I opened solution for pair of equations in cos and sin · Issue #28804 ·
sympy/sympy <https://github.com/sympy/sympy/issues/28804>
On Friday, December 19, 2025 at 11:05:20 AM UTC-6 Chris Smith wrote:
> Nevermind...I suspect you just let b be called c in the first case and
> then let c = a+b in the second case. If you can solve for a and c you can
> solve for a and a+b, hence a and b in the second case.
>
> /c
>
> On Thursday, December 18, 2025 at 12:13:12 AM UTC-6 Chris Smith wrote:
>
>> What substitution were you thinking? I would expand the 2nd set and
>> rewrite in terms of exponentials and solve for them.
>>
>> a,b,x,y,l1,l2=symbols('a b x y l1 l2')
>> eq = [l1*cos(a)+l2*cos(a+b)-x,l1*sin(a)+l2*sin(a+b)-y]
>> eq=[i.expand(trig=1).rewrite(exp).subs(exp(I*a),var('A')).subs(exp(I*b),var("B")).as_numer_denom()[0]
>>
>> for i in eq]
>> g = groebner(eq,(A,B))
>> solve(g,(A,B),simplify=0,verify=0)
>>
>> which gives 3 solutions (but the one with A=0 is not valid, so just 2)
>>
>> /c
>>
>>
>>
>>
>> On Tuesday, November 11, 2025 at 4:02:36 AM UTC-6 [email protected]
>> wrote:
>>
>>> this works (A)(although takes awfully long)
>>> a,b,x,y,l1,l2=symbols('a b x y l1 l2')
>>> eq = [l1*cos(a)+l2*cos(b)-x,l1*sin(a)+l2*sin(b)-y]
>>> res = solve(eq,[a,b])
>>>
>>> this does not (B):
>>> a,b,x,y,l1,l2=symbols('a b x y l1 l2')
>>> eq = [l1*cos(a)+l2*cos(a+b)-x,l1*sin(a)+l2*sin(a+b)-y]
>>> res = solve(eq,[a,b])
>>>
>>> solution from (A) could be used to find solution for (B) by a simple
>>> substitution, so, analytic solution exists, sympy, however, produces error
>>>
>>> python 3.11.13
>>> sympy 1.14.0
>>>
>>>
>>> ---------------------------------------------------------------------------
>>> TypeError Traceback (most recent call last) Cell In[11], line 1 ----> 1
>>> res2 = solve(eq,[a,b]) File
>>> ~\Documents\FH\Python\venv\control\Lib\site-packages\sympy\solvers\solvers.py:1170
>>>
>>> <http://localhost:8888/lab/tree/Teaching/RobReg/R2D/~/Documents/FH/Python/venv/control/Lib/site-packages/sympy/solvers/solvers.py#line=1169>,
>>>
>>> in solve(f, *symbols, **flags) 1168 solution = _solve(f[0], *symbols,
>>> **flags)
>>> 1169 else: -> 1170 linear, solution = _solve_system(f, symbols, **flags)
>>> 1171 assert type(solution) is list 1172 assert not solution or type
>>> (solution[0]) is dict, solution File
>>> ~\Documents\FH\Python\venv\control\Lib\site-packages\sympy\solvers\solvers.py:1963
>>>
>>> <http://localhost:8888/lab/tree/Teaching/RobReg/R2D/~/Documents/FH/Python/venv/control/Lib/site-packages/sympy/solvers/solvers.py#line=1962>,
>>>
>>> in _solve_system(exprs, symbols, **flags) 1961 for s in ok_syms: 1962
>>> try: -> 1963 soln = _vsolve(eq2, s, **flags) 1964 except
>>> NotImplementedError: 1965 continue File
>>> ~\Documents\FH\Python\venv\control\Lib\site-packages\sympy\solvers\solvers.py:2638
>>>
>>> <http://localhost:8888/lab/tree/Teaching/RobReg/R2D/~/Documents/FH/Python/venv/control/Lib/site-packages/sympy/solvers/solvers.py#line=2637>,
>>>
>>> in _vsolve(e, s, **flags) 2636 def _vsolve(e, s, **flags): 2637 """return
>>> list of scalar values for the solution of e for symbol s""" -> 2638
>>> return [i[s] for i in _solve(e, s, **flags)] File
>>> ~\Documents\FH\Python\venv\control\Lib\site-packages\sympy\solvers\solvers.py:1743
>>>
>>> <http://localhost:8888/lab/tree/Teaching/RobReg/R2D/~/Documents/FH/Python/venv/control/Lib/site-packages/sympy/solvers/solvers.py#line=1742>,
>>>
>>> in _solve(f, *symbols, **flags) 1738 result = [r for r in result if 1739
>>> not any(checksol(d, r, **flags) 1740 for d in dens)] 1741 if check: 1742 #
>>> keep only results if the check is not False -> 1743 result = [r for r in
>>> result if 1744 checksol(f_num, r, **flags) is not False] 1745 return
>>> result File
>>> ~\Documents\FH\Python\venv\control\Lib\site-packages\sympy\solvers\solvers.py:1744
>>>
>>> <http://localhost:8888/lab/tree/Teaching/RobReg/R2D/~/Documents/FH/Python/venv/control/Lib/site-packages/sympy/solvers/solvers.py#line=1743>,
>>>
>>> in <listcomp>(.0) 1738 result = [r for r in result if 1739 not
>>> any(checksol(d,
>>> r, **flags) 1740 for d in dens)] 1741 if check: 1742 # keep only
>>> results if the check is not False 1743 result = [r for r in result if ->
>>> 1744 checksol(f_num, r, **flags) is not False] 1745 return result File
>>> ~\Documents\FH\Python\venv\control\Lib\site-packages\sympy\solvers\solvers.py:354
>>>
>>> <http://localhost:8888/lab/tree/Teaching/RobReg/R2D/~/Documents/FH/Python/venv/control/Lib/site-packages/sympy/solvers/solvers.py#line=353>,
>>>
>>> in checksol(f, symbol, sol, **flags) 352 return val == 0 353 if
>>> numerical and val.is_number: --> 354 return (abs(val.n(18).n(12, chop=
>>> True)) < 1e-9) is S.true 356 if flags.get('warn', False): 357 warnings.
>>> warn("\n\tWarning: could not verify solution %s." % sol) File
>>> ~\Documents\FH\Python\venv\control\Lib\site-packages\sympy\core\decorators.py:248
>>>
>>> <http://localhost:8888/lab/tree/Teaching/RobReg/R2D/~/Documents/FH/Python/venv/control/Lib/site-packages/sympy/core/decorators.py#line=247>,
>>>
>>> in _SympifyWrapper.make_wrapped.<locals>._func(self, other) 246 if not
>>> isinstance(other, expectedcls): 247 return retval --> 248 return func(
>>> self, other) File
>>> ~\Documents\FH\Python\venv\control\Lib\site-packages\sympy\core\expr.py:400
>>> <http://localhost:8888/lab/tree/Teaching/RobReg/R2D/~/Documents/FH/Python/venv/control/Lib/site-packages/sympy/core/expr.py#line=399>,
>>>
>>> in Expr.__lt__(self, other) 397 @sympify_return([('other', 'Expr')],
>>> NotImplemented) 398 def __lt__(self, other): 399 from .relational import
>>> StrictLessThan --> 400 return StrictLessThan(self, other) File
>>> ~\Documents\FH\Python\venv\control\Lib\site-packages\sympy\core\relational.py:850
>>>
>>> <http://localhost:8888/lab/tree/Teaching/RobReg/R2D/~/Documents/FH/Python/venv/control/Lib/site-packages/sympy/core/relational.py#line=849>,
>>>
>>> in _Inequality.__new__(cls, lhs, rhs, **options) 848 raise
>>> TypeError("Invalid
>>> comparison of non-real %s" % me) 849 if me is S.NaN: --> 850 raise
>>> TypeError("Invalid NaN comparison") 851 # First we invoke the
>>> appropriate inequality method of `lhs` 852 # (e.g., `lhs.__lt__`). That
>>> method will try to reduce to 853 # boolean or raise an exception. It
>>> may keep calling (...) 857 # exception). In that case, it must call us
>>> with 858 # `evaluate=False` to prevent infinite recursion. 859 return
>>> cls._eval_relation(lhs, rhs, **options) TypeError: Invalid NaN
>>> comparison
>>>
>>
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