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These are replies submitted by Kitonum

In Maple 2018.2 on Windows 10, there is no such problem. All settings are the same as yours:

@lcz  See my upgraded answer.



eq := cos(2*x)-sin(x)*(sin(x)^2+1)^(1/2)+cos(x)^2*sin(x)/(sin(x)^2+1)^(1/2):
subs([cos(2*x)=1-2*sin(x)^2, cos(x)^2=1-sin(x)^2], eq);


Or more automatically

eq := cos(2*x)-sin(x)*(sin(x)^2+1)^(1/2)+cos(x)^2*sin(x)/(sin(x)^2+1)^(1/2):
expand(subs(x=arcsin(y), eq));
subs(y=sin(x), %);


@OscarSteenstrup  These are simply parametric equations of a conical surface, where polar coordinates on the yOz plane are taken as parameters.
If the equation of a curve on a plane  xOy  in Cartesian coordinates is  f(x,y)=0  and this curve rotates around the  Ox-axis, then the equation of a surface of revolution is  f(x, sqrt(y^2+z^2))=0 . Your cone can also be defined by this equation. The point is that if the surface is specified parametrically, then the quality of the drawing is usually much higher. See

plots:-implicitplot3d(eval(f, y=sqrt(y^2+z^2)), x=0..12, y=-5..5, z=-5..5);


@Scot Gould  You missed  the theta variable before the tangent.

@Johan159  Should be

plot(x->f(x,5), 0..10, labels=[x,z]); # The curve in 2D
plot3d(f, 0..10, 5..5, axes=normal, labels=[x,y,z]); # The same curve in 3D


@Preben Alsholm  Thank you.  In general case, you are right, but in this example there is no error, because

series(cos(x),x=0, 6) = series(cos(x),x=0, 5);

@Qruze  This is strange. I do not have Maple 2020 and cannot test this behavior. But this code in Maple versions 2015 - 2018 solves the example without any problems.


This issue has been discussed in detail in this thread

@AHSAN  You have to do something yourself. For other questions, see the help on the  commands LinearAlgebra:-Pivot  and  LinearAlgebra:-RowOperation

@nm  I don't know the answer to the last question. I've never used parentheses when working with matrices.

@oggsait  Use the  legend  option for this.

@ik74  In your drawing, the center of the circle should be to the right and not to the left of the centerline.
The code for this case:

a:=2: R:=-8: z:=sqrt(R^2-(r-a+R)^2):
plot3d([[r,phi,z],[r,phi,-z]], phi=0..2*Pi, r=2..3, coords=cylindrical, scaling=constrained);


@daljit97  Sorry, I find it difficult to explain such obvious things (English is not my native language). Probably the best option for you is to take a tutorial and read about the simplest for-loops  do ... od  and conditionals  if ... fi .

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