## problem : vind domein x0 dot , y0  dot   => g(x0 dot , y0 dot ) > 0 = | M|= matrix 
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tips : denis => relation cos(2*θ0) term

p1:c-fv0*(y0dot)^2/v0^3;
p2:fv0*(x0dot*y0dot)/v0^3;
p3:fp*sin(θ0);
p4:fv0*(x0dot*y0dot)/v0^3;
p5:c-fv0*(x0dot)^2/v0^3;
p6:-fp*cos(θ0);
p7:kr*(y0dot)/(x0dot)^2;
p8: -kr/(x0dot);
p9:ct;

M:matrix(
 [p1,p2,p3], 
 [p4,p5,p6], 
 [p7,p8,p9]
);

p10: v0^2 =(x0dot)^2+(y0dot)^2;

assume(x0dot >0,y0dot >0,v0>0);

assume(c>0,ct>0,fv0>0,fp>0,kr>0);

domain : 0 < θ0 < %pi/2

p11:tg(θ0) = (y0dot)/(x0dot);

?   g(x0dot,y0dot) = |M| = determinant
question :find domain for x0dot en y0dot  so that  g(x0dot,y0dot) > 0

M1:subst([x0dot=v0*cos(θ0),y0dot=v0*sin(θ0)],M);

M2:determinant(M1);

M3:combine(M2);

M3a:ratsimp(M3);

numerator
========

M4:collectterms(M3a,cos(2*θ0));

intermezzo
========

sumexpand:true;

M5:expand(M4);

M6:ratsimp(M5);

info: same M4

numerator
========

M7:num(M6);

cos(θ0)^2,cos(θ0)^4,sin(θ0)^2    =>  cos(4*θ0),cos(2*θ0)   with trigreduce

M8:trigreduce(M7);

M9:combine(M8);

M10:expand(M9);

general:    0 < θ0 < π/2
=================

input  : θ0 = 0
===========

M11a:subst((θ0 = 0),M10);

input  : θ0 =Π/2
============

M11b:subst((θ0 =π/2),M10);

function :plot of angle : θ0
====================

G(θ0) := ''M10;

theoretical list
===========
step value : π/10

kl:makelist(G(θ0),θ0,0,π/2,π/10);