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TrigNSolve (1.0.0) current version: 1.0.1 »

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Solve a system of trigonometric or hyperbolic equations

Contributed by: Daniel Lichtblau

ResourceFunction["TrigNSolve"][eqns,vars]

numerically solves the trigonometric system eqns in vars.

Details and Options

ResourceFunction["TrigNSolve"] also handles hyperbolic functions in the input.

Input can be in the form of trigonometric or hyperbolic polynomials or equations.

ResourceFunction["TrigNSolve"] creates regular polyomials from its input, using a new set of internal variables. It uses NSolve to obtain intermediate solutions in these variables. It then post-processes using arc trigs and arc hyperbolics to solve for the original variables.

ResourceFunction["TrigNSolve"] takes Method and WorkingPrecision options, which it passes to NSolve.

Examples are taken from the Mathematica.StackExchange forum.

Because ResourceFunction["TrigNSolve"] ultimately creates polynomials, it cannot handle mixed variables that appear both in polynomial and in trig form (e.g. x^2+Cos[x]).

Examples

Basic Examples (2)

Find real solutions for a system of five equations:

In[1]:=

$sys = {1 - Cos[theta1] - 2/3 Sin[phi + Pi/6] == 0.227746, h + 2/3 Cos[phi + Pi/6] - Sin[theta1] == -0.714585, -1. - Cos[theta2] - 2/3 Sin[phi - Pi/6] == (3 Cos[psi2])/4, 2/5 + h + 2/3 Cos[phi - \[Pi]/6] - Sin[theta2] == (3 Sin[psi2])/4, 0.635187 Cos[phi - \[Pi]/6 - psi2] + 2/3 Cos[1.78586 + phi] Sin[psi2] == 0}; solns = ResourceFunction["TrigNSolve"][ sys, {h, theta1, theta2, phi, psi2}]; rsolns = Select[solns, FreeQ[#, Complex] &]$

Out[3]=

Check that residuals are small:

In[4]:=

Out[4]=

Scope (4)

A system can have variables appearing in trigs or as ordinary variables and can also have radical expressions:

In[5]:=

$sys = {1/(4 Sin[x]) - (a Sin[x])/(Sqrt[1 + a^2 + 2 a Cos[x]])^3, r^3 - 3/a^3 + (2 (a + Cos[x]))/(Sqrt[1 + a^2 + 2 a Cos[x]])^3, r^3 - 3 + (1 + a Cos[x])/(Sqrt[1 + a^2 + 2 a Cos[x]])^3 + 1/(4 Sin[x])}; solns = ResourceFunction["TrigNSolve"][sys, {r, a, x}, Method -> Automatic]; Select[solns, FreeQ[#, Complex] &]$

Out[6]=

Solve a mixed trig-hyperbolic system:

In[7]:=

eqns = {Cosh[1]*Cosh[r3] - Sinh[1]*Sinh[r3]*Cos[-a3] == 1/2*(Cosh[1]*Cosh[2] - Sinh[1]*Sinh[2]*Cos[-1]), 1/2*(Cosh[1]*Cosh[2] - Sinh[1]*Sinh[2]*Cos[-1]) == Cosh[2]*Cosh[r3] - Sinh[2]*Sinh[r3]*Cos[1 - a3]};
ResourceFunction["TrigNSolve"][eqns, {r3, a3}]

Out[8]=

Solve a system in trigs and ordinary variables:

In[9]:=

tsys = {Cos[ArcTan[1.1496/h]] == Cos[0.0591 Pi]*Cos[y]*Cos[0.44167 Pi - x] + Sin[0.0591 Pi]*Sin[y],
Cos[ArcTan[1.501428/h]] == Cos[0.0591 Pi]*Cos[y]*Cos[0.4 Pi - x] + Sin[0.0591 Pi]*Sin[y], Cos[ArcTan[1.927918/h]] == Cos[0.0591 Pi]*Cos[y]*Cos[0.358333 Pi - x] + Sin[0.0591 Pi]*Sin[y]};
solns = ResourceFunction["TrigNSolve"][tsys, {h, x, y}];
rsolns = Select[solns, FreeQ[#, Complex] &]

Out[10]=

Numerically solve a difficult trigonometric system and extract real-valued solutions:

In[11]:=

dvalRules = {da -> 1.12, db -> 1.23, dc -> 0.84};
eqns = {2 a b/(a + b) Sin[(aA + aB)/2] == dc, 2 b c/(b + c) Cos[aA/2] == da, 2 c a (c + a) Cos[aB/2] == db, Sin[aA]/Sin[aA + aB] == da/dc, Sin[aB]/Sin[aA + aB] == db/dc} /. dvalRules /. {aA -> 2 aA, aB -> 2 aB};
solns = ResourceFunction["TrigNSolve"][eqns, {a, b, c}];
realSolns = Select[solns, FreeQ[#, Complex] &]

Out[12]=

Options (5)

Method (5)

Use the "Endomorphism" method from NSolve on a large trigonometric system:

In[13]:=

tpolys = {-(41/4) + Cos[a1] (10 + 3 Cos[b1]) - Cos[b2] (4 + (7 Cos[c2])/2) + 1/4 (50 - 10 Cos[a1] - 3 Cos[a1] Cos[b1] - 10 Sin[a1] - 3 Cos[b1] Sin[a1] - 3 Sin[b1]) + 1/16 (-159 + 16 Cos[b2] + 14 Cos[b2] Cos[c2] + 16 Sin[b2] + 14 Cos[c2] Sin[b2] + 14 Sin[c2]),
(10 + 3 Cos[b1]) Sin[a1] + 1/4 (50 - 10 Cos[a1] - 3 Cos[a1] Cos[b1] - 10 Sin[a1] - 3 Cos[b1] Sin[a1] - 3 Sin[b1]) + 1/16 (-159 + 16 Cos[b2] + 14 Cos[b2] Cos[c2] + 16 Sin[b2] + 14 Cos[c2] Sin[b2] + 14 Sin[c2]),
1/4 (50 - 10 Cos[a1] - 3 Cos[a1] Cos[b1] - 10 Sin[a1] - 3 Cos[b1] Sin[a1] - 3 Sin[b1]) + 3 Sin[b1] - (4 + (7 Cos[c2])/2) Sin[b2] + 1/16 (-159 + 16 Cos[b2] + 14 Cos[b2] Cos[c2] + 16 Sin[b2] + 14 Cos[c2] Sin[b2] + 14 Sin[c2]),
Cos[a1] (10 + 3 Cos[b1]) - Cos[a3] (33/2 + (47 Cos[c3])/8) + 1/4 (50 - 10 Cos[a1] - 3 Cos[a1] Cos[b1] - 10 Sin[a1] - 3 Cos[b1] Sin[a1] - 3 Sin[b1]) + 1/32 (-400 + 132 Cos[a3] + 47 Cos[a3] Cos[c3] + 132 Sin[a3] + 47 Cos[c3] Sin[a3] + 47 Sin[c3]),
(10 + 3 Cos[b1]) Sin[a1] - (33/2 + (47 Cos[c3])/8) Sin[a3] + 1/4 (50 - 10 Cos[a1] - 3 Cos[a1] Cos[b1] - 10 Sin[a1] - 3 Cos[b1] Sin[a1] - 3 Sin[b1]) + 1/32 (-400 + 132 Cos[a3] + 47 Cos[a3] Cos[c3] + 132 Sin[a3] + 47 Cos[c3] Sin[a3] + 47 Sin[c3]),
1/4 (50 - 10 Cos[a1] - 3 Cos[a1] Cos[b1] - 10 Sin[a1] - 3 Cos[b1] Sin[a1] - 3 Sin[b1]) + 3 Sin[b1] + 1/32 (-400 + 132 Cos[a3] + 47 Cos[a3] Cos[c3] + 132 Sin[a3] + 47 Cos[c3] Sin[a3] + 47 Sin[c3])};
First[AbsoluteTiming[
tsolsE = ResourceFunction["TrigNSolve"][tpolys, {a1, a3, b1, b2, c2, c3}, Method -> "EndomorphismMatrix"]]]

Out[14]=

Extract real-valued solutions:

In[15]:=

Out[15]=

Compare to the default Method→Automatic timing:

In[16]:=

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And to the "Monodromy" method:

In[17]:=

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Timing results strongly suggest that "Homotopy" was used as the Automatic default for this example:

In[18]:=

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Possible Issues (1)

TrigNSolve requires that variables appearing in trig functions not also appear as polynomial variables:

In[19]:=

$ResourceFunction["TrigNSolve", ResourceVersion->"1.0.0"][{x^2 + y - Cos[y], Sin[x] + y^2} == 0, {x, y}]$

Out[19]=

Requirements

Wolfram Language 13.0 (December 2021) or above

Version History

1.0.1 – 20 June 2024
1.0.0 – 17 June 2024

Related Resources

Author Notes

This was written in to fill a need as seen from several questions that have appeared on the Mathematica StackExchange forum over the years. A few were used in the examples above.

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License

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TrigNSolve (1.0.0) current version: 1.0.1 »

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