Expressing sum of sines and cosines as a complex exponential

[MuIotaTau]

If I'm given a function $f(x) = A cos (x) + B sin (x)$, is there any way to turn this into an expression of the form $F(x) = C e^{i(x + \phi)}$? I know how to use Euler's formula to turn this into $\alpha e^{i(x + \phi)} + \beta e^{-i(x + \phi)}$, but is there a way to incorporate the second term into the first somehow, maybe with a change in the constants?

 

[mfb]

You can use Euler's formula to express F(x) as sum of cos and sin and then find relations for the constants C and ϕ as function of A and B.