# Trigonometric manipulation

1. Nov 1, 2013

### Fluidman117

Hello,

Probably a simple problem, but Im not able to figure it out.

$$a \cos (\epsilon) - b \sin (\epsilon) = c$$ in-phase part
$$a \sin (\epsilon) - b \cos (\epsilon) = d$$ out-of-phase part

In order to find the phase shift, the in-phase term has to be divided by the out-of-phase term?

$$\frac{a \cos (\epsilon) - b \sin (\epsilon) = c}{a \sin (\epsilon) - b \cos (\epsilon) = d}$$

And the phase shift is the arctan of the out of phase and the in-phase term to my knowledge. But I am not able to manipulate the formula in a way that I'll get to an arctan. Can someone point me in the right direction?

Thanks

2. Nov 1, 2013

### voko

$$a \cos x + b \sin x = \sqrt {a^2 + b^2 } ( \frac a {\sqrt {a^2 + b^2} } \cos x + b \frac b {\sqrt {a^2 + b^2} } \sin x) \\ = \sqrt {a^2 + b^2 } ( \sin c \cos x + \cos c \sin x) = \ ...$$