Illustration for article Tangent half-angle formula. Created by Gustavb using Euklides.

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box(-1.6, -1.6, 1.6, 1.6, 2.5)
thickness(.8)

% Axis
X1 = point(-1.7, 0); X2 = point(1.6, 0)
Y1 = point(0, -1.7); Y2 = point(0, 1.6)
color(darkgray);
draw(segment(X1,X2), full, arrow)
draw(segment(Y1,Y2), full, arrow)
color(black);

O = point(0,0);  % origo

% Unit circle
A = point(-1,0); B = point(1,0)
C = circle(A, B)

% Angle
phi interactive(48, -3, -180, 180, "", right)
P = point(C, phi:) 

% Where A->P intersects the Y-axis
I = intersection(line(A,P), line(Y1,Y2)) 

% P projected on X-axis
D = projection(P,line(X1,X2));

draw(C);
draw(P); draw(O,P)
draw(A); draw(I)
mark(B,O,P, simple, .5)
mark(O,A,P, simple, .6)
mark(D,P,B, simple, .6)
thickness(1.3); draw(line(A,P))
thickness(1.0); draw(P,D,A); draw(P,D,B)

% Draw labels
draw("$(0, t)$",I,.1,140:)
draw("$(-1,0)$",A,.08,145:)
draw("$\varphi$",O,.3,20:)
draw("$\varphi/2$",A,.4,13:)
draw("$\varphi \over 2$",P,.35,-78:)
draw("$(\cos \varphi,\sin \varphi)$",P,.15,-12:)
draw("$y$",Y2,.1,-30:); draw("$x$",X2,.1,-130:)

1. Create EPS
   

   $ euk2eps Tangent_half-angle_formula.euk

2. Render EPS to PNG
   

   $ convert -density 250 Tangent_half-angle_formula.eps Tangent_half-angle_formula.png

3. Optimize PNG
   

   $ optipng -zc1-9 -zm1-9 -zs0-2 -f0-5 Tangent_half-angle_formula.png

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