The visualization shows that Maple returns the correct result, which can be simplified (see the last line of the code):

Download positive_function_new2.mw

This can be done in many ways. Here are 2 ways:

Download heightofcylinder-new.mw

Your matrix equation  S*A*inverse(S)=A  is equivalent to  S*A-A*S=0  provided that  det(S)<>0 .
The solution is below:

Download ME.mw

Let  k=theta/V  so  theta=k*V .

isolate(subs(theta = k*V, (1/3)*m*a^2*theta*s^2+(1/16)*m*g*a*sqrt(2)*(4+4*theta-Pi)+theta*B*s = (-K__m^2*s*theta+K__m*V)/(L*s+R)), k);
 ;

In real domain use the  surd  function for this:

The reason that the blue curve is not completely plotted in your document  is that Maple, for a negative number of the three possible values of the cubic root, returns the principal value of the cubic root (with the smallest argument). Compare:
 

simplify((-8)^(1/3)); 
surd(-8, 3)

Download inverseExample_new.mw

In fact, all your functions depend only on the variable  t . I made all the necessary simplifications and changes. In particular, I replaced  combine  with  simplify  with the option  size , because it turns out a more compact expression:

Download equations_new.mw

You must specify a sequence step, for example:

f := unapply(3*x^2-2*x^3-1.080674649*x^2*(x-1)^2-.8118769171*x^2*(x-1)^3+.4147046974*x^2*(x-1)^4+.4585681954*x^2*(x-1)^5, x):
ma := seq(f(x), x = 0..1, 0.1);

             ma := 0., 0.02517819625, 0.09374594496, 0.1954298021, 0.3207056009, 0.4607261850, 0.6065902371, 0.7481833253, 0.8728222816, 0.9639340323, 1.000

restart;	
f:=(x,y)->sin(x)*cos(y):
g:=(x,y)->sin(y)*cos(x):
v:=(x,y)->combine(f(x,y)-g(x,y)):
v(x,y);

alpha:=1: dT:=Th-Tc: n:=1:
plot([[Th-273,subs(Tc=400,n*alpha*dT/2),Th=300..700],[Th-273,subs(Tc=500,n*alpha*dT/2),Th=300..700]]);

Eq1.mw

restart;
V:=Vector[row]([x,y,z]):
M:=Matrix([seq(seq([`if`(y<>0,eval(solve(x*'y'+'z'=0,x),['y'=y,'z'=z]),undefined),y,z], y=-1..5),z=0..5)]):
interface(rtablesize=50):
<V; M>;

PS. "undefined" means that if  y=0  then there is no unique value  x  satisfying this equation. However, this case splits into 2 subcase: if  z = 0 , then  x  can be any number, and if  z<>0 , then this  x  does not exist. Below is a version of the code that takes these options into account:

restart;
V:=Vector[row]([x,y,z]):
M:=Matrix([seq(seq([`if`(y<>0,eval(solve(x*'y'+'z'=0,x),['y'=y,'z'=z]),`if`(z<>0,`not exist`,any)),y,z], y=-1..5),z=0..5)]):
interface(rtablesize=50):
<V; M>;

Use the tgonometric form of complex numbers:

z1 := cos(alpha)+I*sin(alpha): 
z2 := cos(beta)+I*sin(beta):
is((z1^2+2*z1*z2+z2^2)/(z1+z2) = z1/z2+z2/z1);
is(z1/z2+z2/z1+2 = 2*(1+cos(alpha-beta)));

                                          false
                                           true

I simplified both of your procedures by making the following changes:
1. The parameter of the procedures is taken as a whole vector, and not its coordinates.
2. Used elementwise operators.
3. Used two-argument  arctan .

Now everything works as expected:
 

Download PP.mw

According to Maple's help, in Maple there is no  Vector[col] , but there is  Vector[column]  (or Vector).
In Maple 2018.2, Vector[col] does not kill the session, but just an error message appears.

Apply the  simplify  command to this expression.

 It’s better to do all this programmatically:

g:=t->t*(cos(t)+I*sin(t));
D(g)(2*Pi);

                                    g := t -> t (cos(t) + I sin(t))
                                              1 + 2 I Pi

saw1.mw

restart;
f:=x->4*(x/2/Pi+1/2-floor(x/2/Pi+1/2)-1/2):
A:=plot(f(x), x=-3*Pi..3*Pi, scaling=constrained, color=red, size=[1000,300]);

a0:=1/Pi*int(f(x),x=-Pi..Pi);
a:=unapply(1/Pi*int(f(x)*cos(n*x),x=-Pi..Pi), n);
b:=unapply(1/Pi*int(f(x)*sin(n*x),x=-Pi..Pi),n);
g:=n->a0/2+add(a(k)*cos(k*x)+b(k)*sin(k*x), k=1..n);
F:=n->plots:-display(A,plot(g(n),x=-3*Pi..3*Pi, color=blue), plots:-textplot([1.2,2.5,N=n], font=[times,bold,18])):
plots:-display(A$5,seq(F(n)$5, n=1..5), seq(F(n), n=6..50), insequence, scaling=constrained, size=[1000,300], axes=none);

Edit.