1 | initial version |

First, even though this particular system can be solved by hand easily enough, giving a nice, tidy family of solutions, I don't suspect that Sage will be able to see that. (Somebody please correct me if I'm wrong on that.) Certainly this is the case if you change your equations a little bit. In general, the best you can hope for is numerical solutions.

Resigning ourselves to that, I found a couple other questions on ASKSAGE with similar issues: here and here. My answer below is heavily cribbed from the one given by DSM at the first link. He recommends using scipy's 'minimize' function.

```
from scipy import optimize
```

Then, given a starting point, e.g. (5,4), we hunt for a nearby solution.

```
sage: f(x,y) = (sin(x) + y, sin(x) - y)
sage: minimize(norm(f), (5,4), disp = 0)
(6.28318530718, 8.79023942731e-19)
```

If you loop over a lattice of points, you will *hopefully* find all the solutions in that region:

```
sage: short = lambda (x,y): (x.n(20), y.n(20)) # To cut down on the number of digits displayed.
sage: for startloc in CartesianProduct([-6,-2 .. 6],[-6,-2 .. 6]):
....: sol = minimize(norm(f), startloc, disp = 0)
....: print startloc," ", short(sol), " ",short(f(*sol))
[-6, -6] (-6.2832, 4.8257e-19) (2.4541e-16, 2.4445e-16)
[-6, -2] (-6.2832, 2.0430e-19) (-6.4304e-16, -6.4345e-16)
[-6, 2] (-6.2832, -2.0430e-19) (-6.4345e-16, -6.4304e-16)
[-6, 6] (-6.2832, -4.8257e-19) (2.4445e-16, 2.4541e-16)
[-2, -6] (-3.1416, -4.7943e-19) (-1.2294e-16, -1.2199e-16)
[-2, -2] (-3.1416, 1.0451e-19) (-1.2236e-16, -1.2257e-16)
[-2, 2] (-3.1416, -1.0451e-19) (-1.2257e-16, -1.2236e-16)
[-2, 6] (-3.1416, 4.7943e-19) (-1.2199e-16, -1.2294e-16)
[2, -6] (3.1416, -4.7943e-19) (1.2199e-16, 1.2294e-16)
[2, -2] (3.1416, 1.0451e-19) (1.2257e-16, 1.2236e-16)
[2, 2] (3.1416, -1.0451e-19) (1.2236e-16, 1.2257e-16)
[2, 6] (3.1416, 4.7943e-19) (1.2294e-16, 1.2199e-16)
[6, -6] (6.2832, 4.8257e-19) (-2.4445e-16, -2.4541e-16)
[6, -2] (6.2832, 2.0430e-19) (6.4345e-16, 6.4304e-16)
[6, 2] (6.2832, -2.0430e-19) (6.4304e-16, 6.4345e-16)
[6, 6] (6.2832, -4.8257e-19) (-2.4541e-16, -2.4445e-16)
```

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