# NSolve not solving simple couple of paired equations

Both Solve[] and NSolve[] seem to be unable to solve this simple system of two equations: $$\sin(\phi_1) + 0.6 \sin(\phi_1 + \phi_2) ==0,$$ $$0.1\sin(\phi_2) + 0.6 \sin(\phi_1 + \phi_2) ==0,$$ with $$\phi_1,\phi_2 \in (-\pi,\pi )$$.

NSolve[{Sin[ϕ1] + 0.6 Sin[ϕ1 + ϕ2] ==
0 && ϕ1 > -π && ϕ1 <= π,
0.1 Sin[ϕ2] + 0.6 Sin[ϕ1 + ϕ2] ==
0 && ϕ2 > -π && ϕ2 <= π}, {ϕ1, ϕ2}]


The output simply prints the input. Am I doing something wrong? I know from graphs that there are indeed solutions

• Try: Reduce[{Sin[\[Phi]1] + 6/10*Sin[\[Phi]1 + \[Phi]2] == 0 && 1/10*Sin[\[Phi]2] + 6/10*Sin[\[Phi]1 + \[Phi]2] == 0 && -Pi < \[Phi]1 < Pi && -Pi < \[Phi]2 < Pi}, {\[Phi]1, \[Phi]2}]? Solve and NSolve deals primarily with linear and polynomial equations.One solution: ContourPlot[{Sin[\[Phi]1] + 6/10*Sin[\[Phi]1 + \[Phi]2] == 0, 1/10*Sin[\[Phi]2] + 6/10*Sin[\[Phi]1 + \[Phi]2] == 0}, {\[Phi]1, -Pi, Pi}, {\[Phi]2, -Pi, Pi}] – Mariusz Iwaniuk Jan 20 at 17:08
• Thanks. Unfortunately, Reduce only gives one solution (0,0) whereas I need to access all solutions, moreover in an automatised way as I'm looping over different values for the coefficients of the Sine functions. – Rudyard Jan 20 at 17:26
• All solution: Reduce[{Sin[\[Phi]1] + 6/10*Sin[\[Phi]1 + \[Phi]2] == 0 && 1/10*Sin[\[Phi]2] + 6/10*Sin[\[Phi]1 + \[Phi]2] == 0 && -Pi < \[Phi]1 <= Pi && -Pi < \[Phi]2 <= Pi}, {\[Phi]1, \[Phi]2}]? – Mariusz Iwaniuk Jan 20 at 17:33
• ah, of course.. – Rudyard Jan 20 at 17:41
• Yes try: Solve[%] – Mariusz Iwaniuk Jan 20 at 17:54

In addition to Marius' comment, you can use Solve with Method -> Reduce:

Solve[
{
Sin[ϕ1] + 0.6 Sin[ϕ1 + ϕ2] == 0 && ϕ1 > -π && ϕ1 <= π,
0.1 Sin[ϕ2] + 0.6 Sin[ϕ1 + ϕ2] == 0 && ϕ2 > -π && ϕ2 <= π
},
{ϕ1, ϕ2},
Method->Reduce
]
`

Solve::ratnz: Solve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result.

{{ϕ1 -> 0, ϕ2 -> 0}, {ϕ1 -> 0, ϕ2 -> 3.14159}, {ϕ1 -> 3.14159, ϕ2 -> 0}, {ϕ1 -> 3.14159, ϕ2 -> 3.14159}}

• Why not use $\dfrac{1}{10}$ and $\dfrac{6}{10}$ instead to remove the error message? – Moo Jan 20 at 17:56