How to find 4 unknowns?

Question

There are 4 unknowns constants: a,b,c,p. The variable is phi and the result is d.

For phi={13,23,45,73} the results are d={32.08754292290807,32.1136624112758,32.1728430675844,32.2430287624866}

The equation is:

(c/0.3048*((1+(a*p/(b*c))*(sin(phi*pi/180))^2))/sqrt(1-sqrt(1-b^2/a^2)*
((sin(phi*pi/180))^2))) = d

How can I solve it in Mathematica?

Answer 1

If I decipher your parentheses correctly, this is just a second order algebraic equation in u if you set u=Sin[phi]^2. So you may directly define a function

f[a_,b_,c_,p_] := Block[{A,B,C,Delta,U},
                         A=...;B=...;C=...;Delta=Sqrt[B^2-4*A C];
                         U=...;
                         phi=ArcSin[Sqrt[U]];]

which first solves the equation in u, then takes the arcsin.