With suitable constants c1,...,c6 and substituting x = sin(sigma_A^EC), your equation can be written as
c1*(x^2-c2) - c3*(x*c4-sqrt(1-x^2)*c5) - c6 = 0
Isolating sqrt(1-x^2) on one side of the equation and squaring gives a polynomial of degree 4 in x. The roots of this polynomial can either be solved for analytically using the "solve" command or numerically using the "root" command.
syms c1 c2 c3 c4 c5 c6 x
eqn = c1*(x^2-c2) - c3*(x*c4-sqrt(1-x^2)*c5) - c6 == 0
solve(eqn,x,'MaxDegree',4)
