Kitonum

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16 years, 223 days

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These are replies submitted by Kitonum

@nm  For problems with a parameter in Mathematica there is a very powerful  Reduce  command that does a full analysis of an equation or inequality:

 

I think developers should pay attention to this post, because none of the natural approaches give the desired result (code in Maple 2018.2):

solve( x^2+(y-2)^2=0, {x,y}) assuming real;  # Incorrect output
RealDomain:-solve( x^2+(y-2)^2=0, {x,y});  # NULL

                                   

@acer Thank you for your efforts in analyzing this situation in detail.

@Carl Love  and  @mmcdara  Thanks for the answers!

We get the following,

is(Or(cos(x) = -sqrt(1-sin(x)^2), cos(x) = sqrt(1-sin(x)^2))) assuming real;

  #  FAIL

It's a shame Maple doesn't know the right answer  true .

@mmcdara  The formula  cos(x)=(-4*sin(x/2)^4+sin(x)^2)/(4*sin(x/2)^4+sin(x)^2)
  may lead to errors for some argument values (i.e. these expressions are not equivalent). In addition, the expression becomes more complex. I still don't understand what the meaning of these transformations is?

eval(cos(x)=(-4*sin(x/2)^4+sin(x)^2)/(4*sin(x/2)^4+sin(x)^2), x=0);

       Error, numeric exception: division by zero

@Andiguys Yes, it is easy to do. Just remove x=10000 from your data. To plot 2 graphs for each vertical axis, use a separate list for each pair of graphs. Read the plot,details help for this.

 vv ,  mmcdara   Thank you for your interest and the solutions provided. My solution below differs from vv's solution in that I use  simplify  with siderals instead of  solve . Since  siderals  does not work with  abs, in order not to miss solutions I had to take into account different signs distributions (4 options in total):

restart;
A:=<x1,a>: B:=<x2,b>: C:=<x3,c>: vA:=<v1,0>: vB:=<v2,0>: vC:=<v3,0>:
f:=t->A+t*vA: g:=t->B+t*vB: h:=t->C+t*vC:
S:=t->1/2*LinearAlgebra:-Determinant(<g(t)-f(t) | h(t)-f(t)>):
P:=combinat:-permute([1,1,-1,-1],2):
{seq(simplify(abs(S(10)), {S(0)=2*p[1], S(5)=3*p[2]}), p=P)};

                                 


PS.  This problem is taken from a wonderful book (in Russian)  В.В. Прасолов "Задачи по планиметрии (часть 2)", which contains many original and non-trivial geometric problems on a variety of topics. This book can probably be found in translations into other languages.

If the number (-1) in the exponent simply means the reciprocal value, then the syntax is very simple

simplify(1/(diff(z, t)-diff(z, x)));

If you mean the inverse function, then clarification is required here, since you have a function of several variables.

@nilswe 

restart;
`&||` := (R::seq(algebraic)) -> `if`(_npassed = 0, infinity, 1/(add(`/`~(1, [R]))));
`&||`(R1,R2);

salim-barzani   Try   eq6 := eval(L, {x = Y/t, t^2 = Z, x^2 = X});

@JAMET  This is a completely different task and you should ask a separate question for it. Moreover, you must use English, because this is an English-language forum.

@JAMET   See my updated answer.

@Andiguys  Maybe so

P := evalf[4]([eval([k1, delta][], s[2]), s[1]]); 
display(plot3d(TRC(k1, delta), k1 = 0 .. 1, delta = 0 .. 1), pointplot3d(P, color = red, symbol = solidcircle, symbolsize = 17), textplot3d([P[], P], font = [times, 12], align = [ABOVE,RIGHT]), orientation = [45, 75], axes = normal);

         

 

 

@Andiguys  I didn't notice that the  0  after the delta should also be a subscript. The following option   labels=[ `&delta;`[0], s[2] ]    works for me.

@Earl  I think that for your purposes a procedure will be useful that will allow you to easily generate a chain of tangent circles of any length. The parameters of the  Circles  procedure are: n - circle's number, R - radius of the original circle, r1 - radius of the larger of the small circles. The procedure returns the center of the n-th small circle:

restart;
Circles:=proc(n,R,r1)
option remember;
if n=1 then return [2*sqrt(R*r1),r1] else
simplify([Circles(n-1,R,r1)[1]*sqrt(R)/(sqrt(R)+sqrt(Circles(n-1,R,r1)[2])),Circles(n-1,R,r1)[2]*R/(sqrt(R)+sqrt(Circles(n-1,R,r1)[2]))^2]) fi;
end proc:

# Examples of use

S0:=seq(Circles(n,4,1), n=1..15);
S:=seq(plottools:-circle(Circles(n,4,1),Circles(n,4,1)[2]), n=1..15):
plots:-display(plottools:-circle([0,4],4),S, scaling=constrained, view=[0..5.5,0..2.5],size=[900,600]);

     

 

 

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