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Ronan

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14 years, 125 days
East Grinstead, United Kingdom
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These are answers submitted by Ronan

Home User...

Ronan 1426
December 31 2024
0 0

If memory serves me correctly, I think I specified Home User. As I have a Personal Edition license.

Possibly...

Ronan 1426
December 27 2024
1 0

Hope this is correct
 

> 

restart
> 

> 

L:=1

1

 (1)
> 

E:=[0,0]

[0, 0]

 (2)
> 

A:=[-1/2*L,L*sqrt(3)/2]
> 

 

[-1/2, (1/2)*3^(1/2)]

 (3)
> 

D1:=[L,0]

[1, 0]

 (4)
> 

C:=[L,L]

[1, 1]

 (5)
> 

AC:=C-A

[3/2, -(1/2)*3^(1/2)+1]

 (6)
> 

AB:=L

1

 (7)
> 

BC:=L

1

 (8)
> 

expn1:=2*L*cos(theta)=sqrt(AC[1]^2+AC[2]^2)

2*cos(theta) = (1/2)*(9+4*(-(1/2)*3^(1/2)+1)^2)^(1/2)

 (9)
> 

theta:=solve(expn1,theta)

arccos((1/4)*(9+4*(-(1/2)*3^(1/2)+1)^2)^(1/2))

 (10)
> 

b:=2*Pi-(Pi-2*theta)

Pi+2*arccos((1/4)*(9+4*(-(1/2)*3^(1/2)+1)^2)^(1/2))

 (11)
> 

simplify( (11) );

2*Pi-2*arcsin((1/2)*(4-3^(1/2))^(1/2))

 (12)
> 

expn2:=L-L*cos(c)=L*sin(2*Pi/3)-L*sin(a+Pi/3)

1-cos(c) = (1/2)*3^(1/2)-sin(a+(1/3)*Pi)

 (13)
> 

expn3:=L-L*sin(c)=L*cos(2*Pi/3)+L(a+Pi/3)

1-sin(c) = 1/2

 (14)
> 

c:=solve(expn3,c)

(1/6)*Pi

 (15)
> 

a:=solve(expn2,a)

-(1/3)*Pi+arcsin(3^(1/2)-1)

 (16)
> 

simplify( (16) );

-(1/3)*Pi+arcsin(3^(1/2)-1)

 (17)
> 

B[1]:=L-L*sin(c)

1/2

 (18)
> 

B[2]:=L-L*cos(c)

-(1/2)*3^(1/2)+1

 (19)
> 

B:=[B[1],B[2]]

[1/2, -(1/2)*3^(1/2)+1]

 (20)
> 


plots:-display(plottools:-polygon([E,D1,C,B,A]),colour=yellow)

1

  

 
> 

 


 

Download Fun-Angles.mw

ln(x-2) becomes complex...

Ronan 1426
December 15 2024
1 0

In you expression

test := subs(a = 2, x^2/2 + x*a + a^2*ln(x - a))

ln(x-a) becomes complex with x<a.

Is this the Clifford packace you are lo...

Ronan 1426
December 11 2024
0 1

I dont know is this is the Clifford package you mean. I found this on last friday. Index of /12/eng/files/st_files/kyrchei/kyr_files/Maple Package

interface(displayprecision=5)...

Ronan 1426
November 22 2024
1 0

You can set the number of digits displayed on the screen using for eample

interface(displayprecision=5)

This does not affect the accuracy of calculation set by Digits

Use a free video downloader....

Ronan 1426
November 15 2024
1 0

I use 4k Video Downloader. There is a free version of it. AFAIK if you are not redestributing the videos so purely for you own use it is not a problem. 

There are  a couple you tube chanels I use. Insights into Mathematics and Wild Egg Maths. I downloaded most of the videos. About 3 weeks ago both chanels were hacked and all the content was gone.So far only Wild egg maths has been restored.

Basically if you value it get you own copy.

A possible workaround...

Ronan 1426
November 09 2024
2 0

I tried setting transparency to 0 but that didn't work. I used plottools:-point as a work around.

L:=plottools:-line([0,0],[0,3],color="blue"):
c1:=plottools:-disk([0,1],.1):
c2:=plottools:-disk([0,2],.1,color="green",transparency=0):
c3:=plottools:-point([0,1],symbol=solidcircle,symbolsize=45,color="red");
plots:-display([L,c1,c2,c3],scaling=constrained,axes=none)

 

Would this answer be of use?...

Ronan 1426
November 06 2024
1 9

I answered something similar in 2016. I don't have the time right now to investicate you example.

How to take derivative of sum? - MaplePrimes

Mentally...

Ronan 1426
October 29 2024
1 0

10:55-10:10 =45
11:58-10:55=63
9 divides 45 and 63

so 9

align and \n...

Ronan 1426
September 22 2024
2 1

Use align and \n (within the text) as an option in textplot or textplot3d. This is just a quick example.

with(plots);with(plottools)
A:=[0,0,0];C[1]:=[1,1,1]
plt1:=point([A,C[1]],symbol=solidcircle,symbolsize=16,color=blue)
plt2:=textplot3d([ [A[],"A",align={above,right}] , [C[1][],typeset("C[1]=\n",C[1]),align={below,left}] ]):

display(plt1,plt2)

Like this...

Ronan 1426
September 17 2024
0 0

One way of doing it.

> 

restart
> 

> 

local D;
> 

BD:=4;
AB:=6;

D

  

4

  

6

 (1)
> 

eq1:=BD*sin(4*theta)=AB*sin(2*theta)

4*sin(4*theta) = 6*sin(2*theta)

 (2)
> 

sol:=solve(eq1,[theta])

[[theta = 0], [theta = (1/2)*Pi], [theta = (1/2)*arctan((1/3)*7^(1/2))], [theta = -(1/2)*arctan((1/3)*7^(1/2))]]

 (3)
> 

assign(sol[3]);
theta:=simplify(theta)

(1/4)*arctan(3*7^(1/2))

 (4)
> 

eq2:=AB*sin(2*theta)=BC*sin(theta)

6*sin((1/2)*arctan(3*7^(1/2))) = BC*sin((1/4)*arctan(3*7^(1/2)))

 (5)
> 

#eq3:=eval(eq2,(sol[3]))
> 

 

> 

simplify(solve(eq2,[BC])[])

[BC = 12*cos((1/4)*arctan(3*7^(1/2)))]

 (6)
> 

assign(%)
> 

BC

12*cos((1/4)*arctan(3*7^(1/2)))

 (7)
> 

DC:=BD*cos(Pi-4*theta)+BC*cos(theta)

-1/2+12*cos((1/4)*arctan(3*7^(1/2)))^2

 (8)
> 

simplify( (8) );

10

 (9)
 

 

Download 2024-09-18_A_Find_DC_geometry.mw

Like this....

Ronan 1426
September 17 2024
1 0

180-2m+60+k = 2n+40+k  implies m+n=100  so m=100-n

m+40+n+x=180

100-n+40+n+x=180

therefore x=40

Doing it visually...

Ronan 1426
September 13 2024
3 2

Looking at it.ABC is half the area and ABE is quater the area. Draw a vertical line from E intersects the diagonal midpoint at say G. Triangle ABF and EGF are similar. with a 4;1 area ratio. So the horizontal coordinate of F is 2/3 AE. also ABF+EGF is quater of the total area.

The triangle EGC area.  is 1/8 of the total area so 3cm^2.

ABF+2* EGF would be 6 (quater of the total area) 

EGF would be 1/6 of that =1. So  total area of 4.

Suggestion:-One Drive or Google docs...

Ronan 1426
August 20 2024
0 3

Could you put the zip file on One Drive Google Docs or Dropbox for example and have a link to it it in your post.

A proof...

Ronan 1426
June 24 2024
0 9

I got curious. I have converted away from the geometry package and used a rational parameterisation for cos and sin

[a cos(t), b sin(t)]  to  [a* (1t^2)/(1+t^2), b*2*t/(1+t^2)]   . In this case t is not the angle.

This proves the angle is constant upto sign changes for signum . But that just changes the reported angle from sfor example 50deg to 130deg.

 

I would like to know the name of this theorem.

 

> 

restart;
with(plots):with(LinearAlgebra):

 
> 

x0 := 100;
y0 := 40;
a := 7;
b := 5;
c := sqrt(a^2 - b^2);
e1:= x^2/a^2 + y^2/b^2 - 1;
F1:=[ -c, 0];
F2:=[ c, 0];
eq := simplify((a^2 - x0^2)*(y - y0)^2 + (b^2 - y0^2)*(x - x0)^2 + 2*x0*y0*(x - x0)*(y - y0)) = 0;
sol := solve({eq}, {y});
tang1:=(lhs-rhs)~(op(sol[1]));

tang2:= (lhs-rhs)~(op(sol[2]));
sol2 := op(solve({op(sol[1]), x^2/a^2 + y^2/b^2 - 1 }, {x, y}));
xM2 := rhs(sol2[1]);
yM2 := rhs(sol2[2]);
A:=[ xM2, yM2];
sol3 := op(solve({op(sol[2]), x^2/a^2 + y^2/b^2 - 1 }, {x, y}));
xM3 := rhs(sol3[1]);
yM3 := rhs(sol3[2]);
B:=[ xM3, yM3];

slpvecAB:=<(B[2]-A[2]),B[1]-A[1]>;  #slpvec
Pol:=-slpvecAB[1]*(x-A[1])+slpvecAB[2]*(y-A[2]);
simplify(Pol);
isolate(%, y);

TANG := proc(t) local xM, yM; xM :=a*(1-t^2)/(1+t^2);# a*cos(t);
               yM :=b*2*t/(1+t^2);# b*sin(t);
               return(expand(1/49*x*xM + 1/25*y*yM - 1 ));
        end proc;

#t := -0.25;

TT:= TANG(t);
#Equation(TT);
M:=[a*(1-t^2)/(1+t^2),b*2*t/(1+t^2)];#[ a*cos(t), b*sin(t)];

100

  

40

  

7

  

5

  

2*6^(1/2)

  

(1/49)*x^2+(1/25)*y^2-1

  

[-2*6^(1/2), 0]

  

[2*6^(1/2), 0]

  

-1575*x^2+(8000*y-5000)*x-9951*y^2-3920*y+328400 = 0

  

{y = (4000/9951)*x-1960/9951+(5/9951)*13087^(1/2)*x-(500/9951)*13087^(1/2)}, {y = (4000/9951)*x-1960/9951-(5/9951)*13087^(1/2)*x+(500/9951)*13087^(1/2)}

  

y-(4000/9951)*x+1960/9951-(5/9951)*13087^(1/2)*x+(500/9951)*13087^(1/2)

  

y-(4000/9951)*x+1960/9951+(5/9951)*13087^(1/2)*x-(500/9951)*13087^(1/2)

  

x = (49/1642)*13087^(1/2)+1225/3284, y = -(125/3284)*13087^(1/2)+245/1642

  

(49/1642)*13087^(1/2)+1225/3284

  

-(125/3284)*13087^(1/2)+245/1642

  

[(49/1642)*13087^(1/2)+1225/3284, -(125/3284)*13087^(1/2)+245/1642]

  

x = -(49/1642)*13087^(1/2)+1225/3284, y = (125/3284)*13087^(1/2)+245/1642

  

-(49/1642)*13087^(1/2)+1225/3284

  

(125/3284)*13087^(1/2)+245/1642

  

[-(49/1642)*13087^(1/2)+1225/3284, (125/3284)*13087^(1/2)+245/1642]

  

slpvecAB := Vector(2, {(1) = (125/1642)*13087^(1/2), (2) = -(49/821)*13087^(1/2)})

  

-(125/1642)*13087^(1/2)*(-(49/1642)*13087^(1/2)+x-1225/3284)-(49/821)*13087^(1/2)*(y+(125/3284)*13087^(1/2)-245/1642)

  

-(1/6568)*13087^(1/2)*(500*x-245+392*y)

  

y = -(125/98)*x+5/8

  

proc (t) local xM, yM; xM := a*(1-t^2)/(1+t^2); yM := 2*b*t/(1+t^2); return expand((1/49)*x*xM+(1/25)*y*yM-1) end proc

  

-(1/7)*x*t^2/(t^2+1)+(1/7)*x/(t^2+1)+(2/5)*y*t/(t^2+1)-1

  

[7*(-t^2+1)/(t^2+1), 10*t/(t^2+1)]

 (1)
> 

 

> 

P:=rhs~(solve( [tang1, TT],[x,y])[]);

[-7*(93*t+56+13087^(1/2))/(93*t-56-13087^(1/2)), -(5/107)*(56+13087^(1/2))*(107*t-13087^(1/2)+56)/(93*t-56-13087^(1/2))]

 (2)
> 

Q:=rhs~(solve( [tang2, TT],[x,y])[]);

[-7*(93*t+56-13087^(1/2))/(93*t-56+13087^(1/2)), (5/107)*(-56+13087^(1/2))*(107*t+13087^(1/2)+56)/(93*t-56+13087^(1/2))]

 (3)
> 

slpvecPF2:=simplify(<(P[2]-F2[2]),P[1]-F2[1]>);
PF2:=simplify(-slpvecPF2[1]*(x-P[1])+slpvecPF2[2]*(y-P[2]));
> 

 

slpvecPF2 := Vector(2, {(1) = -5*(56+13087^(1/2))*(-107*t+13087^(1/2)-56)/(-9951*t+5992+107*13087^(1/2)), (2) = ((-186*t+2*13087^(1/2)+112)*6^(1/2)-651*t-7*13087^(1/2)-392)/(93*t-56-13087^(1/2))})

  

(2*(465+(-5*t+y)*13087^(1/2)+(-93*y-280)*t+56*y)*6^(1/2)+(5*t*x-7*y)*13087^(1/2)+7*(40*x-93*y)*t-465*x-392*y)/(93*t-56-13087^(1/2))

 (4)
> 

slpvecQF2:=simplify(<(Q[2]-F2[2]),Q[1]-F2[1]>);

QF2:=simplify(-slpvecQF2[1]*(x-Q[1])+slpvecQF2[2]*(y-Q[2]));

slpvecQF2 := Vector(2, {(1) = 5*(-56+13087^(1/2))*(107*t+13087^(1/2)+56)/(9951*t-5992+107*13087^(1/2)), (2) = ((-2*6^(1/2)+7)*13087^(1/2)+(-186*t+112)*6^(1/2)-651*t-392)/(93*t-56+13087^(1/2))})

  

(((10*t-2*y)*13087^(1/2)+(-186*y-560)*t+112*y+930)*6^(1/2)+(-5*t*x+7*y)*13087^(1/2)+(280*x-651*y)*t-465*x-392*y)/(93*t-56+13087^(1/2))

 (5)
> 

 

> 

 

> 

alpha :=(simplify(VectorAngle(slpvecQF2,slpvecPF2),assume=real))

(1/2)*Pi-signum(93*t-56+13087^(1/2))*signum(93*t-56-13087^(1/2))*arcsin((93/14)*(1302*6^(1/2)*t^2+2057*t^2+1498*6^(1/2)-2743)/(((-100*t^2+112*6^(1/2)-292)*13087^(1/2)+8649*6^(1/2)*t^2+37034*t^2-16223*6^(1/2)+50018)^(1/2)*((100*t^2-112*6^(1/2)+292)*13087^(1/2)+8649*6^(1/2)*t^2+37034*t^2-16223*6^(1/2)+50018)^(1/2)))

 (6)
> 

Setof:=frontend(indets,[alpha])

{Pi, arcsin((93/14)*(1302*6^(1/2)*t^2+2057*t^2+1498*6^(1/2)-2743)/(((-100*t^2+112*6^(1/2)-292)*13087^(1/2)+8649*6^(1/2)*t^2+37034*t^2-16223*6^(1/2)+50018)^(1/2)*((100*t^2-112*6^(1/2)+292)*13087^(1/2)+8649*6^(1/2)*t^2+37034*t^2-16223*6^(1/2)+50018)^(1/2))), signum(93*t-56-13087^(1/2)), signum(93*t-56+13087^(1/2))}

 (7)
> 

(sin(Setof[2]))^2

(8649/196)*(1302*6^(1/2)*t^2+2057*t^2+1498*6^(1/2)-2743)^2/(((-100*t^2+112*6^(1/2)-292)*13087^(1/2)+8649*6^(1/2)*t^2+37034*t^2-16223*6^(1/2)+50018)*((100*t^2-112*6^(1/2)+292)*13087^(1/2)+8649*6^(1/2)*t^2+37034*t^2-16223*6^(1/2)+50018))

 (8)
> 

simpascn:=arcsin(sqrt(factor( (8) ))); # the t's eliminate

arcsin((1/58693852)*(1451954478752954-68582717957500*6^(1/2))^(1/2))

 (9)
> 

 alpha1:=(subs(Setof[2]=simpascn,alpha))*180/Pi

180*((1/2)*Pi-signum(93*t-56+13087^(1/2))*signum(93*t-56-13087^(1/2))*arcsin((1/58693852)*(1451954478752954-68582717957500*6^(1/2))^(1/2)))/Pi

 (10)
> 

plot(alpha1,t=-10..10,view=[-10..10,0..180])

 
> 

 
> 

 

> 

 

> 

 


 

Download 2024-06-24_constant_angle.mw

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