How to expand one exponential complex equation to two trigonometric ones in sympy?

Question

I have one exponential equation with two unknowns, say:

y*exp(ix) = sqrt(2) + i * sqrt(2)

Manually, I can transform it to system of trigonometric equations:

y * cos x = sqrt(2)
y * sin x = sqrt(2)

How can I do it automatically in sympy?

I tried this:

from sympy import *
x = Symbol('x', real=True)
y = Symbol('y', real=True)
eq = Eq(y * cos(I * x), sqrt(2) + I * sqrt(2))
print([e.trigsimp() for e in eq.as_real_imag()])

but only got two identical equations except one had "re" before it and another one "im".

Answer 1

You can call the method .rewrite(sin) or .rewrite(cos) to obtain the desired form of your equation. Unfortunately, as_real_imag cannot be called on an Equation directly but you could do something like this:

from sympy import *


def eq_as_real_imag(eq):
    lhs_ri = eq.lhs.as_real_imag()
    rhs_ri = eq.rhs.as_real_imag()    
    return Eq(lhs_ri[0], rhs_ri[0]), Eq(lhs_ri[1], rhs_ri[1])

x = Symbol('x', real=True)
y = Symbol('y', real=True)

original_eq = Eq(y*exp(I*x), sqrt(2) + I*sqrt(2))
trig_eq = original_eq.rewrite(sin)  # Eq(y*(I*sin(x) + cos(x)), sqrt(2) + sqrt(2)*I)

eq_real, eq_imag = eq_as_real_imag(trig_eq) 
print(eq_real)  # Eq(y*cos(x), sqrt(2))
print(eq_imag)  # Eq(y*sin(x), sqrt(2)) 

(You might also have more luck just working with expressions (implicitly understood to be 0) instead of an Equation eg eq.lhs - eq.rhs in order to call the method as_real_imag directly)