Ask Your Question
1

Trigonometric equation solving

asked 2020-03-14 17:02:51 -0500

Ademorcerf gravatar image

How to solve this equation in sagemath and get a solution t1=f(t2)
(cos(t1)cos(t2) - sin(t1)sin(t2) - cos(t1))(cos(t2)sin(t1) + (cos(t1) + 1)sin(t2)) - ((cos(t1) - 1)cos(t2) - sin(t1)sin(t2))(cos(t2)sin(t1) + cos(t1)sin(t2) + sin(t1)) ==0

edit retag flag offensive close merge delete

Comments

HI found 3 solutions graphically using SageMath, but I admit that I do not know the method to solve this kind of problem. if someone can explain to me the method to solve this kind of problem analytically, i am interested.I looked for the analytical solution all evening without success ;-(

solution on sagecell

@Ademorcerf: I suppose its an homework ? do you have the solution by hand ?

ortollj gravatar imageortollj ( 2020-03-16 17:24:41 -0500 )edit

2 answers

Sort by ยป oldest newest most voted
4

answered 2020-03-17 07:09:51 -0500

rburing gravatar image

updated 2020-03-18 06:53:00 -0500

This equation $e=0$ is polynomial in $c_1=\cos(t_1), s_1=\sin(t_1), c_2=\cos(t_2), s_2=\sin(t_2)$. Explicitly:

sage: var('t1,t2')
sage: eqn = (cos(t1)*cos(t2) - sin(t1)*sin(t2) - cos(t1))*(cos(t2)*sin(t1) + (cos(t1) + 1)*sin(t2)) - ((cos(t1) - 1)*cos(t2) - sin(t1)*sin(t2))*(cos(t2)*sin(t1) + cos(t1)*sin(t2) + sin(t1)) ==0
sage: R.<c1,s1,c2,s2> = PolynomialRing(QQ, order='lex')
sage: trig_to_poly = lambda expr: R(str(expr).replace('cos(t1)','c1').replace('sin(t1)','s1').replace('cos(t2)','c2').replace('sin(t2)','s2'))
sage: e = trig_to_poly(eqn.lhs()); e
-c1^2*s2 - 2*c1*s1*c2 + 2*c1*c2*s2 - c1*s2 + s1^2*s2 + s1*c2^2 + s1*c2 - s1*s2^2

Calculate a Groebner basis of the ideal $\langle e, c_1^2+s_1^2-1, c_2^2+s_2^2-1 \rangle$ in $\mathbb{Q}[c_1,s_1,c_2,s_2]$ with respect to the lexicographic monomial ordering $c_1 > s_1 > c_2 > s_2$:

sage: GB = R.ideal(e,c1^2+s1^2-1,c2^2+s2^2-1).groebner_basis(); GB
[c1^2 + s1^2 - 1, c1*s1 + 8*c1*s2^5 - 6*c1*s2^3 - 1/2*c1*s2 + 4*s1^3*c2 - 8*s1^3*s2^2 + 6*s1^3 - 16*s1^2*c2*s2^3 + 8*s1^2*c2*s2 - 8*s1^2*s2^3 + 6*s1^2*s2 + 8*s1*c2*s2^4 - 10*s1*c2*s2^2 + 1/2*s1*c2 + 8*s1*s2^4 - 4*s1*s2^2 - 3/2*s1 + 4*c2*s2^3 - c2*s2 + 8*s2^5 - 8*s2^3 + 3/2*s2, c1*c2*s2 - 8*c1*s2^5 + 8*c1*s2^3 - c1*s2 - 4*s1^3*c2 + 8*s1^3*s2^2 - 8*s1^3 + 16*s1^2*c2*s2^3 - 12*s1^2*c2*s2 + 8*s1^2*s2^3 - 6*s1^2*s2 - 8*s1*c2*s2^4 + 12*s1*c2*s2^2 - 2*s1*c2 - 8*s1*s2^4 + 5*s1*s2^2 + 2*s1 - 4*c2*s2^3 + 3*c2*s2 - 8*s2^5 + 10*s2^3 - 3*s2, c1*s2^7 - 5/4*c1*s2^5 + 5/16*c1*s2^3 + 1/2*s1^3*c2*s2^2 - 1/4*s1^3*c2 - s1^3*s2^4 + 5/4*s1^3*s2^2 - 1/4*s1^3 - 2*s1^2*c2*s2^5 + 2*s1^2*c2*s2^3 - 3/8*s1^2*c2*s2 - s1^2*s2^5 + 5/4*s1^2*s2^3 - 3/8*s1^2*s2 + s1*c2*s2^6 - 7/4*s1*c2*s2^4 + 11/16*s1*c2*s2^2 + s1*s2^6 - s1*s2^4 + 1/16*s1*s2^2 + 1/2*c2*s2^5 - 3/8*c2*s2^3 + s2^7 - 3/2*s2^5 + 9/16*s2^3, s1^4 + 2*s1^3*c2*s2 - s1^3*s2 - 2*s1^2*c2*s2^2 + 1/2*s1^2*c2 - 1/2*s1^2 - 3/2*s1*c2*s2 - s1*s2^3 + 3/2*s1*s2 + c2*s2^2 + s2^4 - s2^2, c2^2 + s2^2 - 1]

This system of equations is equivalent to the original one, but it is "more triangular". The (familiar) last equation GB[-1] depends only on $c_2, s_2$ (hence $t_2$). The second-to-last equation GB[-2] depends only on $s_1,c_2,s_2$; it is quartic in $s_1$ with leading coefficient $1$, so we can solve for $s_1$ in terms of $c_2,s_2$ (hence $t_2$). In fact:

sage: GB[-2].factor()
(1/2) * (-s1 + s2) * (-2*s1^3 - 4*s1^2*c2*s2 - s1*c2 + s1 + 2*c2*s2 + 2*s2^3 - 2*s2)

So either $s_1=s_2$ or $s_1$ is a (real) root of that cubic. The discriminant of the cubic is:

sage: D = GB[-2].factor()[1][0].discriminant(s1); D
512*c2^4*s2^4 + 16*c2^4*s2^2 + 512*c2^3*s2^6 - 512*c2^3*s2^4 - 320*c2^3*s2^2 - 8*c2^3 - 288*c2^2*s2^4 + 160*c2^2*s2^2 + 24*c2^2 - 576*c2*s2^4 + 576*c2*s2^2 - 24*c2 - 432*s2^6 + 864*s2^4 - 432*s2^2 + 8

On the circle $c_2^2+s_2^2=1$ it has only one zero, at $(1,0)$:

sage: S = PolynomialRing(QQ,names='c2,s2')
sage: S.ideal([c2^2+s2^2-1,D]).variety()
[{s2: 0, c2: 1}]

Even though it looks a bit like it has more:

sage: var('x,y,z,t')
sage: implicit_plot3d(z==0,(x,-1,1),(y,-1,1),(z,-1,1),color='yellow') + parametric_plot3d([cos(t),sin(t),D.subs({c2:cos(t),s2:sin(t)})], (t,0,2*pi)) + point3d((1,0,0))

discriminant

In any case, the discriminant of the cubic is $\geqslant 0$ on the circle $c_2^2+s_2^2=1$, so all roots of the cubic are real, and they are distinct when the discriminant is $\neq 0$; that is, away from $(1,0)$.

In conclusion: either $s_1 = s_2$ or $s_1$ is a real root of a cubic, which can be given by the cubic formula:

sage: SR(GB[-2]).solve(SR(s1))
[s1 == -2/3*c2*s2 - 1/36*(8*c2^2*s2^2 - 3*c2 + 3)*(-I*sqrt(3) + 1)/(-8/27*c2^3*s2^3 + 1/6*(c2 - 1)*c2*s2 + 1/2*s2^3 + 1/2*c2*s2 - 1/2*s2 + 1/6*sqrt(-1/3*(32*c2^3 - 27)*s2^6 - 2/3*(16*c2^4 - 16*c2^3 - 9*c2^2 - 18*c2 + 27)*s2^4 + 1/6*c2^3 - 1/3*(c2^4 - 20*c2^3 + 10*c2^2 + 36*c2 - 27)*s2^2 - 1/2*c2^2 + 1/2*c2 - 1/6))^(1/3) - 1/2*(-8/27*c2^3*s2^3 + 1/6*(c2 - 1)*c2*s2 + 1/2*s2^3 + 1/2*c2*s2 - 1/2*s2 + 1/6*sqrt(-1/3*(32*c2^3 - 27)*s2^6 - 2/3*(16*c2^4 - 16*c2^3 - 9*c2^2 - 18*c2 + 27)*s2^4 + 1/6*c2^3 - 1/3*(c2^4 - 20*c2^3 + 10*c2^2 + 36*c2 - 27)*s2^2 - 1/2*c2^2 + 1/2*c2 - 1/6))^(1/3)*(I*sqrt(3) + 1), s1 == -2/3*c2*s2 - 1/36*(8*c2^2*s2^2 - 3*c2 + 3)*(I*sqrt(3) + 1)/(-8/27*c2^3*s2^3 + 1/6*(c2 - 1)*c2*s2 + 1/2*s2^3 + 1/2*c2*s2 - 1/2*s2 + 1/6*sqrt(-1/3*(32*c2^3 - 27)*s2^6 - 2/3*(16*c2^4 - 16*c2^3 - 9*c2^2 - 18*c2 + 27)*s2^4 + 1/6*c2^3 - 1/3*(c2^4 - 20*c2^3 + 10*c2^2 + 36*c2 - 27)*s2^2 - 1/2*c2^2 + 1/2*c2 - 1/6))^(1/3) - 1/2*(-8/27*c2^3*s2^3 + 1/6*(c2 - 1)*c2*s2 + 1/2*s2^3 + 1/2*c2*s2 - 1/2*s2 + 1/6*sqrt(-1/3*(32*c2^3 - 27)*s2^6 - 2/3*(16*c2^4 - 16*c2^3 - 9*c2^2 - 18*c2 + 27)*s2^4 + 1/6*c2^3 - 1/3*(c2^4 - 20*c2^3 + 10*c2^2 + 36*c2 - 27)*s2^2 - 1/2*c2^2 + 1/2*c2 - 1/6))^(1/3)*(-I*sqrt(3) + 1), s1 == -2/3*c2*s2 + 1/18*(8*c2^2*s2^2 - 3*c2 + 3)/(-8/27*c2^3*s2^3 + 1/6*(c2 - 1)*c2*s2 + 1/2*s2^3 + 1/2*c2*s2 - 1/2*s2 + 1/6*sqrt(-1/3*(32*c2^3 - 27)*s2^6 - 2/3*(16*c2^4 - 16*c2^3 - 9*c2^2 - 18*c2 + 27)*s2^4 + 1/6*c2^3 - 1/3*(c2^4 - 20*c2^3 + 10*c2^2 + 36*c2 - 27)*s2^2 - 1/2*c2^2 + 1/2*c2 - 1/6))^(1/3) + (-8/27*c2^3*s2^3 + 1/6*(c2 - 1)*c2*s2 + 1/2*s2^3 + 1/2*c2*s2 - 1/2*s2 + 1/6*sqrt(-1/3*(32*c2^3 - 27)*s2^6 - 2/3*(16*c2^4 - 16*c2^3 - 9*c2^2 - 18*c2 + 27)*s2^4 + 1/6*c2^3 - 1/3*(c2^4 - 20*c2^3 + 10*c2^2 + 36*c2 - 27)*s2^2 - 1/2*c2^2 + 1/2*c2 - 1/6))^(1/3), s1 == s2]

Of course, $c_1 = \pm\sqrt{1-s_1}$. To obtain $t_1$, restrict the range and apply $\arccos$ or $\arcsin$.

In case $(c_2,s_2) = (1,0)$ you get:

sage: GB[-2].subs({c2:1,s2:0})
s1^4

so $s_1 = 0$ and hence $c_1 = \pm 1$.

Edit: It still needs to be checked/proved that the roots of the cubic have magnitude $\leq 1$.

Edit 2: The numerical solution by @ortollj suggests that the solutions are in fact very simple expressions. One could try to improve the above approach by adding extra variables for the sine and cosine of double and half the angles, and trigonometric identities such as double-angle and half-angle formulae.

edit flag offensive delete link more
2

answered 2020-03-18 20:07:20 -0500

Juanjo gravatar image

updated 2020-03-18 20:07:57 -0500

It seems that the opening post is homework, since the equation can be solved by hand, if I am right. Anyway, let us see how.

Let $E$ be the left-hand side of the equation given in the opening post. It can be rewritten as follows: $$E= \begin{aligned}[t] & ( c_1c_2-s_1s_2-c_1)(s_1c_2+c_1s_2+s_2) \\ &-(c_1c_2-s_1s_2-c_2)(s_1c_2+c_1s_2+s_1), \end{aligned} $$ with $c_i=\cos(t_i)$, $s_i=\sin(t_i)$, for $i=1,2$. Since $$c_1c_2-s_1s_2 = \cos(t_1+t_2)$$ and $$s_1c_2+c_1s_2=\sin(t_1+t_2),$$ we get $$E= \begin{aligned}[t] & \Bigl( \cos(t_1+t_2)-\cos(t_1)\Bigr) \Bigl( \sin(t_1+t_2)+\sin(t_2)\Bigr) \\ &-\Bigl( \cos(t_1+t_2)-\cos(t_2)\Bigr) \Bigl( \sin(t_1+t_2)+\sin(t_1)\Bigr). \end{aligned} $$ Once again, using well know trigonometric formulae, we obtain $$E= \begin{aligned}[t] & \Bigl(-2\sin\frac{2t_1+t_2}{2}\sin\frac{t_2}{2}\Bigr) \Bigl(2\sin\frac{t_1+2t_2}{2}\cos\frac{t_1}{2}\Bigr) \\ &-\Bigl(-2\sin\frac{t_1+2t_2}{2}\sin\frac{t_1}{2}\Bigr) \Bigl(2\sin\frac{2t_1+t_2}{2}\cos\frac{t_2}{2}\Bigr). \end{aligned} $$ Hence $$E= 4 \sin\frac{2t_1+t_2}{2} \sin\frac{t_1+2t_2}{2} \Bigl(\sin\frac{t_1}{2}\cos\frac{t_2}{2} -\cos\frac{t_1}{2}\sin\frac{t_2}{2}\Bigr),$$ that is, $$E= 4 \sin\frac{2t_1+t_2}{2}\ \sin\frac{t_1+2t_2}{2} \ \sin\frac{t_1-t_2}{2}.$$ Therefore, $E=0$ if and only if, for some $k\in\mathbb{Z}$, $2t_1+t_2=2k\pi$ or $t_1+2t_2=2k\pi$ or $t_1-t_2=2k\pi$. It is now quite obvious how to get $t_1$ in terms of $t_2$.

edit flag offensive delete link more

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools

1 follower

Stats

Asked: 2020-03-14 17:02:18 -0500

Seen: 72 times

Last updated: Mar 18