The Unit Circle is really just a circle drawn with its center at the origin and the radius fixed to 1. This means we can algebraically represent the unit circle with the equation \(x^2+y^2=1\). Note that we are deviating from the notion of a function here, and are instead focusing on a relationship between \(x\) and \(y\) described by an equation. The solutions to that equation just so happen to plot out points on a graph that are identically a circle with a radius of one centered at the origin. Below is a graph of the unit circle with some coordinates shown.
Pick 4 points (one from each quadrant) and verify that \(x^2+y^2=1\) by plugging the point values into the equation and simplifying.
x^2+y^2 = 1 ;{color:black, lineWidth:5}\ (1,0); {showLabel:true,label:points[0],color:black, labelOrientation:Desmos.LabelOrientations.RIGHT}\ (\cos(\pi/6),\sin(\pi/6)) ; {showLabel:true,label:points[1],color:black,labelOrientation:Desmos.LabelOrientations.RIGHT}\ (\cos(\pi/4),\sin(\pi/4)) ; {showLabel:true,label:points[2],color:black}\ (\cos(2\pi/6),\sin(2\pi/6)) ; {showLabel:true,label:points[3],color:black, labelOrientation:Desmos.LabelOrientations.RIGHT}\ (\cos(2\pi/4),\sin(2\pi/4)) ; {showLabel:true,label:points[4],color:black, labelOrientation:Desmos.LabelOrientations.ABOVE}\ (\cos(4\pi/6),\sin(4\pi/6)) ; {showLabel:true,label:points[5],color:black, labelOrientation:Desmos.LabelOrientations.LEFT}\ (\cos(3\pi/4),\sin(3\pi/4)) ; {showLabel:true,label:points[6],color:black}\ (\cos(5\pi/6),\sin(5\pi/6)) ; {showLabel:true,label:points[7],color:black, labelOrientation:Desmos.LabelOrientations.LEFT}\ (\cos(\pi),\sin(\pi)) ; {showLabel:true,label:points[8],color:black, labelOrientation:Desmos.LabelOrientations.LEFT}\ (\cos(7\pi/6),\sin(7\pi/6)) ; {showLabel:true,label:points[9],color:black, labelOrientation:Desmos.LabelOrientations.LEFT}\ (\cos(5\pi/4),\sin(5\pi/4)) ; {showLabel:true,label:points[10],color:black}\ (\cos(8\pi/6),\sin(8\pi/6)) ; {showLabel:true,label:points[11],color:black, labelOrientation:Desmos.LabelOrientations.LEFT}\ (\cos(3\pi/2),\sin(3\pi/2)) ; {showLabel:true,label:points[12],color:black, labelOrientation:Desmos.LabelOrientations.BELOW}\ (\cos(10\pi/6),\sin(10\pi/6)) ; {showLabel:true,label:points[13],color:black, labelOrientation:Desmos.LabelOrientations.RIGHT}\ (\cos(7\pi/4),\sin(7\pi/4)) ; {showLabel:true,label:points[14],color:black}\ (\cos(11\pi/6),\sin(11\pi/6)) ; {showLabel:true,label:points[15],color:black, labelOrientation:Desmos.LabelOrientations.RIGHT}