Eliminate keeps the variable

Question

I have the following equations:

x == 10*Sin[i/10] + j + 0.008*j*j - 30
y == h/2 + j/2*Sin[j/50 - t/50 - i/118] + (i*0.9)*Cos[j/25 - (i + t)/65]

i, h, and t are constant factors.

I'm trying to get y in terms of x and no j in the resulting equation.

Eliminate[{%1, %2}, j]

Gives me a huge set of equations that still has j in them.

What am I doing wrong?

Answer 1

Limited scope of Eliminate

The only thing you are doing wrong is to have expectations beyond the capability of Eliminate.

You get the warning

Eliminate::ifun: Inverse functions are being used by Eliminate, so some solutions may not be found; use Reduce for complete solution information.

The documentation reads :

Eliminate works primarily with linear and polynomial equations.

Sometimes Eliminate will work with complicated expressions, but it's only supposed to work with polynomial equations. Other tools like Solve and Reduce sometimes will not find solutions either.

Workaround.

Borrowing from @Nasser in the comments, you can solve for j on one equation and replace on the other. Notice there are two solutions, so I Apply (@@) Or to keep this as a single logical expression.

Reduce[
   Or@@ReplaceAll[
      y ==  h/2 + j/2*Sin[j/50 - t/50 - i/118] + (i*9/10)* Cos[j/25 - (i + t)/65]
      , Echo[Solve[ x == 10*Sin[i/10] + j + 8/1000*j*j - 30, j], "Solutions:", Length]
   ]
   ,{x,y}
]