Definition of Phase Shifts (Coupled Oscillators)

Question

I was wondering if maybe someone could look at this excerpt from a textbook (Attached). It states that “the displacement of the two masses will be in opposite directions (out of phase by pi)” but I thought that the Definition of Phase was:
Phase = theta in the following formula, cos(wt+theta).
However, the two equations in this book excerpt have the exact same expression in parenthesis, and both are nested in the same function, cosine - so how is it that they have different phases? Many thanks.

garyp · Accepted Answer

$$-4cos{(sqrt{5}t)} = +4cos{(sqrt{5}t + pi)}$$

Philip · Answer

Here's a simple GIF to complement @garyp's succinct answer:

I've plotted the two functions $-cos{(sqrt{5}t)}$ (blue) and $cos{(sqrt{5}t + phi)}$ (orange), and I've varied $phi$ between 0 and $2pi$. As you can see, they start off being exactly exactly reflected, as you'd expect, and now as I tune $phi$, the orange curve moves "closer" to the blue one until it matches it exactly when $phi = pi/2$, as one would expect.

Definition of Phase Shifts (Coupled Oscillators)

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