In Eq.1 the second term is an addition to cos ⁡ ⁡ ( ω ω t ) that resolves the ambiguity. In Eq.2 the second term looks like an addition, but it is actually a cancellation that reduces a 2-dimensional vector to just one dimension, resulting in the ambiguity. Eq.2 also shows why the Fourier transform has responses at both ± ± ω ω , even though ω ω can have only one sign. What the false response does is enable the inverse transform to distinguish between a real-valued function and a complex one.