenticationMethod {     FORMS = 1,     WINDOWSAUTHENTICATION = 2,     SINGLESIGNON = 3 }   The problem however is that I need the word "FORMS" when I ask for AuthenticationMethod.FORMS and not the id 1.  I have found the following solution for this problem ():  First I need to create a custom attribute called "StringValue":  public class StringValue : System.Attribute {     private readonly string _value;      public StringValue(string value)     {         _value = value;     }      public string Value     {         get { return _value; }     }  }   Then I can add this attribute to my enumerator:  public enum AuthenticationMethod {     [StringValue("FORMS")]     FORMS = 1,     [StringValue("WINDOWS")]     WINDOWSAUTHENTICATION = 2,     [StringValue("SSO")]     SINGLESIGNON = 3 }   And of course I need something to retrieve that StringValue:  public static class StringEnum {     public static string GetStringValue(Enum value)     {         string output = null;         Type type = value.GetType();          //Check first in our cached results...          //Look for our 'StringValueAttribute'           //in the field's custom attributes          FieldInfo fi = type.GetField(value.ToString());         StringValue[] attrs =            fi.GetCustomAttributes(typeof(StringValue),                                    false) as StringValue[];         if (attrs.Length &gt; 0)         {             output = attrs[0].Value;         }          return output;     } }   Good now I've got the tools to get a string value for an enumerator. I can then use it like this:  string valueOfAuthenticationMethod = StringEnum.GetStringValue(AuthenticationMethod.FORMS);   Okay now all of these work like a charm but I find it a whole lot of work. I was wondering if there is a better solution for this.  I also tried something with a dictionary and static properties but that wasn't better either. Explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$ Do you know an explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$? Using the axiom of choice, every vector space admits a norm but have you an explicit formula on  $\mathcal{C}^0(\mathbb{R},\mathbb{R})$?  A related question is: Can we proved that $\mathcal{C}^0(\mathbb{R},\mathbb{R})$ has a norm without the axiom of choice? What does an ampersand after this assignment operator mean? I have a doubt on something. I have a theorem that says that if $R$ is a ring, then $R/I$ is a domain iff $I$ is prime. Since $X^2-1$ is note prime, $\mathbb Z[X]/(X^2-1)$ is not a domain. But, $X^2-1=(X-1)(X+1)$, and since $X-1$ and $X+1$ are prime in $\mathbb Z[X]$, there are coprime, and thus, by the chinese remainder $$\mathbb Z[X]/(X^2-1)\cong \mathbb Z[X]/(X-1)\times \mathbb Z[X]/(X+1),$$ since $\mathbb Z[X]/(X-1)\times \mathbb Z[X]/(X+1)$ is a domain, then so is $\mathbb Z[X/(X^2-1)$. What is the mistakes here ? Why did Dumbledore 'put on' the Resurrection Stone (Ring)? Generating random results by weight in PHP? I know how to generate a random number in PHP but lets say I want a random number between 1-10 but I want more 3,4,5's then 8,9,10's. How is this possible? I would post what I have tried but honestly, I don't even know where to start. Give me a blank Comment box when I am Improving a Suggested Edit The suggested edit screen includes both the suggested edit, and the reason for the edit:        When you click to improve the edit, that comment is left as-is and hidden at the bottom of the post above the save edit button:        At the moment, when I Improve a Suggested Edit, the Comment text for that Edit is retained (though editable). It's unlikely that the comment would still apply (e.g. Suggester added a tag and Approver corrected a typo), so I would rather have the field blank.  Also, it's better to have a blank Comment than an incorrect one... many people don't bother changing the original Comment.  So that box should be blank like this instead: Choosing my name as an author when publishing a scientific paper, can I use my "unofficial" first name? I have one official first name which appears on my ID card. I have another first name which friends call me by. Is it possible to use my unofficial first name and official family name to publish scientific papers? How can I prove to someone that I am one of the authors? Does one need to adjust for multiple comparisons when using MANOVA? There are 10 dependent variables and 2 groups. When I wanted to compare 10 dependent variables between two groups, I performed MANOVA.  I wonder if there is a multiple comparison problem in that case. Should I have performed Bonferroni correction because of 10 dependent variables? Or is Bonferroni useless because MANOVA just does analysis once? Adding Custom Menu "Run Script" Button Causes Duplicate Is there a way to use the Blender Text Editor for adding a custom menu and not get a duplicate every time I click on "Run Script" to test the menu? For example, in the text editor, load the template like so:  Template -> Python -> UI Menu  Then notice every time you press "Run Script" you get another menu in your main menu bar. How can I update my menu without getting a duplicate?  I tried simply commenting out register() but then new updates don't show up. How to make a circle with grid on the outside ring I made something like this:    I did this with the Ellipse Tool and the Line Tool.  Now the circle is accurate, but the lines inside are very inaccurate. Is there another tool that does the same thing but with a lot more accuracy? "How to format" block mentions syntax highlighting even if it is not available on the site When you edit the body of a post, the "How to Format" widget is shown in the right sidebar. It mentions the syntax highlighting function, even on  like :    It's probably also a good idea to move the part about code fences down for non-technical sites (like ); other formatting options are more important there. Simplest or nicest proof that $1+x \le e^x$ The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or canonical proof? I would ideally like a proof which fits into a few lines, is accessible to students with limited calculus experience, and does not involve too much analysis of different cases. How to setup two completely separate partitions on one SSD I have a MacBook with 1TB SSD. I want to have two separate partitions with two independent macOS installations. I want to make sure that none of the partitions have any kind of possibility to interact with the other, I want them to be as separated as if they were physically separate. Is this achievable with, for example, FileVault for each of them? Edit: My question is neither &quot;how can I install two different macOS versions on two different partitions on one SDD&quot;, neither &quot;how do I use FileVault&quot;, etc. I know how to do all of that. I want to make sure that there is no interaction between the partitions possible. Can this be made sure with the above approach? Is it sufficient? Installing 2 versions of Ubuntu on same hard drive? Why isn't my String comparison working in iOS 6? Excel function that evaluates a string as if it were a formula?