Items tagged with trigonometric

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Hi everybody,

I'm kind of new to Maple and i'm trying to solve a system of trigonometric equations inequality as follow:

f:= {((2*a*sin(S)*cos(S)^(2)))/(1-sin(S)^(3))<1,90> S>-90,a>1};

solve(f,{a,S});
Error, (in PiecewiseTools:-Convert) unable to convert

 

How can I solve the system?

Thanks a lot.

It is evident that by repeated applications of the double-angle and product trigonometric identities, one may transform any monomial of the form sin(x)^p * cos(y)^q, where p and q are positive integers, to a linear combination of only first powers of sines and cosines.

Example 1:  The monomial  4*sin(x)*cos(y)^2 is equivalent to

 

Example 2: The monomial 16*sin(x)^2*cos(y)^3 is equivalent to             

How does one write a Maple procedure to do that transformation in the general case of sin(x)^p * cos(y)^q?

 

Hi, I have the following equation defined in Maple:

f:=-4*a*sin(2*x)+4*b*cos(2*x)=-8*sin(2*x)+20*cos(2*x);

And I want to equate the coefficients of the similar terms (sin with sin and cos with cos), in order to do this:

-4a=-8

+4b=20

And then solve for the variables a, b in that 2 new equations.

Any ideas?

Hello, 

I have just started using Maple, and it seems very powerful. I am trying to solve trigonometric equations and get all the solutions in a range, but when I use fsolve I only get one solution. 

Is this by design of the function or is there another way to do this? 

 

Tom

Hello,

I'm a quiet perplexed  in front of the result of the function solve for trigonometric equations.

The result of this equation solve(cos(x)=a,x); is arccos(a) and the solution -arcos(a) is not given.

In order to have the other solution (-arccos(a)), I try this solve({cos(x)=a,x>Pi/2,x<3*(Pi/2)},x); but without success.

1) How can I obtain all the solution with the solve function with trigonometric equation and only symbolic equations (no numerical value)?

2) Is it possible to obtain a specific solution by defining the definition domain of the variables in the equation ?

Thanks a lot for your help

Hello,

I try to use the Explore function on a trigonometric expressions depending of 8 parameters. My aim is to study the influence of these parameters on the results.

However, i receive an error message and I didn't manage to troubleshoot it.

May you have a look of an extract of my code and see if you see the mistake ?

TestExplore.mw

Thanks a lot for your help.

Hello,

I would like to silmplify a trigonometric equation with some squares. I'm sure this equation can be reduced but i didn't manage to simplify it with Maple.

Here is the equation (named condition in the maple file) that I would like to simplify:

simplification_condition_de_compatibilité.mw

May you help me to simplify this trigonometric equation?

I would like simplification if time the pattern cos()^2+sin()^2 appears.

Thanks a lot for your help

Hi,

It might be a silly question but here it goes. I have a sin function in terms of sin(omega*(T0-T)+Phi) and i need to expand it by keeping omega*T0+phi as a single term. One way is by subs omega*T0+phi as a constant 'c' and then after expanding we can back substitute. But is there any option in expand function itself?

 

Regards

Sunit

Hello,

In some trigonometric equations, I have variables depending of time like Psi(t). I don't why after some manipulation my variable is transformed in :-Psi(t).

Have some ideas why some variables can be change with :- before ?

Here an example

ResolTrig.mw

It may come because of the code "Local Psi". I added this because it seemed to me that Psi was protected. Do I right? If not can I remove Local Psi?

Thanks a lot for your help

Hello,

I would like to determine a closed form solution (=analytical solution) of the following trigonometric equations system.

TrigonometricSystem2.mw

The unknowns are :

ListAllUnknowns := [Psi(t), Theta[1](t), Theta[2](t), x[1](t), x[2](t), z[1](t), z[2](t)]

Do you have ideas so as to conduct the symbolic resolution of this trigonometric equations system ?

I have been told that the use of Grobner basis could be useful but I have never try this.

Thanks a lot for yours feedbacks.

Hello,

In the context of solving mechanical constraint equations, I often need to simplify trigonometric equation. In mathematica, the FullSimplify function makes the simplification I need. But, i'm using Maple for a long time and I would rather contnue my calculation with Mathematica.

May you see if so can help me to simplify this equation ?

Here the equation I would like to simplify with Maple :

TrigonometricEquation.mw

Here the result obtained with mathematica

résultatMma.pdf

Thanks a lot for your help

Hello,

After trigonometric manipulations in a mechanical problem, I can obtain the desired angles but defined with modulo 2Pi.

I would like to program or find a function which can do this operation :

While angle doesn't belong to [-Pi, Pi]

do 

  If angle > Pi then do angle = angle - 2Pi

  If angle < - Pi then do angle = angle + 2Pi

end

Is there an existing function which can do this operation ?

Otherwise, may you help me to program it ?

Thanks a lot for your help

Hello,

I would like to simplify a trigonometric equation that I obtain with a vectorial closure (in mechanics)

Here the equation that I would like to simplify 

eq_liaison :=(-sin(p(t)+g(t))*cos(a(t))-sin(b(t))*sin(a(t))*cos(p(t)+g(t)))*l2[1]+((-sin(p(t)+g(t))*cos(a(t))-sin(b(t))*sin(a(t))*cos(p(t)+g(t)))*cos(th(t))+(-cos(p(t)+g(t))*cos(a(t))+sin(a(t))*sin(b(t))*sin(p(t)+g(t)))*sin(th(t)))*l3[1] = 0

Do you have ideas so as to simplify again this expression ?

This expression can still be simplified. You can find here the result expected :

I find surprising that I have so many difficulties to make trigonometric simplications with the trigonometric functions.

Thank you for your help

PS : Sorry for duplicating posts. As I didn't receive any answer, I have tried to simplified my post to isolate the difficulty.

Hello,

Still on the thematic on simplification of trigonometric expression.

I would like to simplify this equation. Normally, for a mecanical point of view, this equation could be simplified a lot and namely the psi[1](t) and theta[1](t) variables should disappear.

The difference with the former posts is the fact that now each term (for example  2*sin(gamma0(t))*z0(t)*cos(beta0(t))*xb[1]) can regroup 2 terms in factor with the trigonometric part.

eq:=l2[1]^2 = 2*sin(gamma0(t))*z0(t)*cos(beta0(t))*xb[1]-2*sin(gamma0(t))*zp[1](t)*cos(beta0(t))*xb[1]+2*sin(gamma0(t))*y0(t)*sin(alpha0(t))*zb[1]-2*sin(gamma0(t))*yp[1](t)*sin(alpha0(t))*zb[1]+2*sin(gamma0(t))*x0(t)*cos(alpha0(t))*zb[1]-2*sin(gamma0(t))*xp[1](t)*cos(alpha0(t))*zb[1]-2*cos(gamma0(t))*z0(t)*cos(beta0(t))*zb[1]+2*cos(gamma0(t))*zp[1](t)*cos(beta0(t))*zb[1]+2*cos(gamma0(t))*y0(t)*sin(alpha0(t))*xb[1]-2*cos(gamma0(t))*yp[1](t)*sin(alpha0(t))*xb[1]+2*cos(gamma0(t))*x0(t)*cos(alpha0(t))*xb[1]-2*cos(gamma0(t))*xp[1](t)*cos(alpha0(t))*xb[1]+2*y0(t)*cos(alpha0(t))*cos(beta0(t))*yb[1]-2*yp[1](t)*cos(alpha0(t))*cos(beta0(t))*yb[1]-2*x0(t)*sin(alpha0(t))*cos(beta0(t))*yb[1]+2*xp[1](t)*sin(alpha0(t))*cos(beta0(t))*yb[1]-2*sin(psi[1](t))*cos(theta[1](t))*l3[1]*xb[1]+2*sin(psi[1](t))*sin(theta[1](t))*l3[1]*zb[1]-2*cos(theta[1](t))*cos(psi[1](t))*l3[1]*zb[1]-2*cos(psi[1](t))*sin(theta[1](t))*l3[1]*xb[1]-2*sin(gamma0(t))*y0(t)*sin(alpha0(t))*cos(theta[1](t))*cos(psi[1](t))*l3[1]-2*sin(gamma0(t))*yp[1](t)*sin(alpha0(t))*sin(psi[1](t))*sin(theta[1](t))*l3[1]+2*sin(gamma0(t))*yp[1](t)*sin(alpha0(t))*cos(theta[1](t))*cos(psi[1](t))*l3[1]+2*sin(gamma0(t))*x0(t)*cos(alpha0(t))*sin(psi[1](t))*sin(theta[1](t))*l3[1]-2*sin(gamma0(t))*x0(t)*cos(alpha0(t))*cos(theta[1](t))*cos(psi[1](t))*l3[1]-2*sin(gamma0(t))*xp[1](t)*cos(alpha0(t))*sin(psi[1](t))*sin(theta[1](t))*l3[1]+2*sin(gamma0(t))*xp[1](t)*cos(alpha0(t))*cos(theta[1](t))*cos(psi[1](t))*l3[1]-2*cos(gamma0(t))*z0(t)*cos(beta0(t))*sin(psi[1](t))*sin(theta[1](t))*l3[1]+2*cos(gamma0(t))*z0(t)*cos(beta0(t))*cos(theta[1](t))*cos(psi[1](t))*l3[1]+2*cos(gamma0(t))*zp[1](t)*cos(beta0(t))*sin(psi[1](t))*sin(theta[1](t))*l3[1]-2*cos(gamma0(t))*zp[1](t)*cos(beta0(t))*cos(theta[1](t))*cos(psi[1](t))*l3[1]-2*cos(gamma0(t))*y0(t)*sin(alpha0(t))*sin(psi[1](t))*cos(theta[1](t))*l3[1]-2*cos(gamma0(t))*y0(t)*sin(alpha0(t))*cos(psi[1](t))*sin(theta[1](t))*l3[1]+2*cos(gamma0(t))*yp[1](t)*sin(alpha0(t))*sin(psi[1](t))*cos(theta[1](t))*l3[1]+2*cos(gamma0(t))*yp[1](t)*sin(alpha0(t))*cos(psi[1](t))*sin(theta[1](t))*l3[1]-2*cos(gamma0(t))*x0(t)*cos(alpha0(t))*sin(psi[1](t))*cos(theta[1](t))*l3[1]-2*cos(gamma0(t))*x0(t)*cos(alpha0(t))*cos(psi[1](t))*sin(theta[1](t))*l3[1]+2*cos(gamma0(t))*xp[1](t)*cos(alpha0(t))*sin(psi[1](t))*cos(theta[1](t))*l3[1]+2*cos(gamma0(t))*xp[1](t)*cos(alpha0(t))*cos(psi[1](t))*sin(theta[1](t))*l3[1]+yb[1]^2+xb[1]^2+zb[1]^2+l3[1]^2+z0(t)^2+zp[1](t)^2+y0(t)^2+yp[1](t)^2+x0(t)^2+xp[1](t)^2+2*z0(t)*sin(beta0(t))*yb[1]-2*zp[1](t)*sin(beta0(t))*yb[1]-2*z0(t)*zp[1](t)-2*y0(t)*yp[1](t)-2*x0(t)*xp[1](t)-2*sin(gamma0(t))*y0(t)*cos(alpha0(t))*sin(beta0(t))*xb[1]+2*sin(gamma0(t))*yp[1](t)*cos(alpha0(t))*sin(beta0(t))*xb[1]+2*sin(gamma0(t))*x0(t)*sin(alpha0(t))*sin(beta0(t))*xb[1]-2*sin(gamma0(t))*xp[1](t)*sin(alpha0(t))*sin(beta0(t))*xb[1]+2*cos(gamma0(t))*y0(t)*cos(alpha0(t))*sin(beta0(t))*zb[1]-2*cos(gamma0(t))*yp[1](t)*cos(alpha0(t))*sin(beta0(t))*zb[1]-2*cos(gamma0(t))*x0(t)*sin(alpha0(t))*sin(beta0(t))*zb[1]+2*cos(gamma0(t))*xp[1](t)*sin(alpha0(t))*sin(beta0(t))*zb[1]-2*sin(gamma0(t))*z0(t)*cos(beta0(t))*sin(psi[1](t))*cos(theta[1](t))*l3[1]-2*sin(gamma0(t))*z0(t)*cos(beta0(t))*cos(psi[1](t))*sin(theta[1](t))*l3[1]+2*sin(gamma0(t))*zp[1](t)*cos(beta0(t))*sin(psi[1](t))*cos(theta[1](t))*l3[1]+2*sin(gamma0(t))*zp[1](t)*cos(beta0(t))*cos(psi[1](t))*sin(theta[1](t))*l3[1]+2*sin(gamma0(t))*y0(t)*sin(alpha0(t))*sin(psi[1](t))*sin(theta[1](t))*l3[1]+2*sin(gamma0(t))*y0(t)*cos(alpha0(t))*sin(beta0(t))*sin(psi[1](t))*cos(theta[1](t))*l3[1]+2*sin(gamma0(t))*y0(t)*cos(alpha0(t))*sin(beta0(t))*cos(psi[1](t))*sin(theta[1](t))*l3[1]-2*sin(gamma0(t))*yp[1](t)*cos(alpha0(t))*sin(beta0(t))*sin(psi[1](t))*cos(theta[1](t))*l3[1]-2*sin(gamma0(t))*yp[1](t)*cos(alpha0(t))*sin(beta0(t))*cos(psi[1](t))*sin(theta[1](t))*l3[1]-2*sin(gamma0(t))*x0(t)*sin(alpha0(t))*sin(beta0(t))*sin(psi[1](t))*cos(theta[1](t))*l3[1]-2*sin(gamma0(t))*x0(t)*sin(alpha0(t))*sin(beta0(t))*cos(psi[1](t))*sin(theta[1](t))*l3[1]+2*sin(gamma0(t))*xp[1](t)*sin(alpha0(t))*sin(beta0(t))*sin(psi[1](t))*cos(theta[1](t))*l3[1]+2*sin(gamma0(t))*xp[1](t)*sin(alpha0(t))*sin(beta0(t))*cos(psi[1](t))*sin(theta[1](t))*l3[1]+2*cos(gamma0(t))*y0(t)*cos(alpha0(t))*sin(beta0(t))*sin(psi[1](t))*sin(theta[1](t))*l3[1]-2*cos(gamma0(t))*y0(t)*cos(alpha0(t))*sin(beta0(t))*cos(theta[1](t))*cos(psi[1](t))*l3[1]-2*cos(gamma0(t))*yp[1](t)*cos(alpha0(t))*sin(beta0(t))*sin(psi[1](t))*sin(theta[1](t))*l3[1]+2*cos(gamma0(t))*yp[1](t)*cos(alpha0(t))*sin(beta0(t))*cos(theta[1](t))*cos(psi[1](t))*l3[1]-2*cos(gamma0(t))*x0(t)*sin(alpha0(t))*sin(beta0(t))*sin(psi[1](t))*sin(theta[1](t))*l3[1]+2*cos(gamma0(t))*x0(t)*sin(alpha0(t))*sin(beta0(t))*cos(theta[1](t))*cos(psi[1](t))*l3[1]+2*cos(gamma0(t))*xp[1](t)*sin(alpha0(t))*sin(beta0(t))*sin(psi[1](t))*sin(theta[1](t))*l3[1]-2*cos(gamma0(t))*xp[1](t)*sin(alpha0(t))*sin(beta0(t))*cos(theta[1](t))*cos(psi[1](t))*l3[1]

Do you have some ideas so as to simplify this equation ?

N.B : Former posts on the topic of trigonometric simplification

http://www.mapleprimes.com/questions/209884-Simplification-Of-Trigonometric-Expression-II

http://www.mapleprimes.com/questions/209721-Simplification-Of-Trigonometric-Expressions

I put a worksheet attached in order to facilitate the troubleshooting.

Thanks a lot for your help

trigonometric_simplification.mw

 

Hello,

In the post :

http://www.mapleprimes.com/questions/209721-Simplification-Of-Trigonometric-Expressions

you have help me to build a procedure so as to simplify trigonometric expressions of the following form, that is to say where each trigonometric expression is in factor with a term :

x0(t)+sin(alpha0(t))*sin(gamma0(t))*sin(beta0(t))*xb[1]-sin(alpha0(t))*sin(beta0(t))*cos(gamma0(t))*zb[1]-sin(alpha0(t))*cos(beta0(t))*yb[1]+cos(alpha0(t))*sin(gamma0(t))*zb[1]+cos(alpha0(t))*cos(gamma0(t))*xb[1]-l2[1]*(sin(psi[1](t))*sin(alpha0(t))*sin(gamma0(t))*sin(beta0(t))-cos(psi[1](t))*sin(alpha0(t))*sin(beta0(t))*cos(gamma0(t))+sin(psi[1](t))*cos(alpha0(t))*cos(gamma0(t))+cos(psi[1](t))*cos(alpha0(t))*sin(gamma0(t)))-l3[1]*(sin(theta[1](t))*sin(psi[1](t))*sin(alpha0(t))*sin(beta0(t))*cos(gamma0(t))+sin(theta[1](t))*cos(psi[1](t))*sin(alpha0(t))*sin(gamma0(t))*sin(beta0(t))+cos(theta[1](t))*sin(psi[1](t))*sin(alpha0(t))*sin(gamma0(t))*sin(beta0(t))-cos(theta[1](t))*cos(psi[1](t))*sin(alpha0(t))*sin(beta0(t))*cos(gamma0(t))-sin(theta[1](t))*sin(psi[1](t))*cos(alpha0(t))*sin(gamma0(t))+sin(theta[1](t))*cos(psi[1](t))*cos(alpha0(t))*cos(gamma0(t))+cos(theta[1](t))*sin(psi[1](t))*cos(alpha0(t))*cos(gamma0(t))+cos(theta[1](t))*cos(psi[1](t))*cos(alpha0(t))*sin(gamma0(t)))-xp[1](t) = 0

From a mechanical point of view, this form of equations comes from the constraint equations obtained with a vectorial closure.

Now I would like to silmplify the constraint equations which come form angular closure.

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 try to treat the right side with the following code :

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(-(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)), size))

The result is :

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

It seems that the result is not simplified enough. I would like to obtain this expression :

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

Have you a idea why the simplification is not conducted once more ? Do you have ideas so as to simplify the equation so as to obtain the result mentioned ?

Thanks a lot for your help

 

 

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