## Finding all solutions to 2 simultaneous trigonomet...

Hey there folks - I have the following headcracker...

I have 2 equations:

cos(a t) = cos(b t + c)
&
sin(a t) = sin(b t + c)

Where a, b and c are known constants, and t is a variable.

I would like to find a way to solve these 2 equations simultaneously, i.e. find the t values that solve both equations at the same time. Of course, there will be an infinite number of solutions, so I also need a way to define an interval that t needs to be restricted to, e.g. t = 0..20 * Pi .

The best I've managed is:

... but I can't seem to make this work for solving the 2 trig. equations simultaneously

nor can I figure out the syntax for getting all solutions compiled as a list, e.g.

- which would be enormously helpful for further calculations.

Can anyone give some help on this?

Regards,  Matthew

## Warning, solutions may have been lost ...

I need some help. I'm trying to solve this system of equations, but maple says the solutions may have been lost. Here are the equations:

phi := alpha+theta;
sigma := b*c/(2*pi*r);
f := 2*arccos(exp(-(1/2)*b*(1-r/R)*R/(r*sin(phi))))/pi;
d := 4*f*sin(phi)*(cos(phi)-lambda*sin(phi))/(sigma*(sin(phi)+lambda*cos(phi)));
e := .1152*alpha+.6634;
x := solve(d = e, alpha)

I am trying to solve for alpha by setting d = e. Any help  would be greatly appreciated.

## How do I convert a series to trigonometry or hyper...

theta := a-(1/2)*beta*a*y^2+(1/24)*beta^2*a*y^4-(1/720)*beta^3*a*y^6+(1/40320)*beta^4*a*y^8-(1/3628800)*beta^5*a*y^10+(1/479001600)*beta^6*a*y^12-(1/87178291200)*beta^7*a*y^14+(1/20922789888000)*beta^8*a*y^16-(1/6402373705728000)*beta^9*a*y^18+(1/2432902008176640000)*beta^10*a*y^20-(1/1124000727777607680000)*beta^11*a*y^22+(1/620448401733239439360000)*beta^12*a*y^24

Pls, anyone with useful informations on how to convert a series just like the one above to trigonometry or hyperbolic form. Need response as soon as possible. Thankin you in anticipation for your favorable response.

## arctan and the csgn function...

Hi,

I was wondering why when I want to equate

Maple returns

I know that the maple arctan(x,y) (or atan2) plots from Pi/..-Pi/2, but at x = 0, depending on the sign of y, it should either return Pi/2 or -Pi/2. but the equation returned wouldnt do that, It will plot only from Pi/2 to 0. see below.

first, why does maple evaluate it to that particular expression of csgn?
second, how to get a "correct" result?

## system of equations...

hi there,

this is my first post here, and first time using maple

I do have trigonometric system of equations, and I like to solve for thetas(1-5).

## issues with angular displacement in given inverse ...

Hi,

I was playing around with the example "Simple Inverse Kinematic Problem" and found somethings to be odd:

the angular motion seems to be calculated from between pi and negative pi and this has some effects when using position block to move a joint.

the original angular displacement is graph below

the angular displacement after ik calculations have been performed:

if you run the simulation it seems to copy and mirror the input pendulum, however if you disable one of the IK solutions you see that infact its motion isn't like the input.

this becomes more prevailant when you use a position block to force rotation on a joint instead of using the 'prescribed rotation' blocks that comes with the example.

My question then:
why does this happen?
how do I work around this?

the importance that the motion is follow precisely becomes more prevailent when we want to extract other values such as vel, accel, torque. they are incorrect and very jumpy. Also simply put the angular displacement is wrong, how do I fix it?

(side note: I'm thinking  it has to do with the way arctan is calculated in maple limits it to stay in range -pi to pi
"For real arguments x, y, the two-argument function arctan(y, x), computes the principal value of the argument of the complex number x+Iy, so −π < arctan(y,x) ≤ π." from https://www.maplesoft.com/support/help/Maple/view.aspx?path=invtrig)

pjf

## solving numerical numerical equations in a specifi...

Hello,

I have to solve numerical trogonometric equations such as :
solve(.3707752782+.1499320455*sin(theta[4](t))+.1117559025*cos(theta[4](t))=0.5,theta[4](t));

But, after, I would like to keep only the solution defined in a specific interval such as : [0,Pi]

1) Is there a possibility to define options with the function solve to limit the solutions belonging to a specific interval ?

2) Otherwise, may you help me to make an systematic process to choose a solution in a specific interval ?

## Integrating an expression...

Hello Everyone

I have an expression which I would like to integrate from x=0 to x=L. The expression is

x1:=(sin(beta*x)*cos(m*Pi*x/L))/(1+alpha*x)

Here, beta, m, alpha are constant. However I want the result in terms of these quantities.

I will be grateful if you could help mw in this regard.

Thanks a lot.

## Solution of trigonometric equation...

Hello,

I would like to solve this equation :

tan(gamma0(t)) = tan(gamma[1](t)-theta[1](t)-psi[1](t)

I would like to select the solution of arctan by assuming the domain of variation of some variables. In my case, if -Pi/2 < gamma[1](t) - theta[1](t) - psi[1](t) < Pi/2, the arctan gives only one solution.

Can you help me to solve this equation ?

eq:=tan(gamma0(t)) = tan(gamma[1](t)-theta[1](t)-psi[1](t))
solve(eq,gamma0(t)) assuming -Pi/2 < gamma[1](t) - theta[1](t) - psi[1](t) < Pi/2

I guess that I don't use properly the "assume" function.

The result that I would like to obain is quiet simple :

gamma0(t)) = gamma[1](t)-theta[1](t)-psi[1](t)

Thanks a lot for your help

## Simplification of trigonometric expression. IV...

Hello,

I have still some difficulties to conduct some specific trigonometric simplications but which are very common in mechanism study.

The equations are in the form :

sin(gamma0(t))*cos(beta0(t)) = -(sin(psi[1](t))*cos(theta[1](t))*cos(gamma[1](t))+sin(psi[1](t))*sin(theta[1](t))*sin(gamma[1](t))-cos(theta[1](t))*cos(psi[1](t))*sin(gamma[1](t))+cos(psi[1](t))*sin(theta[1](t))*cos(gamma[1](t)))*cos(beta[1](t))

I would like to obtain this equation after simplifications :

sin(gamma0(t))*cos(beta0(t)) = cos(beta[1](t))*sin(gamma[1](t)-theta[1](t)-psi[1](t))

I try to make a procedure to automatize the simplification of this kind of trigonometric equation.

Strangely, I noticed that the simplification is done only if there is a minus before the combine function. The simplification works but the result is wrong because i didn't obtain the good sign.

For you information, I try to make these simplifications with MMA and the FullSimplify function of MMA gives directly the expected result that is to say :

I'm sure that it shoud exist a good way to conduct this kind of simplications in Maple.

Can you help me to correct my procedure so to obtain the good result and be enough general, adaptative ?

Code here and attached in this post :

Initialisation
restart:
with(LinearAlgebra):
with(Student[MultivariateCalculus]):
with(plots):
with(MathML):
with(ListTools):
constants:= ({constants} minus {gamma})[]:
`evalf/gamma`:= proc() end proc:
`evalf/constant/gamma`:= proc() end proc:
unprotect(gamma);
Angular Constraint equations
eq_liaison:=sin(gamma0(t))*cos(beta0(t)) = -(sin(gamma[1](t))*sin(psi[1](t))*sin(theta[1](t))-sin(gamma[1](t))*cos(theta[1](t))*cos(psi[1](t))+cos(gamma[1](t))*sin(psi[1](t))*cos(theta[1](t))+cos(gamma[1](t))*cos(psi[1](t))*sin(theta[1](t)))*cos(beta[1](t));
Traitement
TrigoTransform2:= proc(Eq)
local S,S1,tt,pp,Eq2,ListVariables,ListVariablesMod,Subs,size,rhsEq2,lhsEq2;
#Construit une liste à plat#
ListVariables:=indets(Eq, function(identical(t)));
ListVariables:=[op(ListVariables)];
ListVariablesMod:=map(f->cat(op(0,f),_),ListVariables);
Subs:=ListVariables=~ListVariablesMod;
#Variables Changement#
Eq2:=Eq:
print("Equation traitée=",Eq2):
Eq2:=subs(Subs, Eq2);
print("Equation après subs=",Eq2):
#Trigonometric transformations#
lhsEq2:=applyrule([
cos(u::anything)*cos(v::anything)-sin(u::anything)*sin(v::anything)=cos(u+v),
cos(u::anything)*sin (v::anything)+sin(u::anything)*cos(v::anything)=sin(u+v),
sin(u::anything)*sin(v::anything)-cos(u::anything)*cos(v::anything)=-cos(u+v),
-sin(v::anything)*cos(u::anything)-sin(u::anything)*cos(v::anything)=-sin(u+v)], simplify(lhs(Eq2), size));
print("Equation lhsEq2 première analyse=",lhsEq2):
rhsEq2:=applyrule([
cos(u::anything)*cos(v::anything)-sin(u::anything)*sin(v::anything)=cos(u+v),
cos(u::anything)*sin (v::anything)+sin(u::anything)*cos(v::anything)=sin(u+v),
sin(u::anything)*sin(v::anything)-cos(u::anything)*cos(v::anything)=-cos(u+v),
-sin(v::anything)*cos(u::anything)-sin(u::anything)*cos(v::anything)=-sin(u+v)], simplify(rhs(Eq2), size));
print("Equation rhsEq2 première analyse=",rhsEq2):
try
lhsEq2:=(trigsubs(2*combine(lhsEq2))[])/2;
print("Equation lhsEq2=",lhsEq2):
catch:
lhsEq2:=lhs(Eq2);
end try;
try
rhsEq2:=(trigsubs(-2*combine(rhsEq2))[])/2;
print("Equation rhsEq2=",rhsEq2):
catch:
rhsEq2:=rhs(Eq2);
end try;
Eq2:= lhsEq2=rhsEq2;
#Variables Changement#
Eq2:=subs(map(t->rhs(t)=lhs(t),Subs),Eq2)
end proc:
TrigoTransform2(eq_liaison);

TrigoTransformEqAng2_anglais.mws

Thanks a lot for your help.

## Solve trigonometric equation...

I don't know if I am doing it right or not because Maple isn't solving the equation. The equation solution is x=2^(1/2).

Here I let a picture of which equation I want to solve and Maple's response.

## Why is my trig failing to compute? ...

So it looks like radians work.

Why does degrees fail and how do I get degrees work?

How do you guys like to access pi? Do you keep a symbol of it around in a random document to open?

http://i.imgur.com/07o1dyH.png

## isosceles right triangle...

Let A(x , 2 , -1) ,B(0, y , 1) ,C(3 , 4 , -2) , AB=AC and ABC is isosceles right triangle. Find x and y ?

## Trigonometry and cosine...

True of False, Explain:

If ∏/2<θ<∏, Then cos θ/2<0

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