The set of all points at unit distance from the origin. In the complex plane, this is {z: |z| = 1}. Astute geometricians will realise that this is indeed the locus of points equidistant from a focus, a set vulgarly referred to as a "circle".

The unit circle is the boundary of the unit disk (of which, however, it is not a part). It is often used to formulate theorems of complex analysis, due to its utmost symmetry (it is, after all, a circle) and its uniticity.

The unit circle is an illustration of a circle with a radius equal to 1, with origin set in point (0,0). This circle can be used as a visible model to base all trigonometric functions, as the unit circle acts similarly to the periodic actions of these functions. For this reason, the unit circle is an important part of trigonometry.

Below is a graphical representation of the unit circle with quadrant I filled in with radian units and their coordinate points.

                                   (0,1)
                                              (1/2,sqrt(3)/2)
                                    π/2 
                                                   π/3   
                             ..od@@**X**@@eu..                       
                         .u@*""      X      ^"#Rb.                    
                       u$""          X         v "#N.                 
                    .d*"             X        e.    "$u    (sqrt(2)/2,sqrt(2)/2)           
                   dP"               X       v        "$u             
                 u$"                 X      e.          "N.  π/4          
                dP                   X      v            d$$c          
               8"                    X     e.         ..@P '#L         
              8"                     X    v        .dP      #L        
             dF                      X   e.      ..@P        $c  (sqrt(3)/2,1/2)     
            J$                       X   v    .dP            '$  π/6     
            $\                       X  $.  ..@P        ...d$*k      
           4$                        X v  .dP       ...d$$    '$      
           M>                        X. ..@P    ...d$$        M>     
           $                         X.pd   ...d$$            ?k     
           $                         XP...d$$                 JR     
(-1,0) 1π  XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX  0π / 2π (1,0)
           $                         X                        4R     
           M>                        X                        M>     
           ?k                        X                        $      
           '$                        X                       J$      
            *k                       X                       $\      
            '$r                      X                      dP       
             '$                      X                     xR        
              "$                     X                    x$         
               ^$.                   X                   JR          
                '$u                  X                  dP           
                  "N.                X                u$"            
                   '#N.              X              u$"              
                     '"No.           X           .d*"                
                        ^"Rb..       X       .u@*"                   
                           '""*@beo__X..ee@@*""                      
                                  '""X"                             
                                 
                                  (3π)/2
        
                                  (0,-1)
The unit circle can be applied to any trigonometric function, as shown below.
x = x coordinate
y = y coordinate
r = radius (1 in the unit circle)

sinθ = y/r    cscθ = r/y

cosθ = x/r    secθ = r/x

tanθ = y/x    cotθ = x/y

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